Abstract: In this paper, we propose an alternative construction of a certain class of Deformed Double Current Algebras. We construct them as spherical subalgebras of symplectic reection algebras in the Deligne category. They can also be thought of as ultraproducts of the corresponding spherical subalgebras in finite rank. We also provide new presentations of DDCA of types A and B by generators and relations.

Publication: Transformation GroupsISSN: 1083-4362

ID: CaltechAUTHORS:20220726-997340000

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Abstract: In their 2011 paper on the AGT conjecture, Alba, Fateev, Litvinov and Tarnopolsky (AFLT) obtained a closed-form evaluation for a Selberg integral over the product of two Jack polynomials, thereby unifying the well-known Kadell and Hua–Kadell integrals. In this paper we use a variety of symmetric functions and symmetric function techniques to prove generalisations of the AFLT integral. These include (i) an A_n analogue of the AFLT integral, containing two Jack polynomials in the integrand; (ii) a generalisation of (i) for γ = 1 (the Schur or GUE case), containing a product of n+1 Schur functions; (iii) an elliptic generalisation of the AFLT integral in which the role of the Jack polynomials is played by a pair of elliptic interpolation functions; (iv) an AFLT integral for Macdonald polynomials.

Publication: Communications in Mathematical Physics Vol.: 388 No.: 2 ISSN: 0010-3616

ID: CaltechAUTHORS:20211110-164135228

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Abstract: We prove that for any autonomous 4-dimensional integral system of Painlevé type, the Jacobian of the generic spectral curve has a unique polarization, and thus by Torelli’s theorem cannot be isomorphic as an unpolarized abelian surface to any other Jacobian. This enables us to identify the spectral curve and any irreducible genus 2 component of the boundary of an affine patch of the Liouville torus.

Publication: International Mathematics Research Notices Vol.: 2021 No.: 18 ISSN: 1073-7928

ID: CaltechAUTHORS:20211217-98152000

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Abstract: Following [Beem C., Peelaers W., Rastelli L., Comm. Math. Phys. 354 (2017), 345-392] and [Etingof P., Stryker D., SIGMA 16 (2020), 014, 28 pages], we undertake a detailed study of twisted traces on quantizations of Kleinian singularities of type A_(n−1). In particular, we give explicit integral formulas for these traces and use them to determine when a trace defines a positive Hermitian form on the corresponding algebra. This leads to a classification of unitary short star-products for such quantizations, a problem posed by Beem, Peelaers and Rastelli in connection with 3-dimensional superconformal field theory. In particular, we confirm their conjecture that for n≤4 a unitary short star-product is unique and compute its parameter as a function of the quantization parameters, giving exact formulas for the numerical functions by Beem, Peelaers and Rastelli. If n=2, this, in particular, recovers the theory of unitary spherical Harish-Chandra bimodules for sl₂. Thus the results of this paper may be viewed as a starting point for a generalization of the theory of unitary Harish-Chandra bimodules over enveloping algebras of reductive Lie algebras [Vogan Jr. D.A., Annals of Mathematics Studies, Vol. 118, Princeton University Press, Princeton, NJ, 1987] to more general quantum algebras. Finally, we derive recurrences to compute the coefficients of short star-products corresponding to twisted traces, which are generalizations of discrete Painlevé systems.

Publication: Symmetry, Integrability and Geometry, Methods and Applications (SIGMA) Vol.: 17ISSN: 1815-0659

ID: CaltechAUTHORS:20210602-134422197

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Abstract: We prove Macdonald-type deformations of a number of well-known classical branching rules by employing identities for elliptic hypergeometric integrals and series. We also propose some conjectural branching rules and allied conjectures exhibiting a novel type of vanishing behaviour involving partitions with empty 2-cores.

Publication: Symmetry, Integrability and Geometry, Methods and Applications (SIGMA) Vol.: 16ISSN: 1815-0659

ID: CaltechAUTHORS:20210122-141416267

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Abstract: We give a construction of an affine Hecke algebra associated to any Coxeter group acting on an abelian variety by reflections; in the case of an affine Weyl group, the result is an elliptic analogue of the usual double affine Hecke algebra. As an application, we use a variant of the C_n version of the construction to construct a flat noncommutative deformation of the nth symmetric power of any rational surface with a smooth anticanonical curve, and give a further construction which conjecturally is a corresponding deformation of the Hilbert scheme of points.

Publication: Symmetry, Integrability and Geometry: Methods and Applications (SIGMA) Vol.: 16ISSN: 1815-0659

ID: CaltechAUTHORS:20170922-134529235

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Abstract: We prove a Macdonald polynomial analogue of the celebrated Nekrasov–Okounkov hook-length formula from the theory of random partitions. As an application we obtain a proof of one of the main conjectures of Hausel and Rodriguez-Villegas from their work on mixed Hodge polynomials of the moduli space of stable Higgs bundles on Riemann surfaces.

Publication: Journal of Algebraic Combinatorics Vol.: 48 No.: 1 ISSN: 0925-9899

ID: CaltechAUTHORS:20170922-130437528

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Abstract: Previous work established a connection between the geometric invariant theory of the third exterior power of a 9-dimensional complex vector space and the moduli space of genus-2 curves with some additional data. We generalize this connection to arbitrary fields, and describe the arithmetic data needed to get a bijection between both sides of this story.

Publication: Algebra and Number Theory Vol.: 12 No.: 4 ISSN: 1937-0652

ID: CaltechAUTHORS:20170922-135107001

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Abstract: Let E be an elliptic curve over a field k. Let R:=End E. There is a functor Hom_R(−,E) from the category of finitely presented torsion-free left R -modules to the category of abelian varieties isogenous to a power of E, and a functor Hom(−,E) in the opposite direction. We prove necessary and sufficient conditions on E for these functors to be equivalences of categories. We also prove a partial generalization in which E is replaced by a suitable higher-dimensional abelian variety over F_p.

Publication: Compositio Mathematica Vol.: 154 No.: 5 ISSN: 0010-437X

ID: CaltechAUTHORS:20170922-140322875

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Abstract: We refine the statement of the denominator and evaluation conjectures for affine Macdonald polynomials proposed by Etingof–Kirillov Jr. (Duke Math J 78(2):229–256, 1995) and prove the first non-trivial cases of these conjectures. Our results provide a q-deformation of the computation of genus 1 conformal blocks via elliptic Selberg integrals by Felder–Stevens–Varchenko (Math Res Lett 10(5–6):671–684, 2003). They allow us to give precise formulations for the affine Macdonald conjectures in the general case which are consistent with computer computations. Our method applies recent work of the second named author to relate these conjectures in the case of U_q(sl_2) to evaluations of certain theta hypergeometric integrals defined by Felder–Varchenko (Int Math Res Not 21:1037–1055, 2004). We then evaluate the resulting integrals, which may be of independent interest, by well-chosen applications of the elliptic beta integral introduced by Spiridonov (Uspekhi Mat Nauk 56(1(337)):181–182, 2001).

Publication: Selecta Mathematica - New Series Vol.: 24 No.: 2 ISSN: 1022-1824

ID: CaltechAUTHORS:20170607-102914977

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Abstract: We prove a number of quadratic transformations of elliptic Selberg integrals (conjectured in an earlier paper of the author), as well as studying in depth the ''interpolation kernel'', an analytic continuation of the author's elliptic interpolation functions which plays a major role in the proof as well as acting as the kernel for a Fourier transform on certain elliptic double affine Hecke algebras (discussed in a later paper). In the process, we give a number of examples of a new approach to proving elliptic hypergeometric integral identities, by reduction to a Zariski dense subset of a formal neighborhood of the trigonometric limit.

Publication: Symmetry, Integrability and Geometry: Methods and Applications (SIGMA) Vol.: 14ISSN: 1815-0659

ID: CaltechAUTHORS:20170922-141735253

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Abstract: We present a linear system of difference equations whose entries are expressed in terms of theta functions. This linear system is singular at 4m + 12 points for m ≥ 1, which appear in pairs due to a symmetry condition. We parameterize this linear system in terms of a set of kernels at the singular points. We regard the system of discrete isomonodromic deformations as an elliptic analogue of the Garnier system. We identify the special case in which m = 1 with the elliptic Painlevé equation; hence, this work provides an explicit form and Lax pair for the elliptic Painlevé equation.

Publication: Communications in Mathematical Physics Vol.: 355 No.: 2 ISSN: 0010-3616

ID: CaltechAUTHORS:20170725-124624205

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Abstract: We present a Lax pair for the sixth Painlevé equation arising as a continuous isomonodromic deformation of a system of linear difference equations with an additional symmetry structure. We call this a symmetric difference-differential Lax pair. We show how the discrete isomonodromic deformations of the associated linear problem gives us a discrete version of the fifth Painlevé equation. By considering degenerations, we obtain symmetric difference-differential Lax pairs for the fifth Painlevé equation and the various degenerate versions of the third Painlevé equation.

Publication: Journal of Integrable Systems Vol.: 2 No.: 1 ISSN: 2058-5985

ID: CaltechAUTHORS:20171120-085000823

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Abstract: We present four classes of nonlinear systems which may be considered discrete analogues of the Garnier system. These systems arise as discrete isomonodromic deformations of systems of linear difference equations in which the associated Lax matrices are presented in a factored form. A system of discrete isomonodromic deformations is completely determined by commutation relations between the factors. We also reparameterize these systems in terms of the image and kernel vectors at singular points to obtain a separate birational form. A distinguishing feature of this study is the presence of a symmetry condition on the associated linear problems that only appears as a necessary feature of the Lax pairs for the least degenerate discrete Painlevé equations.

Publication: Symmetry, Integrability and Geometry: Methods and Applications (SIGMA) Vol.: 12ISSN: 1815-0659

ID: CaltechAUTHORS:20161215-111704158

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Abstract: Generalized power sums are linear combinations of ith powers of coordinates. We consider subalgebras of the polynomial algebra generated by generalized power sums, and study when such algebras are Cohen–Macaulay. It turns out that the Cohen–Macaulay property of such algebras is rare, and tends to be related to quantum integrability and representation theory of Cherednik algebras. Using representation theoretic results and deformation theory, we establish Cohen–Macaulayness of the algebra of q, t-deformed power sums defined by Sergeev and Veselov, and of some generalizations of this algebra, proving a conjecture of Brookner, Corwin, Etingof, and Sam. We also apply representation-theoretic techniques to studying m-quasi-invariants of deformed Calogero–Moser systems. In an appendix to this paper, M. Feigin uses representation theory of Cherednik algebras to compute Hilbert series for such quasi-invariants, and show that in the case of one light particle, the ring of quasi-invariants is Gorenstein.

