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: 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: 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 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: 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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