76 research outputs found
Multivariable Christoffel-Darboux Kernels and Characteristic Polynomials of Random Hermitian Matrices
We study multivariable Christoffel-Darboux kernels, which may be viewed as
reproducing kernels for antisymmetric orthogonal polynomials, and also as
correlation functions for products of characteristic polynomials of random
Hermitian matrices. Using their interpretation as reproducing kernels, we
obtain simple proofs of Pfaffian and determinant formulas, as well as Schur
polynomial expansions, for such kernels. In subsequent work, these results are
applied in combinatorics (enumeration of marked shifted tableaux) and number
theory (representation of integers as sums of squares).Comment: This is a contribution to the Vadim Kuznetsov Memorial Issue on
Integrable Systems and Related Topics, published in SIGMA (Symmetry,
Integrability and Geometry: Methods and Applications) at
http://www.emis.de/journals/SIGMA
Applications of minor summation formula III, Plücker relations, lattice paths and Pfaffian identities
Kyushu University 21st Century COE Program Development of Dynamic Mathematics with High Functionality九州大学21世紀COEプログラム「機能数理学の構築と展開」The initial purpose of the present paper is to provide a combinatorial proof of the minor summation formula of Pfaffians in [Ishikawa, Wakayama, Minor summation formula of Pfaffians, Linear and Multilinear Algebra 39 (1995) 285–305] based on the lattice path method. The second aim is to study applications of the minor summation formula for obtaining several identities. Especially, a simple proof of Kawanaka\u27s formula concerning a q-series identity involving the Schur functions [Kawanaka, A q-series identity involving Schur functions and related topics, Osaka J. Math. 36 (1999) 157–176] and of the identity in [Kawanaka, A q-Cauchy identity involving Schur functions and imprimitive complex reflection groups, Osaka J. Math. 38 (2001) 775–810] which is regarded as a determinant version of the previous one are given
Generalizations of Cauchy's Determinant and Schur's Pfaffian
We present several generalizations of Cauchy's determinant and Schur's
Pfaffian by considering matrices whose entries involve some generalized
Vandermonde determinants. Special cases of our formulae include previuos
formulae due to S.Okada and T. Sundquist. As an application, we give a relation
for the Littlewood--Richardson coefficients involving a rectangular partition.Comment: 26 page
Transition between characters of classical groups, decomposition of Gelfand-Tsetlin patterns and last passage percolation
We study the combinatorial structure of the irreducible characters of the
classical groups , ,
, and the
"non-classical" odd symplectic group , finding new
connections to the probabilistic model of Last Passage Percolation (LPP).
Perturbing the expressions of these characters as generating functions of
Gelfand-Tsetlin patterns, we produce two families of symmetric polynomials that
interpolate between characters of and and between characters of
and . We identify the first family as a
one-parameter specialization of Koornwinder polynomials, for which we thus
provide a novel combinatorial structure; on the other hand, the second family
appears to be new. We next develop a method of Gelfand-Tsetlin pattern
decomposition to establish identities between all these polynomials that, in
the case of characters, can be viewed as describing the decomposition of
irreducible representations of the groups when restricted to certain subgroups.
Through these formulas we connect orthogonal and symplectic characters, and
more generally the interpolating polynomials, to LPP models with various
symmetries, thus going beyond the link with classical Schur polynomials
originally found by Baik and Rains [BR01a]. Taking the scaling limit of the LPP
models, we finally provide an explanation of why the Tracy-Widom GOE and GSE
distributions from random matrix theory admit formulations in terms of both
Fredholm determinants and Fredholm Pfaffians.Comment: 60 pages, 11 figures. Typos corrected and a few remarks adde
Rogers-Szego polynomials and Hall-Littlewood symmetric functions
We use Rogers-Szego polynomials to unify some well-known identities for
Hall-Littlewood symmetric functions due to Macdonald and Kawanaka.Comment: 18 page
- …