Publication: Communications in Mathematical Physics Vol.: 347 No.: 1 ISSN: 0010-3616

ID: CaltechAUTHORS:20161006-112242015

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Abstract: We give a new construction of noncommutative surfaces via elliptic difference operators, attaching a 1-parameter noncommutative deformation to any projective rational surface with smooth anticanonical curve. The construction agrees with one implicit in work of Van den Bergh (iterated blowups of noncommutative Hirzebruch surfaces), but the construction enables one to prove a number of new facts about these surfaces. We show that they are noncommutative smooth proper surfaces in the sense of Chan and Nyman, with projective Quot schemes, that moduli spaces of simple sheaves are Poisson and that moduli spaces classifying semistable sheaves of rank 0 or 1 are projective. We further show that the action of SL_2(Z) as derived autoequivalences of rational elliptic surfaces extends to an action as derived equivalences of surfaces in our family with K^2=0. We also discuss applications to the theory of special functions arising by interpreting moduli spaces of 1-dimensional sheaves as moduli spaces of difference equations. When the moduli space is a single point, the equation is rigid, and we give an integral representation for the solutions. More generally, twisting by line bundles corresponds to isomonodromy deformations, so this gives rise to Lax pairs. When the moduli space is 2-dimensional, one obtains Lax pairs for the elliptic Painlevé equation; this associates a Lax pair to any rational number, of order twice the denominator. There is also an elliptic analogue of the Riemann-Hilbert correspondence: an analytic equivalence between categories of elliptic difference equations, swapping the role of the shift of the equation and the nome of the curve.

ID: CaltechAUTHORS:20170922-135420800

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Abstract: Given a complex curve C of genus 2, there is a well-known relationship between the moduli space of rank 3 semistable bundles on C and a cubic hypersurface known as the Coble cubic. Some of the aspects of this is known to be related to the geometric invariant theory of the third exterior power of a 9-dimensional complex vector space. We extend this relationship to arbitrary fields and study some of the connections to invariant theory, which will be studied more in-depth in a followup paper.

ID: CaltechAUTHORS:20170922-135945128

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Abstract: We construct the elliptic Painlevé equation and its higher dimensional analogs as the action of line bundles on 1 -dimensional sheaves on noncommutative surfaces.

Publication: Algebra and Number Theory Vol.: 9 No.: 6 ISSN: 1937-0652

ID: CaltechAUTHORS:20151016-074552683

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Abstract: Using maximal isotropic submodules in a quadratic module over ℤ_p, we prove the existence of a natural discrete probability distribution on the set of isomorphism classes of short exact sequences of cofinite type ℤ_p-modules, and then conjecture that as E varies over elliptic curves over a fixed global field k, the distribution of 0 → E(k)⊗ ℚ_p/ ℤ_p → Selp∞ E → Ш[p^∞ ] → 0 is that one. We show that this single conjecture would explain many of the known theorems and conjectures on ranks, Selmer groups, and Shafarevich–Tate groups of elliptic curves. We also prove the existence of a discrete probability distribution on the set of isomorphism classes of finite abelian p-groups equipped with a nondegenerate alternating pairing, defined in terms of the cokernel of a random alternating matrix over ℤ_p, and we prove that the two probability distributions are compatible with each other and with Delaunay’s predicted distribution for Ш. Finally, we prove new theorems on the fppf cohomology of elliptic curves in order to give further evidence for our conjecture.

Publication: Cambridge Journal of Mathematics Vol.: 3 No.: 3 ISSN: 2168-0930

ID: CaltechAUTHORS:20151119-085554759

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Abstract: We describe a method, based on the theory of Macdonald-Koornwinder polynomials, for proving bounded Littlewood identities. Our approach provides an alternative to Macdonald's partial fraction technique and results in the first examples of bounded Littlewood identities for Macdonald polynomials. These identities, which take the form of decomposition formulas for Macdonald polynomials of type (R,S) in terms Macdonald polynomials of type A, are q,t-analogues of known branching formulas for characters of the symplectic, orthogonal and special orthogonal groups, important in the theory of plane partitions. As applications of our results we obtain combinatorial formulas for characters of affine Lie algebras, Rogers-Ramanujan identities for such algebras complementing recent results of Griffin et al., and transformation formulas for Kaneko-Macdonald-type hypergeometric series.

ID: CaltechAUTHORS:20170922-141021522

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Abstract: The purpose of this article is to bring structure to (basic) hypergeometric biorthogonal systems, in particular to the q-Askey scheme of basic hypergeometric orthogonal polynomials. We aim to achieve this by looking at the limits as p→0 of the elliptic hypergeometric biorthogonal functions from Spiridonov (2003), with parameters which depend in varying ways on p. As a result we get 38 systems of biorthogonal functions with for each system at least one explicit measure for the bilinear form. Amongst these we indeed recover the q-Askey scheme. Each system consists of (basic hypergeometric) rational functions or polynomials.

Publication: Journal of Approximation Theory Vol.: 193ISSN: 0021-9045

ID: CaltechAUTHORS:20150501-080444288

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Abstract: We study birational morphisms between smooth projective surfaces that respect a given Poisson structure, with particular attention to induced birational maps between the (Poisson) moduli spaces of sheaves on those surfaces. In particular, to any birational morphism, we associate a corresponding "minimal lift" operation on sheaves of homological dimension ≤ 1, and study its properties. In particular, we show that minimal lift induces a stratification of the moduli space of simple sheaves on the codomain by open subspaces of the moduli space of simple sheaves on the domain, compatibly with the induced Poisson structures.

ID: CaltechAUTHORS:20170922-144027436

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Abstract: By analogy with work of Hitchin on integrable systems, we construct natural relaxations of several kinds of moduli spaces of difference equations, with special attention to a particular class of difference equations on an elliptic curve (arising in the theory of elliptic special functions). The common feature of the relaxations is that they can be identified with moduli spaces of sheaves on rational surfaces. Not only does this make various natural questions become purely geometric (rigid equations correspond to -2-curves), it also establishes a number of nontrivial correspondences between different moduli spaces, since a given moduli space of sheaves is typically the relaxation of infinitely many moduli spaces of equations. In the process of understanding this, we also consider a number of purely geometric questions about rational surfaces with anticanonical curves; e.g., we give an essentially combinatorial algorithm for testing whether a given divisor is the class of a -2-curve or is effective with generically integral representative.

ID: CaltechAUTHORS:20170922-142713387

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Abstract: The Sklyanin algebra Sη has a well-known family of infinite-dimensional representations D(μ),μ∈C∗ , in terms of difference operators with shift η acting on even meromorphic functions. We show that for generic η the coefficients of these operators have solely simple poles, with linear residue relations depending on their locations. More generally, we obtain explicit necessary and sufficient conditions on a difference operator for it to belong to D(μ) . By definition, the even part of D(μ) is generated by twofold products of the Sklyanin generators. We prove that any sum of the latter products yields a difference operator of van Diejen type. We also obtain kernel identities for the Sklyanin generators. They give rise to order-reversing involutive automorphisms of D(μ) , and are shown to entail previously known kernel identities for the van Diejen operators. Moreover, for special μ they yield novel finite-dimensional representations of Sη .

Publication: Communications in Mathematical Physics Vol.: 320 No.: 3 ISSN: 0010-3616

ID: CaltechAUTHORS:20130723-114503880

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Abstract: The permutation representation afforded by a Coxeter group W acting on the cosets of a standard parabolic subgroup inherits many nice properties from W such as a shellable Bruhat order and a flat deformation over ℤ[q] to a representation of the corresponding Hecke algebra. In this paper we define a larger class of “quasiparabolic” subgroups (more generally, quasiparabolic W-sets), and show that they also inherit these properties. Our motivating example is the action of the symmetric group on fixed-point-free involutions by conjugation.

Publication: Journal of Algebraic Combinatorics Vol.: 37 No.: 3 ISSN: 0925-9899

ID: CaltechAUTHORS:20130426-134058196

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Abstract: We prove analogues for elliptic interpolation functions of Macdonaldʼs version of the Littlewood identity for (skew) Macdonald polynomials, in the process developing an interpretation of general elliptic “hypergeometric” sums as skew interpolation functions. One such analogue has an interpretation as a “vanishing integral”, generalizing a result of Rains and Vazirani (2007) [17]; the structure of this analogue gives sufficient insight to enable us to conjecture elliptic versions of most of the other vanishing integrals of Rains and Vazirani (2007) [17] as well. We are thus led to formulate ten conjectures, each of which can be viewed as a multivariate quadratic transformation, and can be proved in a number of special cases.

Publication: Journal of Combinatorial Theory. Series A Vol.: 119 No.: 7 ISSN: 0097-3165

ID: CaltechAUTHORS:20120730-134423880

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Abstract: We characterize averages of ∏^N_(l=1)│x−t_l│^(ɑ−1) with respect to the Selberg density, further constrained so that t_l є [0,x](l=1,...,q) and t_l є [x,1](l=q^+1,...,N), in terms of a basis of solutions of a particular Fuchsian matrix differential equation. By making use of the Dotsenko-Fateev integrals, the explicit form of the connection matrix from the Frobenius type power series basis to this basis is calculated, thus allowing us to explicitly compute coefficients in the power series expansion of the averages. From these we are able to compute power series for the marginal distributions of the t_j(j=1,...,N) . In the case q = 0 and α < 1 we compute the explicit leading order term in the x → 0 asymptotic expansion, which is of interest to the study of an effect known as singularity dominated strong fluctuations. In the case q = 0 and ɑ є Z^+, and with the absolute values removed, the average is a polynomial, and we demonstrate that its zeros are highly structured.

Publication: Communications in Mathematical Physics Vol.: 309 No.: 3 ISSN: 0010-3616

ID: CaltechAUTHORS:20120314-084236569

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Abstract: Under suitable hypotheses, we construct a probability measure on the set of closed maximal isotropic subspaces of a locally compact quadratic space over F_p. A random subspace chosen with respect to this measure is discrete with probability 1, and the dimension of its intersection with a fixed compact open maximal isotropic subspace is a certain nonnegative-integer-valued random variable. We then prove that the p-Selmer group of an elliptic curve is naturally the intersection of a discrete maximal isotropic subspace with a compact open maximal isotropic subspace in a locally compact quadratic space over F_p. By modeling the first subspace as being random, we can explain the known phenomena regarding distribution of Selmer ranks, such as the theorems of Heath-Brown, Swinnerton-Dyer, and Kane for 2-Selmer groups in certain families of quadratic twists, and the average size of 2- and 3-Selmer groups as computed by Bhargava and Shankar. Our model is compatible with Delaunay's heuristics for p-torsion in Shafarevich-Tate groups, and predicts that the average rank of elliptic curves over a fixed number field is at most 1/2. Many of our results generalize to abelian varieties over global fields.

Publication: Journal of the American Mathematical Society Vol.: 25 No.: 1 ISSN: 0894-0347

ID: CaltechAUTHORS:20111207-112922811

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Abstract: In this article we consider the elliptic Selberg integral, which is a BC_n symmetric multivariate extension of the elliptic beta integral. We categorize the limits that are obtained as p → 0, for given behavior of the parameters as p → 0. This article is therefore the multivariate version of our earlier paper "Basic Hypergeometric Functions as Limits of Elliptic Hypergeometric Functions". The integrand of the elliptic Selberg integral is the measure for the BC_n symmetric biorthogonal functions introduced by the second author, so we also consider the limits of the associated bilinear form. We also provide the limits for the discrete version of this bilinear form, which is related to a multivariate extension of the Frenkel-Turaev summation.

ID: CaltechAUTHORS:20170922-144400917

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Abstract: In this article we extend the results of our article "Limits of elliptic hypergeometric biorthogonal functions" to the multivariate setting. In that article we determined which families of biorthogonal functions arise as limits from the elliptic hypergeometric biorthogonal functions from Spiridonov when p → 0. Here we show that the classification of the possible limits of the BC_n type multivariate biorthogonal functions previously introduced by the second author is identical to the univariate classification. That is, for each univariate limit family there exists a multivariate extension, and in particular we obtain multivariate versions for all elements of the q-Askey scheme. For the Askey-Wilson polynomials these are the Koornwinder polynomials, and the multivariate versions of the Pastro polynomials form a two-parameter family which include the Macdonald polynomials.

ID: CaltechAUTHORS:20170922-152223616

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Abstract: We construct a family of second-order linear difference equations parametrized by the hypergeometric solution of the elliptic Painlevé equation (or higher-order analogues), and admitting a large family of monodromy-preserving deformations. The solutions are certain semiclassical biorthogonal functions (and their Cauchy transforms), biorthogonal with respect to higher-order analogues of Spiridonov's elliptic beta integral.

Publication: Symmetry, Integrability and Geometry, Methods and Applications (SIGMA) Vol.: 7ISSN: 1815-0659

ID: CaltechAUTHORS:20110929-113549134

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Abstract: We study differential operators on an elliptic curve of order higher than 2 which are algebraically integrable (i.e., finite gap). We discuss classification of such operators of order 3 with one pole, discovering exotic operators on special elliptic curves defined over Q which do not deform to generic elliptic curves. We also study algebraically integrable operators of higher order with several poles and with symmetries, and (conjecturally) relate them to crystallographic elliptic Calogero-Moser systems (which is a generalization of the results of Airault, McKean, and Moser).

Publication: Symmetry, Integrability and Geometry, Methods and Applications (SIGMA) Vol.: 7ISSN: 1815-0659

ID: CaltechAUTHORS:20110725-113214387

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Abstract: Perhaps the nicest multivariate orthogonal polynomials are the Macdonald and Koornwinder polynomials, respectively 2-parameter deformations of Schur functions and 6-parameter deformations of orthogonal and symplectic characters, satisfying a trio of nice properties known as the Macdonald “conjectures”. In recent work, the author has constructed elliptic analogues: a family of multivariate functions on an elliptic curve satisfying analogues of the Macdonald conjectures, and degenerating to Macdonald and Koornwinder polynomials under suitable limits. This article will discuss the two main constructions for these functions, focusing on the more algebraic/combinatorial of the two approaches.

ID: CaltechAUTHORS:20170925-124323959

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Abstract: We give a general formula relating self cup products in cohomology to connecting maps in nonabelian cohomology, and apply it to obtain a formula for the self cup product associated to the Weil pairing.

Publication: Mathematical Research Letters Vol.: 18 No.: 6 ISSN: 1073-2780

ID: CaltechAUTHORS:20130328-103531472

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Abstract: We introduce elliptic weights of boxed plane partitions and prove that they give rise to a generalization of MacMahon’s product formula for the number of plane partitions in a box. We then focus on the most general positive degenerations of these weights that are related to orthogonal polynomials; they form three 2-D families. For distributions from these families, we prove two types of results. First, we construct explicit Markov chains that preserve these distributions. In particular, this leads to a relatively simple exact sampling algorithm. Second, we consider a limit when all dimensions of the box grow and plane partitions become large and prove that the local correlations converge to those of ergodic translation invariant Gibbs measures. For fixed proportions of the box, the slopes of the limiting Gibbs measures (that can also be viewed as slopes of tangent planes to the hypothetical limit shape) are encoded by a single quadratic polynomial.

Publication: Selecta Mathematica - New Series Vol.: 16 No.: 4 ISSN: 1022-1824

ID: CaltechAUTHORS:20101123-092946208

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Abstract: We compute the Poincaré polynomial and the cohomology algebra with rational coefficients of the manifold M_n of real points of the moduli space of algebraic curves of genus 0 with n labeled points. This cohomology is a quadratic algebra, and we conjecture that it is Koszul. We also compute the 2-local torsion in the cohomology of M_n. As was shown by the fourth author, the cohomology of M_n does not have odd torsion, so that the above determines the additive structure of the integral homology and cohomology. Further, we prove that the rational homology operad of M_n is the operad of 2-Gerstenhaber algebras, which is closely related to the Hanlon-Wachs operad of 2-Lie algebras (generated by a ternary bracket). Finally, using Drinfeld’s theory of quantization of coboundary Lie quasibialgebras, we show that a large series of representations of the quadratic dual Lie algebra L_n of H^*(M_n,Q) (associated to such quasibialgebras) factors through the the natural projection of L_n to the associated graded Lie algebra of the prounipotent completion of the fundamental group of M_n. This leads us to conjecture that the said projection is an isomorphism, which would imply a formula for lower central series ranks of the fundamental group. On the other hand, we show that the spaces M_n are not formal starting from n = 6.

Publication: Annals of Mathematics Vol.: 171 No.: 2 ISSN: 0003-486X

ID: CaltechAUTHORS:20100804-142058742

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Abstract: We prove a pair of transformations relating elliptic hypergeometric integrals of different dimensions, corresponding to the root systems BC_n and A_n; as a special case, we recover some integral identities conjectured by van Diejen and Spiridonov. For BC_n, we also consider their “Type II” integral. Their proof of that integral, together with our transformation, gives rise to pairs of adjoint integral operators; a different proof gives rise to pairs of adjoint difference operators. These allow us to construct a family of biorthogonal abelian functions generalizing the Koornwinder polynomials, and satisfying the analogues of the Macdonald conjectures. Finally, we discuss some transformations of Type II-style integrals. In particular, we find that adding two parameters to the Type II integral gives an integral invariant under an appropriate action of the Weyl group E_7.

Publication: Annals of Mathematics Vol.: 171 No.: 1 ISSN: 0003-486X

ID: CaltechAUTHORS:20100611-112526706

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Abstract: Associated to any subspace arrangement is a ‘De Concini–Procesi model’, a certain smooth compactification of its complement, which in the case of the braid arrangement produces the Deligne–Mumford compactification of the moduli space of genus 0 curves with marked points. In the present work, we calculate the integral homology of real De Concini–Procesi models, extending earlier work of Etingof, Henriques, Kamnitzer and the author on the (2-adic) integral cohomology of the real locus of the moduli space. To be precise, we show that the integral homology of a real De Concini–Procesi model is isomorphic modulo its 2-torsion to a sum of cohomology groups of subposets of the intersection lattice of the arrangement. As part of the proof, we construct a large family of natural maps between De Concini–Procesi models (generalizing the operad structure of moduli space), and determine the induced action on poset cohomology. In particular, this determines the ring structure of the cohomology of De Concini–Procesi models (modulo 2-torsion).

Publication: Journal of Topology Vol.: 3 No.: 4 ISSN: 1753-8416

ID: CaltechAUTHORS:20101221-082008592

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Abstract: We start from an interpretation of the BC_(2)-symmetric “Type I” (elliptic Dixon) elliptic hypergeometric integral evaluation as a formula for a Casoratian of the elliptic hypergeometric equation and then generalize this construction to higher-dimensional integrals and higher-order hypergeometric functions. This allows us to prove the corresponding formulas for the elliptic beta integral and symmetry transformation in a new way, by proving that both sides satisfy the same difference equations and that these difference equations satisfy a needed Galois-theoretic condition ensuring the uniqueness of the simultaneous solution.

Publication: Functional Analysis and its Applications Vol.: 43 No.: 4 ISSN: 0016-2663

ID: CaltechAUTHORS:20100415-102226300

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Abstract: The theory of zonal polynomials is used to compute the average of a Schur polynomial of argument AX, where A is a fixed matrix and X is from the real Ginibre ensemble. This generalizes a recent result of Sommers and Khoruzhenko (2009 J. Phys. A: Math. Theor. 42 222002), and furthermore allows analogous results to be obtained for the complex and real quaternion Ginibre ensembles. As applications, the positive integer moments of the general variance Ginibre ensembles are computed in terms of generalized hypergeometric functions; these are written in terms of averages over matrices of the same size as the moment to give duality formulas, and the averages of the power sums of the eigenvalues are expressed as finite sums of zonal polynomials.

Publication: Journal of Physics A: Mathematical and Theoretical Vol.: 42 No.: 38 ISSN: 1751-8113

ID: CaltechAUTHORS:20090923-143135904

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Abstract: The Knop-Sahi interpolation Macdonald polynomials are inhomogeneous and nonsymmetric generalisations of the well-known Macdonald polynomials. In this paper we apply the interpolation Macdonald polynomials to study a new type of basic hypergeometric series of type gl_n. Our main results include a new q-binomial theorem, a new q-Gauss sum, and several transformation formulae for gl_n series.

Publication: Transformation Groups Vol.: 14 No.: 3 ISSN: 1083-4362

ID: CaltechAUTHORS:20090911-153557865

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Abstract: We describe a uniform way of obtaining basic hypergeometric functions as limits of the elliptic beta integral. This description gives rise to the construction of a polytope with a different basic hypergeometric function attached to each face of this polytope. We can subsequently obtain various relations, such as transformations and three-term relations, of these functions by considering geometrical properties of this polytope. The most general functions we describe in this way are sums of two very-well-poised _10φ_9's and their Nassrallah-Rahman type integral representation.

Publication: Symmetry, Integrability and Geometry, Methods and Applications (SIGMA) Vol.: 5 No.: 059 ISSN: 1815-0659

ID: CaltechAUTHORS:20090904-142309890

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Abstract: In recent work by Etingof, Henriques, Kamnitzer, and the author, a presentation and explicit basis was given for the rational cohomology of the real locus M_(0,n)(R) of the moduli space of stable genus 0 curves with n marked points. We determine the graded character of the action of S_n on this space (induced by permutations of the marked points), both in the form of a plethystic formula for the cycle index, and as an explicit product formula for the value of the character on a given cycle type.

Publication: Selecta Mathematica - New Series Vol.: 15 No.: 1 ISSN: 1022-1824

ID: CaltechAUTHORS:20090922-113504889

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Abstract: In Ann. Math., to appear, 2008, the author proved a number of multivariate elliptic hypergeometric integrals. The purpose of the present note is to explore more carefully the various limiting cases (hyperbolic, trigonometric, rational, and classical) that exist. In particular, we show (using some new estimates of generalized gamma functions) that the hyperbolic integrals (previously treated as purely formal limits) are indeed limiting cases. We also obtain a number of new trigonometric (q-hypergeometric) integral identities as limits from the elliptic level.

Publication: Ramanujan Journal Vol.: 18 No.: 3 ISSN: 1382-4090

ID: CaltechAUTHORS:20090423-141245409

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Abstract: The Kruskal Count is a card trick invented by Martin D. Kruskal (who is well known for his work on solitons) which is described in Fulves and Gardner (1975) and Gardner (1978, 1988). In this card trick a magician “guesses” one card in a deck of cards which is determined by a subject using a special counting procedure that we call Kruskal's counting procedure. The magician has a strategy which with high probability will identify the correct card, explained below. Kruskal's counting procedure goes as follows. The subject shuffles a deck of cards as many times as he likes. He mentally chooses a (secret) number between one and ten. The subject turns the cards of the deck face up one at a time, slowly, and places them in a pile. As he turns up each card he decreases his secret number by one and he continues to count this way till he reaches zero. The card just turned up at the point when the count reaches zero is called the first key card and its value is called the first key number. Here the value of an Ace is one, face cards are assigned the value five, and all other cards take their numerical value. The subject now starts the count over, using the first key number to determine where to stop the count at the second key card. He continues in this fashion, obtaining successive key cards until the deck is exhausted. The last key card encountered, which we call the tapped card, is the card to be “guessed” by the magician.

ISSN: 1614-0311

ID: CaltechAUTHORS:20170925-112254500

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Abstract: This paper is a sequel of [ER]. Specifically, let W be a Coxeter group, generated by s_i, i ∈ I. Then, following [ER], one can define a new deformation A_+ = A_+(W) of the group algebra Z[W_+] of the group W_+ of even elements in W. This deformation is an algebra over the ring R = Z[t^(±1)_(ijk)] = Z[T] of regular functions on a certain torus T of deformation parameters. The main result of [ER] implies that this deformation is flat (i.e. A_+ is a flat R-module) if and only if for every triple of indices Δ = {i, j, k} ⊂ I the corresponding rank 3 parabolic subgroup WΔ ⊂ W is infinite. (To be more precise, in [ER] we work over C, but the results routinely extend to the case of ground ring Z.)

Publication: Geometric and Functional Analysis Vol.: 17 No.: 6 ISSN: 1016-443X

ID: CaltechAUTHORS:20171101-153438612

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Abstract: We study the rational cohomology groups of the real De Concini–Procesi model corresponding to a finite Coxeter group, generalizing the type-A case of the moduli space of stable genus 0 curves with marked points. We compute the Betti numbers in the exceptional types, and give formulae for them in types B and D. We give a generating-function formula for the characters of the representations of a Coxeter group of type B on the rational cohomology groups of the corresponding real De Concini–Procesi model, and deduce the multiplicities of one-dimensional characters in the representations, and a formula for the Euler character. We also give a moduli space interpretation of this type-B variety, and hence show that the action of the Coxeter group extends to a slightly larger group.

Publication: International Mathematics Research Notices Vol.: 2008ISSN: 1073-7928

ID: CaltechAUTHORS:20090709-105804586

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Abstract: This note gives an explicit proof that the scalar subgroup of the Clifford-Weil group remains unchanged when passing to the quotient representation filling a gap in [3]. For other current and future errata to [3] see http://www.research.att.com/~njas/doc/cliff2.html/.

Publication: Albanian Journal of Mathematics Vol.: 2 No.: 3 ISSN: 1930-1235

ID: CaltechAUTHORS:20170925-133038865

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Abstract: It has been shown that the last passage time in certain symmetrized models of directed percolation can be written in terms of averages over random matrices from the classical groups U(l), Sp(2l) and O(l). We present a theory of such results based on non-intersecting lattice paths, and integration techniques familiar from the theory of random matrices. Detailed derivations of probabilities relating to two further symmetrizations are also given.

Publication: Journal of Statistical Physics Vol.: 129 No.: 5-6 ISSN: 0022-4715

ID: CaltechAUTHORS:20171003-095341751

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Abstract: When one expands a Schur function in terms of the irreducible characters of the symplectic (or orthogonal) group, the coefficient of the trivial character is 0 unless the indexing partition has an appropriate form. A number of q,t-analogues of this fact were conjectured in [10]; the present paper proves most of those conjectures, as well as some new identities suggested by the proof technique. The proof involves showing that a nonsymmetric version of the relevant integral is annihilated by a suitable ideal of the affine Hecke algebra, and that any such annihilated functional satisfies the desired vanishing property. This does not, however, give rise to vanishing identities for the standard nonsymmetric Macdonald and Koornwinder polynomials; we discuss the required modification to these polynomials to support such results.

Publication: Transformation Groups Vol.: 12 No.: 4 ISSN: 1083-4362

ID: CaltechAUTHORS:20171010-113734662

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Abstract: Based on Spiridonov’s analysis of elliptic generalizations of the Gauss hypergeometric function, we develop a common framework for 7-parameter families of generalized elliptic, hyperbolic and trigonometric univariate hypergeometric functions. In each case we derive the symmetries of the generalized hypergeometric function under the Weyl group of type E_7 (elliptic, hyperbolic) and of type E_6 (trigonometric) using the appropriate versions of the Nassrallah-Rahman beta integral, and we derive contiguous relations using fundamental addition formulas for theta and sine functions. The top level degenerations of the hyperbolic and trigonometric hypergeometric functions are identified with Ruijsenaars’ relativistic hypergeometric function and the Askey-Wilson function, respectively. We show that the degeneration process yields various new and known identities for hyperbolic and trigonometric special functions. We also describe an intimate connection between the hyperbolic and trigonometric theory, which yields an expression of the hyperbolic hypergeometric function as an explicit bilinear sum in trigonometric hypergeometric functions.

Publication: Communications in Mathematical Physics Vol.: 275 No.: 1 ISSN: 0010-3616

ID: CaltechAUTHORS:20171003-101141109

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Abstract: Let D be a simply laced Dynkin diagram of rank r whose affinization has the shape of a star (i.e., D_4, E_6, E_7, E_8). To such a diagram one can attach a group G whose generators correspond to the legs of the affinization, have orders equal to the leg lengths plus 1, and the product of the generators is 1. The group G is then a 2-dimensional crystallographic group: G=Z_ℓ⋉Z^2, where ℓ is 2, 3, 4, and 6, respectively. In this paper, we define a flat deformation H(t, q) of the group algebra C[G] of this group, by replacing the relations saying that the generators have prescribed orders by their deformations, saying that the generators satisfy monic polynomial equations of these orders with arbitrary roots (which are deformation parameters). The algebra H(t, q) for D4 is the Cherednik algebra of type C∨C_1, which was studied by Noumi, Sahi, and Stokman, and controls Askey–Wilson polynomials. We prove that H(t, q) is the universal deformation of the twisted group algebra of G, and that this deformation is compatible with certain filtrations on C[G]. We also show that if q is a root of unity, then for generic t the algebra H(t, q) is an Azumaya algebra, and its center is the function algebra on an affine del Pezzo surface. For generic q, the spherical subalgebra eH(t, q)e provides a quantization of such surfaces. We also discuss connections of H(t, q)with preprojective algebras and Painlevé VI.

Publication: Advances in Mathematics Vol.: 212 No.: 2 ISSN: 0001-8708

ID: CaltechAUTHORS:20171018-155901135

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Abstract: We show that the Hilbert polynomial P(t) of the trace space A/[A,A] of the centrally extended preprojective algebra A of an ADE quiver is equal to the Hilbert series of the maximal nilpotent subalgebra of the corresponding simple Lie algebra under the principal gradation. This implies that the Hilbert polynomial of the center of A is t^(2h−4)P(1/t), where h is the Coxeter number.

Publication: Journal of Algebra Vol.: 313 No.: 1 ISSN: 0021-8693

ID: CaltechAUTHORS:20171004-092059005

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Abstract: For a positive integer n we introduce quadratic Lie algebras tr_n, qtr_n and finitely discrete groups Tr_n, QTr_n naturally associated with the classical and quantum Yang–Baxter equation, respectively. We prove that the universal enveloping algebras of the Lie algebras tr_n, qtr_n are Koszul, and compute their Hilbert series. We also compute the cohomology rings for these Lie algebras (which by Koszulity are the quadratic duals of the enveloping algebras). Finally, we construct a basis of U(tr_n). We construct cell complexes which are classifying spaces of the groups Tr_n and QTr_n, and show that the boundary maps in them are zero, which allows us to compute the integral cohomology of these groups. We show that the Lie algebras tr_n, qtr_n map onto the associated graded algebras of the Malcev Lie algebras of the groups Tr_n, QTr_n, respectively. In the case of Tr_n, we use quantization theory of Lie bialgebras to show that this map is actually an isomorphism. At the same time, we show that the groups Tr_n and QTr_n are not formal for n⩾4.

Publication: Journal of Algebra Vol.: 305 No.: 2 ISSN: 0021-8693

ID: CaltechAUTHORS:20171002-155152258

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Abstract: Motivated by the discovery that the eighth root of the theta series of the E_8 lattice and the 24th root of the theta series of the Leech lattice both have integer coefficients, we investigate the question of when an arbitrary element f∈R (where R=1+xZ〚x〛) can be written as f=g^n for g∈R, n⩾2. Let P_n:={g^n|g∈R} and let μ_n:=n∏_p|_np. We show among other things that (i) for f∈R, f∈P_n⇔f(mod μ_n)∈P_n, and (ii) if f∈P_n, there is a unique g∈P_n with coefficients mod μ_n/n such that f≡g^n(mod μ_n). In particular, if f≡1(mod μ_n) then f∈P_n. The latter assertion implies that the theta series of any extremal even unimodular lattice in R^n (e.g. E_8 in R^8) is in P_n if n is of the form 2^i3^j5^k (i⩾3). There do not seem to be any exact analogues for codes, although we show that the weight enumerator of the rth order Reed–Muller code of length 2^m is in P_2r(and similarly that the theta series of the Barnes–Wall lattice BW_2m is in P_2m). We give a number of other results and conjectures, and establish a conjecture of Paul D. Hanna that there is a unique element f∈P_n (n⩾2) with coefficients restricted to the set {1,2,…,n}.

Publication: Journal of Combinatorial Theory. Series A Vol.: 113 No.: 8 ISSN: 0097-3165

ID: CaltechAUTHORS:20171027-091207506

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Abstract: When one expands a Schur function in terms of the irreducible characters of the symplectic (or orthogonal) group, the coefficient of the trivial character is 0 unless the indexing partition has an appropriate form. A number of q-analogues of this fact were conjectured in [8]; the present paper proves most of those conjectures, as well as some new identities suggested by the proof technique. The proof involves showing that a nonsymmetric version of the relevant integral is annihilated by a suitable ideal of the affine Hecke algebra, and that any such annihilated functional satisfies the desired vanishing property. This does not, however, give rise to vanishing identities for the standard nonsymmetric Macdonald and Koornwinder polynomials; we discuss the required modification to these polynomials to support such results.

ID: CaltechAUTHORS:20171010-113111196

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Abstract: Preprojective algebras of quivers were introduced in 1979 by Gelfand and Ponomarev [GP], because for quivers of finite ADE type, they are models for indecomposable representations (they contain each indecomposable exactly once). Twenty years later, these algebras and their deformed versions introduced in [CBH] (for arbitrary quivers) became a subject of intense interest, since their representation varieties, called quiver varieties, played an important role in geometric representation theory. Ironically, it is exactly for quivers of finite ADE type that preprojective algebras fail to have good properties—they are not Koszul and their deformed versions are not flat.

Publication: Journal of Algebra Vol.: 299 No.: 2 ISSN: 0021-8693

ID: CaltechAUTHORS:20171002-152707569

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Abstract: Our interest is in the scaled joint distribution associated with k-increasing subsequences for random involutions with a prescribed number of fixed points. We proceed by specifying in terms of correlation functions the same distribution for a Poissonized model in which both the number of symbols in the involution and the number of fixed points are random variables. From this, a de-Poissonization argument yields the scaled correlations and distribution function for the random involutions. These are found to coincide with the same quantities known in random matrix theory from the study of ensembles interpolating between the orthogonal and symplectic universality classes at the soft edge, the interpolation being due to a rank 1 perturbation.

Publication: International Mathematics Research Notices Vol.: 2006ISSN: 1073-7928

ID: CaltechAUTHORS:20171106-151415738

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Abstract: Killip and Nenciu gave random recurrences for the characteristic polynomials of certain unitary and real orthogonal upper Hessenberg matrices. The corresponding eigenvalue probability density functions (p.d.f's) are β-generalizations of the classical groups. Left open was the direct calculation of certain Jacobians. We provide the sought direct calculation. Furthermore, we show how a multiplicative rank 1 perturbation of the unitary Hessenberg matrices provides a joint eigenvalue p.d.f. generalizing the circular β-ensemble, and we show how this joint density is related to known interrelations between circular ensembles. Projecting the joint density onto the real line leads to the derivation of a random three-term recurrence for polynomials with zeros distributed according to the circular Jacobi β-ensemble.

Publication: International Mathematics Research Notices Vol.: 2006ISSN: 1073-7928

ID: CaltechAUTHORS:20171106-151903451

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Abstract: We construct new families of (q-) difference and (contour) integral operators having nice actions on Koornwinder's multivariate orthogonal polynomials. We further show that the Koornwinder polynomials can be constructed by suitable sequences of these operators applied to the constant polynomial 1, giving the difference-integral representation of the title. Macdonald's conjectures (as proved by van Diejen and Sahi) for the principal specialization and norm follow immediately, as does a Cauchy-type identity of Mimachi.

No.: 417
ID: CaltechAUTHORS:20171010-105435263

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Abstract: We construct a family of BC_n-symmetric biorthogonal abelian functions generalizing Koornwinder's orthogonal polynomials (see [10]) and prove a number of their properties, most notably analogues of Macdonald's conjectures. The construction is based on a direct construction for a special case generalizing Okounkov's interpolation polynomials (see [13]). We show that these interpolation functions satisfy a collection of generalized hypergeometric identities, including new multivariate elliptic analogues of Jackson's summation and Bailey's transformation.

Publication: Duke Mathematical Journal Vol.: 135 No.: 1 ISSN: 0012-7094

ID: CaltechAUTHORS:20171003-154931716

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Abstract: [No abstract]

Vol.: 17
ID: CaltechAUTHORS:20171002-101719306

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Abstract: We give simple linear algebraic proofs of the Eynard–Mehta theorem, the Okounkov-Reshetikhin formula for the correlation kernel of the Schur process, and Pfaffian analogs of these results. We also discuss certain general properties of the spaces of all determinantal and Pfaffian processes on a given finite set.

Publication: Journal of Statistical Physics Vol.: 121 No.: 3-4 ISSN: 0022-4715

ID: CaltechAUTHORS:20171003-102643862

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Abstract: In recent work (math.QA/0309252) on multivariate hypergeometric integrals, the author generalized a conjectural integral formula of van Diejen and Spiridonov to a ten parameter integral provably invariant under an action of the Weyl group E_7. In the present note, we consider the action of the affine Weyl group, or more precisely, the recurrences satisfied by special cases of the integral. These are of two flavors: linear recurrences that hold only up to dimension 6, and three families of bilinear recurrences that hold in arbitrary dimension, subject to a condition on the parameters. As a corollary, we find that a codimension one special case of the integral is a tau function for the elliptic Painlevé equation.

ID: CaltechAUTHORS:20171009-142453684

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Abstract: We consider two important families of BC_n-symmetric polynomials, namely Okounkov's interpolation polynomials and Koornwinder's orthogonal polynomials. We give a family of difference equations satisfied by the former as well as generalizations of the branching rule and Pieri identity, leading to a number of multivariate q-analogues of classical hypergeometric transformations. For the latter, we give new proofs of Macdonald's conjectures, as well as new identities, including an inverse binomial formula and several branching rule and connection coefficient identities. We also derive families of ordinary symmetric functions that reduce to the interpolation and Koornwinder polynomials upon appropriate specialization. As an application, we consider a number of new integral conjectures associated to classical symmetric spaces.

Publication: Transformation Groups Vol.: 10 No.: 1 ISSN: 1083-4362

ID: CaltechAUTHORS:20171002-161044221

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Abstract: We define new deformations of group algebras of Coxeter groups W and of subgroups of even elements in them by deforming the braid relations. We show that these deformations are algebraically flat if and only if they are formally flat, and that this happens if and only if the group W has no finite parabolic subgroups of rank 3. We explain the connection of our deformations with the Hecke algebras of orbifolds defined by the first author and with generalized double affine Hecke algebras defined by the authors and A. Oblomkov.

Publication: International Mathematics Research Notices Vol.: 2005 No.: 10 ISSN: 1073-7928

ID: CaltechAUTHORS:20171106-145535121

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Abstract: Two types of parameter dependent generalizations of classical matrix ensembles are defined by their probability density functions (PDFs). As the parameter is varied, one interpolates between the eigenvalue PDF for the superposition of two classical ensembles with orthogonal symmetry and the eigenvalue PDF for a single classical ensemble with unitary symmetry, while the other interpolates between a classical ensemble with orthogonal symmetry and a classical ensemble with symplectic symmetry. We give interpretations of these PDFs in terms of probabilities associated to the continuous Robinson-Schensted-Knuth correspondence between matrices, with entries chosen from certain exponential distributions, and non-intersecting lattice paths, and in the course of this probability measures on partitions and pairs of partitions are identified. The latter are generalized by using Macdonald polynomial theory, and a particular continuum limit – the Jacobi limit – of the resulting measures is shown to give PDFs related to those appearing in the work of Anderson on the Selberg integral, and also in some classical work of Dixon. By interpreting Anderson’s and Dixon’s work as giving the PDF for the zeros of a certain rational function, it is then possible to identify random matrices whose eigenvalue PDFs realize the original parameter dependent PDFs. This line of theory allows sampling of the original parameter dependent PDFs, their Dixon-Anderson-type generalizations and associated marginal distributions, from the zeros of certain polynomials defined in terms of random three term recurrences.

Publication: Probability Theory and Related Fields Vol.: 131 No.: 1 ISSN: 0178-8051

ID: CaltechAUTHORS:20171003-160005064

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Abstract: This paper studies the behavior under iteration of the maps T_(ab)(x,y) = (F_(ab)(x)−y,x) of the plane ℝ^2 in which F_(ab)(x) = ax if x ≥ 0 and bx if x < 0. The orbits under iteration correspond to solutions of the nonlinear difference equation x_(n+2) = 1/2(a−b)|x_(n+1)|+1/2(a+b)x_(n+1)–x_n. This family of piecewise-linear maps has the parameter space(a,b)∈ ℝ^2. These maps are area-preserving homeomorphisms of ℝ^2 that map rays from the origin into rays from the origin. The action on rays gives an auxiliary map S_(ab) : S^1 → S^1 of the circle, which has a well-defined rotation number. This paper characterizes the possible dynamics under iteration of T_(ab) when the auxiliary map S_(ab) has rational rotation number. It characterizes cases where the map T_(ab) is a periodic map.

Publication: Journal of Difference Equations and Applications Vol.: 11 No.: 12 ISSN: 1023-6198

ID: CaltechAUTHORS:20170927-154918316

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Abstract: This paper studies the behavior under iteration of the maps T_(ab) (x, y) = (F_(ab) (x) − y, x) of the plane ℝ^2, in which F_(ab) (x) = ax if x ≥ 0 and bx if x < 0. The orbits under iteration correspond to solutions of the difference equation x_(n+2) = ½(a-b)|x_(n+1)| + ½(a+b)x_(n+1) – x_n. This family of piecewise-linear maps of the plane has the parameter space (a,b) ϵ ℝ^2. These maps are area-preserving homeomorphisms of ℝ^2 that map rays from the origin into rays from the origin. We show the existence of special parameter values where T_(ab) has every nonzero orbit contained in an invariant circle with an irrational rotation number, with invariant circles that are piecewise unions of arcs of conic sections. Numerical experiments indicate the possible existence of invariant circles for many other parameter values.

Publication: Journal of Difference Equations and Applications Vol.: 11 No.: 13 ISSN: 1023-6198

ID: CaltechAUTHORS:20171002-100946430

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Abstract: This paper studies the behavior under iteration of the maps T_(ab) (x,y) = (F_(ab) (x) − y, x) of the plane ℝ^2, in which F_(ab) (x) = ax if x ≥ 0 and bx if x < 0. These maps are area-preserving homeomorphisms of ℝ^2 that map rays from the origin to rays from the origin. Orbits of the map correspond to solutions of the nonlinear difference equation x_(n+2) = 1/2(a − b)|x_(n+1)|+1/2(a+b)x_(n+1) – x_n . This difference equation can be rewritten in an eigenvalue form for a nonlinear difference operator of Schrödinger type – x_(n+2)+2x_(n+1) – x_n +V_μ(x_(n+1))x_(n+1) = Ex_(n+1), in which μ = (1/2)(a − b) is fixed, and V_μ(x) = μ(sgn(x)) is an antisymmetric step function potential, and the energy E = 2 − 1/2(a+b). We study the set Ω_(SB) of parameter values where the map T_(ab) has at least one bounded orbit, which correspond to l∞-eigenfunctions of this difference operator. The paper shows that for transcendental μ the set Spec∞[μ] of energy values E having a bounded solution is a Cantor set. Numerical simulations suggest the possibility that these Cantor sets have positive (one-dimensional) measure for all real values of μ.

Publication: Journal of Difference Equations and Applications Vol.: 11 No.: 14 ISSN: 1023-6198

ID: CaltechAUTHORS:20171002-100214237

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Abstract: A superposition of a matrix ensemble refers to the ensemble constructed from two independent copies of the original, while a decimation refers to the formation of a new ensemble by observing only every second eigenvalue. In the cases of the classical matrix ensembles with orthogonal symmetry, it is known that forming superpositions and decimations gives rise to classical matrix ensembles with unitary and symplectic symmetry. The basic identities expressing these facts can be extended to include a parameter, which in turn provides us with probability density functions which we take as the definition of special parameter dependent matrix ensembles. The parameter dependent ensembles relating to superpositions interpolate between superimposed orthogonal ensembles and a unitary ensemble, while the parameter dependent ensembles relating to decimations interpolate between an orthogonal ensemble with an even number of eigenvalues and a symplectic ensemble of half the number of eigenvalues. By the construction of new families of biorthogonal and skew orthogonal polynomials, we are able to compute the corresponding correlation functions, both in the finite system and in various scaled limits. Specializing back to the cases of orthogonal and symplectic symmetry, we find that our results imply different functional forms to those known previously.

Publication: Probability Theory and Related Fields Vol.: 130 No.: 4 ISSN: 0178-8051

ID: CaltechAUTHORS:20171003-102014270

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Abstract: The main theorem in this paper is a far-reaching generalization of Gleason's theorem on the weight enumerators of codes which applies to arbitrary-genus weight enumerators of self-dual codes defined over a large class of finite rings and modules. The proof of the theorem uses a categorical approach, and will be the subject of a forthcoming book. However, the theorem can be stated and applied without using category theory, and we illustrate it here by applying it to generalized doubly-even codes over fields of characteristic 2, doubly-even codes over ℤ/2fℤ, and self-dual codes over the noncommutative ring F_q + F_qu, where u^2 = 0.

Publication: Mathematische Nachrichten Vol.: 274-275 No.: 1 ISSN: 0025-584X

ID: CaltechAUTHORS:20171004-091341996

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Abstract: For any q which is a power of 2 we describe a finite subgroup of GL_q(ℂ) under which the complete weight enumerators of generalized doubly-even self-dual codes over F_q are invariant. An explicit description of the invariant ring and some applications to extremality of such codes are obtained in the case q=4.

Publication: Finite Fields and Their Applications Vol.: 10 No.: 4 ISSN: 1071-5797

ID: CaltechAUTHORS:20171004-090734197

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Abstract: We present an efficient addition circuit, borrowing techniques from the classical carry-lookahead arithmetic circuit. Our quantum carry-lookahead (QCLA) adder accepts two n-bit numbers and adds them in O(log n) depth using O(n) ancillary qubits. We present both in-place and out-of-place versions, as well as versions that add modulo 2^n and modulo 2^n - 1. Previously, the linear-depth ripple-carry addition circuit has been the method of choice. Our work reduces the cost of addition dramatically with only a slight increase in the number of required qubits. The QCLA adder can be used within current modular multiplication circuits to reduce substantially the run-time of Shor's algorithm.

ID: CaltechAUTHORS:20171010-104601338

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Abstract: With any integral lattice Λ in n-dimensional Euclidean space we associate an elementary abelian 2-group I(Λ) whose elements represent parts of the dual lattice that are similar to Λ. There are corresponding involutions on modular forms for which the theta series of Λis an eigenform; previous work has focused on this connection. In the present paper I(Λ) is considered as a quotient of some finite 2-subgroup of O_n(ℝ). We establish upper bounds, depending only on n, for the order of I(Λ), and we study the occurrence of similarities of specific types.

Publication: Journal of Number Theory Vol.: 101 No.: 1 ISSN: 0022-314X

ID: CaltechAUTHORS:20170925-135839254

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Abstract: We give an independent proof of the Krasikov-Litsyn bound d/n ≾ (1-5/^(-1/4))/2 on doubly-even self-dual binary codes. The technique used (a refinement of the Mallows-Odlyzko-Sloane approach) extends easily to other families of self-dual codes, modular lattices, and quantum codes; in particular, we show that the Krasikov-Litsyn bound applies to singly-even binary codes, and obtain an analogous bound for unimodular lattices. We also show that in each case, our bound differs from the true optimum by an amount growing faster than O(√n).

Publication: IEEE Transactions on Information Theory Vol.: 49 No.: 5 ISSN: 0018-9448

ID: CaltechAUTHORS:20170925-142327910

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Abstract: An asymmetric covering D(n,R) is a collection of special subsets S of an n-set such that every subset T of the n-set is contained in at least one special S with |S| - |T| ≤ R. In this paper we compute the smallest size of any D(n,1) for n ≤ 8. We also investigate “continuous” and “banded” versions of the problem. The latter involves the classical covering numbers C(n, k, k-1), and we determine the following new values: C(10, 5, 4) = 51, C(11, 7, 6) = 84, C(12, 8, 7) = 126, C(13, 9, 8) = 185 and C(14, 10, 9) = 259. We also find the number of non-isomorphic minimal covering designs in several cases.

Publication: Journal of Combinatorial Designs Vol.: 11 No.: 3 ISSN: 1063-8539

ID: CaltechAUTHORS:20171011-152858004

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Abstract: In [9], it was shown that if U is a random n×n unitary matrix, then for any p≥n, the eigenvalues of U^p are i.i.d. uniform; similar results were also shown for general compact Lie groups. We study what happens when p

ID: CaltechAUTHORS:20171106-141427328

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Abstract: Recently van der Geer and Schoof [11, Prop. 1] formulated an “exact” analogue of the Riemann-Roch theorem for an algebraic number field K, based on Arakelov divisors. They used this result to formally express the completed zeta function ζ_K(s) of K as an integral over the Arakelov divisor class group Pic(K) of K. They introduced a two-variable zeta function attached to a number field K, also given as an integral over the Arkelov class group, which we call either the Arakelov zeta function or the two-variable zeta function. This zeta function was modelled after a two-variable zeta function attached to a function field over a finite filed, introduced in 1996 by Pellikaan [18]. For convenience we review the Arakelov divisor interpretation of the two-variable zeta function and the Riemann-Roch theorem for number fields in an appendix.

Publication: Annales de l'Institut Fourier Vol.: 53 No.: 1 ISSN: 1777-5310

ID: CaltechAUTHORS:20171027-085620532

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Abstract: Self-dual codes are important because many of the best codes known are of this type and they have a rich mathematical theory. Topics covered in this survey include codes over F_2, F_3, F_4, F_q, Z_4, Z_m, shadow codes, weight enumerators, Gleason-Pierce theorem, invariant theory, Gleason theorems, bounds, mass formulae, enumeration, extremal codes, open problems. There is a comprehensive bibliography.

ID: CaltechAUTHORS:20171122-102726979

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Abstract: If the number of runs in a (mixed-level) orthogonal array of strength 2 is specified, what numbers of levels and factors are possible? The collection of possible sets of parameters for orthogonal arrays with N runs has a natural lattice structure, induced by the “expansive replacement” construction method. In particular the dual atoms in this lattice are the most important parameter sets, since any other parameter set for an N-run orthogonal array can be constructed from them. To get a sense for the number of dual atoms, and to begin to understand the lattice as a function of N, we investigate the height and the size of the lattice. It is shown that the height is at most ⌊c(N−1)⌋, where c=1.4039…, and that there is an infinite sequence of values of N for which this bound is attained. On the other hand, the number of nodes in the lattice is bounded above by a superpolynomial function of N (and superpolynomial growth does occur for certain sequences of values of N). Using a new construction based on “mixed spreads”, all parameter sets with 64 runs are determined. Four of these 64-run orthogonal arrays appear to be new.

Publication: Journal of Statistical Planning and Inference Vol.: 102 No.: 2 ISSN: 0378-3758

ID: CaltechAUTHORS:20170927-153859292

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Abstract: The EKC or electrocardiogram sequence is defined by a(1) = 1, a(2) = 2 and, for n ≥ 3, a(n) is the smallest natural number not already in the sequence with the property that gcd{a(n − 1),a(n)} > 1. In spite of its erratic local behavior, which when plotted resembles an electrocardiogram, its global behavior appears quite regular. We conjecture that almost all a(n) satisfy the asymptotic formula a(n) = n(1+1/(3logn)) + o(n/log n) as n → ∞ and that the exceptional values a(n) = p and a(n) = 3p, for p a prime, produce the spikes in the EKG sequence. We prove that {a(n) : n ≥ 1) is a permutation of the natural numbers and that c_1 n ≤ a(n) ≤ c_2 n for constants c_1,c_2. There remains a large gap between what is conjectured and what is proved.

Publication: Experimental Mathematics Vol.: 11 No.: 3 ISSN: 1058-6458

ID: CaltechAUTHORS:20171011-152214607

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Abstract: We obtain an identity between Fredholm determinants of two kinds of operators, one acting on functions on the unit circle and the other acting on functions on a subset of the integers. This identity is a generalization of an identity between a Toeplitz determinant and a Fredholm determinant that has appeared in the random permutation context. Using this identity, we prove, in particular, convergence of moments for arbitrary rows of a random Young diagram under Plancherel measure.

Publication: Communications in Mathematical Physics Vol.: 223 No.: 3 ISSN: 0010-3616

ID: CaltechAUTHORS:20171003-074824665

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Abstract: We show that the maximum fidelity obtained by a positive partial transpose (p.p.t.) distillation protocol is given by the solution to a certain semidefinite program. This gives a number of new lower and upper bounds on p.p.t. distillable entanglement (and thus new upper bounds on 2-locally distillable entanglement). In the presence of symmetry, the semidefinite program simplifies considerably, becoming a linear program in the case of isotropic and Werner states. Using these techniques, we determine the p.p.t. distillable entanglement of asymmetric Werner states and "maximally correlated" states. We conclude with a discussion of possible applications of semidefinite programming to quantum codes and 1-local distillation.

Publication: IEEE Transactions on Information Theory Vol.: 47 No.: 7 ISSN: 0018-9448

ID: CaltechAUTHORS:20170925-141527816

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Abstract: The automorphism group of the Barnes-Wall lattice L_m in dimension 2^m(m ≠ 3) is a subgroup of index 2 in a certain “Clifford group” C_m of structure 2_+^(1+2m). O^+(2m,2). This group and its complex analogue X_m of structure (2^(1+2m)_+YZ_8). Sp(2m, 2) have arisen in recent years in connection with the construction of orthogonal spreads, Kerdock sets, packings in Grassmannian spaces, quantum codes, Siegel modular forms and spherical designs. In this paper we give a simpler proof of Runge@apos;s 1996 result that the space of invariants for C_m degree 2k is spanned by the complete weight enumerators of the codes C⊗F_(2m), where Cranges over all binary self-dual codes of length 2k; these are a basis if m ≥ k - 1. We also give new constructions for L_m and C_m: let M be the Z[√2]-lattice with Gram matrix [2 √2 √2 2}. Then L_m is the rational part of M^(⊗ m), and C_m = Aut(M^(⊗m)). Also, if C is a binary self-dual code not generated by vectors of weight 2, then C_m is precisely the automorphism group of the complete weight enumerator of C⊗F_(2m). There are analogues of all these results for the complex group X_m, with “doubly-even self-dual code” instead of “self-dual code.”

Publication: Designs, Codes and Cryptography Vol.: 24 No.: 1 ISSN: 0925-1022

ID: CaltechAUTHORS:20171106-144248894

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Abstract: We show that in the case of 2-dimensional lattices, Quebbemann's notion of modular and strongly modular lattices has a natural extension to the class group of a given discriminant, in terms of a certain set of translations. In particular, a 2-dimensional lattice has “extra” modularities essentially when it has order 4 in the class group. This allows us to determine the conditions on D under which there exists a strongly modular 2-dimensional lattice of discriminant D, as well as how many such lattices there are. The technique also applies to the question of when a lattice can be similar to its even sublattice.

Publication: Journal of Number Theory Vol.: 88 No.: 2 ISSN: 0022-314X

ID: CaltechAUTHORS:20171002-152216783

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Abstract: We present a number of results relating partial Cauchy-Littlewood sums, integrals over the compact classical groups, and increasing subsequences of permutations. These include: integral formulae for the distribution of the longest increasing subsequence of a random involution with constrained number of fixed points; new formulae for partial Cauchy-Littlewood sums, as well as new proofs of old formulae; relations of these expressions to orthogonal polynomials on the unit circle; and explicit bases for invariant spaces of the classical groups, together with appropriate generalizations of the straightening algorithm.

Publication: Duke Mathematical Journal Vol.: 109 No.: 1 ISSN: 0012-7094

ID: CaltechAUTHORS:20171004-073359453

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Abstract: We consider the following problem: When do alternate eigenvalues taken from a matrix ensemble themselves form a matrix ensemble? More precisely, we classify all weight functions for which alternate eigenvalues from the corresponding orthogonal ensemble form a symplectic ensemble, and similarly classify those weights for which alternate eigenvalues from a union of two orthogonal ensembles forms a unitary ensemble. Also considered are the k-point distributions for the decimated orthogonal ensembles.

No.: 40
ID: CaltechAUTHORS:20170926-090553813

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Abstract: Selecting N random points in a unit square corresponds to selecting a random permutation. By putting 5 types of symmetry restrictions on the points, we obtain subsets of permutations : involutions, signed permutations and signed involutions. We are interested in the statistics of the length of the longest up/right path of random points selections in each symmetry type as the number of points increases to infinity. The limiting distribution functions are expressed in terms of Painlevé II equation. Some of them are Tracy-Widom distributions in random matrix theory, while there are two new classes of distribution functions interpolating GOE and GSE, and GUE and GOE^2 Tracy-Widom distribution functions. Also some applications and related topics are discussed.

No.: 40
ID: CaltechAUTHORS:20170926-160331200

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Abstract: We compute the limiting distributions of the lengths of the longest monotone subsequences of random (signed) involutions with or without conditions on the number of fixed points (and negated points) as the sizes of the involutions tend to infinity. The resulting distributions are, depending on the number of fixed points, (1) the Tracy-Widom distributions for the largest eigenvalues of random GOE, GUE, GSE matrices, (2) the normal distribution, or (3) new classes of distributions which interpolate between pairs of the Tracy-Widom distributions. We also consider the second rows of the corresponding Young diagrams. In each case the convergence of moments is also shown. The proof is based on the algebraic work of J. Baik and E. Rains in [7] which establishes a connection between the statistics of random involutions and a family of orthogonal polynomials, and an asymptotic analysis of the orthogonal polynomials which is obtained by extending the Riemann-Hilbert analysis for the orthogonal polynomials by P. Deift, K. Johansson, and Baik in [3].

Publication: Duke Mathematical Journal Vol.: 109 No.: 2 ISSN: 0012-7094

ID: CaltechAUTHORS:20171106-152554249

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Abstract: The purpose of this paper is to investigate the limiting distribution functions for a polynuclear growth model with two external sources which was considered by Prähofer and Spohn. Depending on the strength of the sources, the limiting distribution functions are either the Tracy–Widom functions of random matrix theory or a new explicit function which has the special property that its mean is zero. Moreover, we obtain transition functions between pairs of the above distribution functions in suitably scaled limits. There are also similar results for a discrete totally asymmetric exclusion process.

Publication: Journal of Statistical Physics Vol.: 100 No.: 3-4 ISSN: 0022-4715

ID: CaltechAUTHORS:20171010-105909686

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Abstract: New 5-designs on 24 points were constructed recently by Harada by the consideration of Z_4-codes. We use Jacobi polynomials as a theoretical tool to explain their existence as resulting of properties of the symmetrized weight enumerator (swe) of the code. We introduce the notion of a colored t-design and we show that the words of any given Lee composition, in any of the 13 Lee-optimal self-dual codes of length 24 over Z_4, form a colored 5-design. New colored 3-designs on 16 points are also constructed in that way.

Publication: Journal of Statistical Planning and Inference Vol.: 86 No.: 2 ISSN: 0378-3758

ID: CaltechAUTHORS:20170927-153153707

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Abstract: New bounds are given for the minimal Hamming and Lee weights of self-dual codes over ℤ_4. For a self-dual code of length n, the Hamming weight is bounded above by 4[n/24]+f(n mod 24), for an explicitly given function f; the Lee weight is bounded above by 8[n/24]+g(n mod 24), for a different function g. These bounds appear to agree with the full linear programming bound for a wide range of lengths. The proof of these bounds relies on a reduction to a problem of binary codes, namely that of bounding the minimum dual distance of a doubly even binary code.

Publication: Finite Fields and Their Applications Vol.: 6 No.: 2 ISSN: 1071-5797

ID: CaltechAUTHORS:20171003-100542142

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Abstract: The weight enumerators (Shor and Laflamme 1997) of a quantum code are quite powerful tools for exploring its structure. As the weight enumerators are quadratic invariants of the code, this suggests the consideration of higher degree polynomial invariants. We show that the space of degree k invariants of a code of length n is spanned by a set of basic invariants in one-to-one correspondence with S^n_k. We then present a number of equations and inequalities in these invariants; in particular, we give a higher order generalization of the shadow enumerator of a code, and prove that its coefficients are nonnegative. We also prove that the quartic invariants of a ((4, 4, 2))_2 code are uniquely determined, an important step in a proof that any ((4, 4, 2))_2 code is additive (Rains 1999).

Publication: IEEE Transactions on Information Theory Vol.: 46 No.: 1 ISSN: 0018-9448

ID: CaltechAUTHORS:20170926-153330743

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Abstract: The expected value of L_n, the length of the longest increasing subsequence of a random permutation of {1, ... , n}, has been studied extensively. This paper presents the results of both Monte Carlo and exact computations that explore the finer structure of the distribution of L_n. The results suggested that several of the conjectures that had been made about L_n were incorrect, and led to new conjectures, some of which have been proved recently by Jinho Baik, Percy Deift, and Kurt Johansson. In particular, the standard deviation of L_n is of order n^(1/6), contrary to earlier conjectures. This paper also explains some regular patterns in the distribution of L_n.

No.: 251 ISSN: 0271-4132

ID: CaltechAUTHORS:20171009-160133131

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Abstract: Partial answers are given to two questions. When does a lattice Λ contain a sublattice Λ′ of index that is geometrically similar to Λ? When is the sublattice "clean", in the sense that the boundaries of the Voronoi cells for Λ' do not intersect Λ?

Publication: Canadian Journal of Mathematics Vol.: 51 No.: 6 ISSN: 0008-414X

ID: CaltechAUTHORS:20170926-102111766

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Abstract: The most powerful technique known at present for bounding the size of quantum codes of prescribed minimum distance is the quantum linear programming bound. Unlike the classical linear programming bound, it is not immediately obvious that if the quantum linear programming constraints are satisfiable for dimension K, then the constraints can be satisfied for all lower dimensions. We show that the quantum linear programming bound is monotonic in this sense, and give an explicitly monotonic reformulation.

Publication: IEEE Transactions on Information Theory Vol.: 45 No.: 7 ISSN: 0018-9448

ID: CaltechAUTHORS:20170926-133944519

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Abstract: In a previous paper, Shor and Laflamme (see Phys. Rev. Lett., vol.78, p.1600-02, 1997) define two "weight enumerators" for quantum error-correcting codes, connected by a MacWilliams (1977) transform, and use them to give a linear-programming bound for quantum codes. We extend their work by introducing another enumerator, based on the classical theory of shadow codes, that tightens their bounds significantly. In particular, nearly all of the codes known to be optimal among additive quantum codes (codes derived from orthogonal geometry) can be shown to be optimal among all quantum codes. We also use the shadow machinery to extend a bound on additive codes to general codes, obtaining as a consequence that any code of length, can correct at most [(n+1)/6] errors.

Publication: IEEE Transactions on Information Theory Vol.: 45 No.: 7 ISSN: 0018-9448

ID: CaltechAUTHORS:20170926-145724451

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Abstract: We present several results on quantum codes over general alphabets (that is, in which the fundamental units may have more than two states). In particular, we consider codes derived from finite symplectic geometry assumed to have additional global symmetries. From this standpoint, the analogs of Calderbank-Shor-Steane codes and of GF(4)-linear codes turn out to be special cases of the same construction. This allows us to construct families of quantum codes from certain codes over number fields; in particular, we get analogs of quadratic residue codes, including a single-error-correcting code encoding one letter in five, for any alphabet size. We also consider the problem of fault-tolerant computation through such codes, generalizing ideas of Gottesman (see Phys. Rev. A, vol.57, no.1, p127-37, 1998).

Publication: IEEE Transactions on Information Theory Vol.: 45 No.: 6 ISSN: 0018-9448

ID: CaltechAUTHORS:20170926-145137075

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Abstract: The optimal minimal Euclidean norm of self-dual codes over ℤ_4 is known through length 24; the purpose of the present note is to determine the optimal minimal Hamming and Lee weights in this range. In the process, we classify all Lee-optimal codes of length 18, 21, 23, and 24. In particular, we find a total of 13 inequivalent codes with the same symmetrized weight enumerator as the Hensel-lifted Golay code.

Publication: Discrete Mathematics Vol.: 203 No.: 1-3 ISSN: 0012-365X

ID: CaltechAUTHORS:20171107-074528830

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Abstract: By using totally isotropic subspaces in an orthogonal space Ω+ (2i, 2), several infinite families of packings of 2^k-dimensional subspaces of real 2i-dimensional space are constructed, some of which are shown to be optimal packings. A certain Clifford group underlies the construction and links this problem with Barnes-Wall lattices, Kerdock sets and quantum-error-correcting codes.

Publication: Journal of Algebraic Combinatorics Vol.: 9 No.: 2 ISSN: 0925-9899

ID: CaltechAUTHORS:20171003-094225780

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Abstract: We exhibit an orthogonal set of product states of two three-state particles that nevertheless cannot be reliably distinguished by a pair of separated observers ignorant of which of the states has been presented to them, even if the observers are allowed any sequence of local operations and classical communication between the separate observers. It is proved that there is a finite gap between the mutual information obtainable by a joint measurement on these states and a measurement in which only local actions are permitted. This result implies the existence of separable superoperators that cannot be implemented locally. A set of states are found involving three two-state particles that also appear to be nonmeasurable locally. These and other multipartite states are classified according to the entropy and entanglement costs of preparing and measuring them by local operations.

Publication: Physical Review A Vol.: 59 No.: 2 ISSN: 1050-2947

ID: CaltechAUTHORS:20170926-092107073

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Abstract: Cayley's 1875 enumerations of centered and bicentered alkanes (unlabeled trees of valency at most 4) are corrected and extended -- possibly for the first time in 124 years.

Publication: Journal of Integer Sequences Vol.: 2ISSN: 1530-7638

ID: CaltechAUTHORS:20170926-154926203

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Abstract: It is reasonable to expect the theory of quantum codes to be simplified in the case of codes of minimum distance 2; thus it makes sense to examine such codes in the hopes that techniques that prove effective there will generalize. With this in mind, we present a number of results on codes of minimum distance 2. We first compute the linear programming bound on the dimension of such a code, then show that this bound can only be attained when the code either is of even length, or is of length 3 or 5. We next consider questions of uniqueness, showing that the optimal code of length 2 or 1 is unique (implying that the well-known one-qubit-in-five single-error correcting code is unique), and presenting nonadditive optimal codes of all greater even lengths. Finally, we compute the full automorphism group of the more important distance 2 codes, allowing us to determine the full automorphism group of any GF(4)-linear code.

Publication: IEEE Transactions on Information Theory Vol.: 45 No.: 1 ISSN: 0018-9448

ID: CaltechAUTHORS:20170926-144218757

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Abstract: It is shown that an n-dimensional unimodular lattice has minimal norm at most 2[n/24]+2, unless n=23 when the bound must be increased by 1. This result was previously known only for even unimodular lattices. Quebbemann had extended the bound for even unimodular lattices to strongly N-modular even lattices for N in {1, 2, 3, 5, 6, 7, 11, 14, 15, 23}, (*) and analogous bounds are established here for odd lattices satisfying certain technical conditions (which are trivial for N=1 and 2). For N>1 in (*), lattices meeting the new bound are constructed that are analogous to the “shorter” and “odd” Leech lattices. These include an odd associate of the 16-dimensional Barnes–Wall lattice and shorter and odd associates of the Coxeter–Todd lattice. A uniform construction is given for the (even) analogues of the Leech lattice, inspired by the fact that (*) is also the set of square-free orders of elements of the Mathieu group M_(23).

Publication: Journal of Number Theory Vol.: 73 No.: 2 ISSN: 0022-314X

ID: CaltechAUTHORS:20171003-105923947

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Abstract: Using the machinery of zonal polynomials, we examine the limiting behavior of random symmetric matrices invariant under conjugation by orthogonal matrices as the dimension tends to infinity. In particular, we give sufficient conditions for the distribution of a fixed submatrix to tend to a normal distribution. We also consider the problem of when the sequence of partial sums of the diagonal elements tends to a Brownian motion. Using these results, we show that if O_n is a uniform random n×n orthogonal matrix, then for any fixed k>0, the sequence of partial sums of the diagonal of O^k_n tends to a Brownian motion as n→∞.

Publication: Probability Theory and Related Fields Vol.: 112 No.: 3 ISSN: 0178-8051

ID: CaltechAUTHORS:20171107-075613066

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Abstract: The problem of finding quantum error correcting codes is transformed into the problem of finding additive codes over the field GF(4) which are self-orthogonal with respect to a certain trace inner product. Many new codes and new bounds are presented, as well as a table of upper and lower bounds on such codes of length up to 30 qubits.

Publication: IEEE Transactions on Information Theory Vol.: 44 No.: 4 ISSN: 0018-9448

ID: CaltechAUTHORS:20170925-160358492

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Abstract: In a recent paper, Shor and Laflamme (see Phys. Rev. Lett., vol.78, p.1600-2, 1997) defined two "weight enumerators" for quantum error-correcting codes, connected by a MacWilliams transform, and used them to give a linear programming bound for quantum codes. We introduce two new enumerators which, while much less powerful at producing bounds, are useful tools nonetheless. The new enumerators are connected by a much simpler duality transform, clarifying the duality between Shor and Laflamme's enumerators. We also use the new enumerators to give a simpler condition for a quantum code to have specified minimum distance, and to extend the enumerator theory to codes with block size greater than 2.

Publication: IEEE Transactions on Information Theory Vol.: 44 No.: 4 ISSN: 0018-9448

ID: CaltechAUTHORS:20170925-154039545

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Abstract: (a) Suppose f(t) ∈ R[t] is a polynomial that maps rationals to rationals and irrationals to irrationals. Show that f(t) = at + b with a and b rational. (b) Does the same conclusion hold under the weaker assumption that f:R → R is an algebraic function (i.e., if there is a polynomial P(x,y) ∈ R[x,y] such that P(t,f(t)) is identically zero)?

Publication: American Mathematical Monthly Vol.: 105 No.: 3 ISSN: 0002-9890

ID: CaltechAUTHORS:20171120-110513579

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Abstract: Conway and Sloane (1990) have previously given an upper bound on the minimum distance of a singly-even self-dual binary code, using the concept of the shadow of a self-dual code. We improve their bound, finding that the minimum distance of a self-dual binary code of length n is at most 4[n/24]+4, except when n mod 24=22, when the bound is 4[n/24]+6. We also show that a code of length a multiple of 24 meeting the bound cannot be singly-even. The same technique gives similar results for additive codes over GF(4) (relevant to quantum coding theory).

Publication: IEEE Transactions on Information Theory Vol.: 44 No.: 1 ISSN: 0018-9448

ID: CaltechAUTHORS:20170927-074638028

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Abstract: We show that the moments of the trace of a random unitary matrix have combinatorial interpretations in terms of longest increasing subsequences of permutations. To be precise, we show that the 2n-th moment of the trace of a random k-dimensional unitary matrix is equal to the number of permutations of length n with no increasing subsequence of length greater than k. We then generalize this to other expectations over the unitary group, as well as expectations over the orthogonal and symplectic groups. In each case, the expectations count objects with restricted "increasing subsequence" length.

Publication: Electronic Journal of Combinatorics Vol.: 5ISSN: 1077-8926

ID: CaltechAUTHORS:20170925-154914546

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Abstract: An elementary construction using binary codes gives new record kissing numbers in dimensions from 32 to 128.

Publication: Electronic Journal of Combinatorics Vol.: 5ISSN: 1077-8926

ID: CaltechAUTHORS:20170925-155652163

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Abstract: We consider the probability distribution on a classical group G which naturally generalizes the normal distribution (the “heat kernel”), defined in terms of Brownian motions on G. As Brownian motion can be defined in terms of the Laplacian on G, much of this work involves the computation of the Laplacian. These results are then used to study the behavior of the normal distribution on U(n) as n↦∞. In addition, we show how the results on computing the Laplacian on the classical groups can be used to compute expectations with respect to Haar measure on those groups.

Publication: Journal of Theoretical Probability Vol.: 10 No.: 3 ISSN: 0894-9840

ID: CaltechAUTHORS:20171106-161236257

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Abstract: If a random unitary matrix U is raised to a sufficiently high power, its eigenvalues are exactly distributed as independent, uniform phases. We prove this result, and apply it to give exact asymptotics of the variance of the number of eigenvalues of U falling in a given arc, as the dimension of U tends to infinity. The independence result, it turns out, extends to arbitrary representations of arbitrary compact Lie groups. We state and prove this more general theorem, paying special attention to the compact classical groups and to wreath products. This paper is excerpted from the author's doctoral thesis, [9].

Publication: Probability Theory and Related Fields Vol.: 107 No.: 2 ISSN: 0178-8051

ID: CaltechAUTHORS:20171107-075059031

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Abstract: A group theoretic framework is introduced that simplifies the description of known quantum error-correcting codes and greatly facilitates the construction of new examples. Codes are given which map 3 qubits to 8 qubits correcting 1 error, 4 to 10 qubits correcting 1 error, 1 to 13 qubits correcting 2 errors, and 1 to 29 qubits correcting 5 errors.

Publication: Physical Review Letters Vol.: 78 No.: 3 ISSN: 0031-9007

ID: CaltechAUTHORS:20170926-101535529

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Abstract: No abstract

Publication: American Mathematical Monthly Vol.: 103 No.: 9 ISSN: 0002-9890

ID: CaltechAUTHORS:20180315-070628395

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Abstract: The authors present the finite-harmonic solution of the constraint equations of the spinor representation of the relativistic string. Choosing a gauge, they make a harmonic decomposition in the form of a product representation. This finite-harmonic approach is then compared with that of Hughston and Shaw (1988). They describe a recursive method for relating series and product parameters, and comment briefly on the question of a generalization for the infinite harmonic case and on the quantization of such systems.

Publication: Classical and Quantum Gravity Vol.: 8 No.: 7 ISSN: 0264-9381

ID: CaltechAUTHORS:20180320-123640344

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