arXiv subjects

Sho Matsumoto

Publications and source records attributed to Sho Matsumoto.

50 records · Page 3Linked to original sources

arXiv2012

Jucys-Murphy Elements and Unitary Matrix Integrals

In this paper, we study the relationship between polynomial integrals on the unitary group and the conjugacy class expansion of symmetric functions in Jucys-Murphy elements. Our main result is an explicit formula for the top coefficients in the class expansion of monomial symmetric functions in Jucys-Murphy elements, from which we recover the first order asymptotics of polynomial integrals over $\U(N)$ as $N \rightarrow \infty$. Our results on class expansion include an analogue of Macdonald's result for the top connection coefficients of the class algebra, a generalization of Stanley and Olshanski's result on the polynomiality of content statistics on Plancherel-random partitions, and an exact formula for the multiplicity of the class of full cycles in the expansion of a complete symmetric function in Jucys-Murphy elements. The latter leads to a new combinatorial interpretation of the Carlitz-Riordan central factorial numbers.

math.CO↗

arXiv2011

General moments of matrix elements from circular orthogonal ensembles

The aim of this paper is to present a systematic method for computing moments of matrix elements taken from circular orthogonal ensembles (COE). The formula is given as a sum of Weingarten functions for orthogonal groups but the technique for its proof involves Weingarten calculus for unitary groups. As an application, explicit expressions for the moments of a single matrix element of a COE matrix are given.

math.PR↗

arXiv2011

Jucys-Murphy elements, orthogonal matrix integrals, and Jack measures

We study symmetric polynomials whose variables are odd-numbered Jucys-Murphy elements. They define elements of the Hecke algebra associated to the Gelfand pair of the symmetric group with the hyperoctahedral group. We evaluate their expansions in zonal spherical functions and in double coset sums. These evaluations are related to integrals of polynomial functions over orthogonal groups. Furthermore, we give an extension of them, based on Jack polynomials.

math.CO↗

arXiv2011

General moments of the inverse real Wishart distribution and orthogonal Weingarten functions

Let $W$ be a random positive definite symmetric matrix distributed according to a real Wishart distribution and let $W^{-1}=(W^{ij})_{i,j}$ be its inverse matrix. We compute general moments $\mathbb{E} [W^{k_1 k_2} W^{k_3 k_4} ... W^{k_{2n-1}k_{2n}}]$ explicitly. To do so, we employ the orthogonal Weingarten function, which was recently introduced in the study for Haar-distributed orthogonal matrices. As applications, we give formulas for moments of traces of a Wishart matrix and its inverse.

math.ST↗

arXiv2010

Primitive factorizations, Jucys-Murphy elements, and matrix models

A factorization of a permutation into transpositions is called "primitive" if its factors are weakly ordered. We discuss the problem of enumerating primitive factorizations of permutations, and its place in the hierarchy of previously studied factorization problems. Several formulas enumerating minimal primitive and possibly non-minimal primitive factorizations are presented, and interesting connections with Jucys-Murphy elements, symmetric group characters, and matrix models are described.

math.CO↗

arXiv2009

On some properties of orthogonal Weingarten functions

We give a Fourier-type formula for computing the orthogonal Weingarten formula. The Weingarten calculus was introduced as a systematic method to compute integrals of polynomials with respect to Haar measure over classical groups. Although a Fourier-type formula was known in the unitary case, the orthogonal counterpart was not known. It relies on the Jack polynomial generalization of both Schur and zonal polynomials. This formula substantially reduces the complexity involved in the computation of Weingarten formulas. We also describe a few more new properties of the Weingarten formula, state a conjecture and give a table of values.

math-ph↗

arXiv2008

Jack deformations of Plancherel measures and traceless Gaussian random matrices

We study random partitions $λ=(λ_1,λ_2,...,λ_d)$ of $n$ whose length is not bigger than a fixed number $d$. Suppose a random partition $λ$ is distributed according to the Jack measure, which is a deformation of the Plancherel measure with a positive parameter $α>0$. We prove that for all $α>0$, in the limit as $n \to \infty$, the joint distribution of scaled $λ_1,..., λ_d$ converges to the joint distribution of some random variables from a traceless Gaussian $β$-ensemble with $β=2/α$. We also give a short proof of Regev's asymptotic theorem for the sum of $β$-powers of $f^λ$, the number of standard tableaux of shape $λ$.

math.CO↗

arXiv2008

Alpha-determinant cyclic modules and Jacobi polynomials

We study the cyclic $U(\mathfrak{gl}_n)$-module generated by the $l$-th power of the $α$-determinant. When $l$ is a non-negative integer, for all but finite exceptional values of $alpha$, one shows that this cyclic module is isomorphic to the $n$-th tensor space $(S^l(\mathbb{C}^n))^{\otimes n}$ of the symmetric $l$-th tensor space of $\mathbb{C}^n$. If $alpha$ is exceptional, then the structure of the module changes drastically, i.e. some irreducible representations which are the irreducible components of the decomposition of $(S^l(\mathbb{C}^n))^{\otimes n}$ disappear in the decomposition of the cyclic module. The degeneration of each isotypic component of the cyclic module is described by a matrix whose size is given by a Kostka number and entries are polynomials in $alpha$ with rational coefficients. As a special case, we determine the matrix in a full of the detail for the case where $n=2$; the matrix becomes a scalar and is essentially given by the classical Jacobi polynomial. Moreover, we prove that these polynomials are unitary.

math.RT↗

arXiv2007

Hyperdeterminantal expressions for Jack functions of rectangular shapes

We derive a Jacobi-Trudi type formula for Jack functions of rectangular shapes. In this formula, we make use of a hyperdeterminant, which is Cayley's simple generalization of the determinant. In addition, after developing the general theory of hyperdeterminants, we give summation formulae for Schur functions involving hyperdeterminants, and evaluate Toeplitz-type hyperdeterminants by using Jack function theory.

math.CO↗

arXiv2007

Moments of characteristic polynomials for compact symmetric spaces and Jack polynomials

We express the averages of products of characteristic polynomials for random matrix ensembles associated with compact symmetric spaces in terms of Jack polynomials or Heckman and Opdam's Jacobi polynomials depending on the root system of the space. We also give explicit expressions for the asymptotic behavior of these averages in the limit as the matrix size goes to infinity.

math.PR↗

arXiv2005

Alpha-determinant cyclic modules of $gl_n(C)$

The alpha-determinant unifies and interpolates the notion of the determinant and permanent. We determine the irreducible decomposition of the cyclic module of $gl_n(C)$ defined by the alpha-determinant. The degeneracy of the irreducible decomposition is determined by the content polynomial of a given partition.

math.RT↗

arXiv2005

Alpha-Pfaffian, pfaffian point process and shifted Schur measure

For any complex number $α$ and any even-size skew-symmetric matrix $B$, we define a generalization $\pfaα(B)$ of the pfaffian $\pf(B)$ which we call the $α$-pfaffian. The $α$-pfaffian is a pfaffian analogue of the $α$-determinant. It gives the pfaffian at $α=-1$. We give some formulas for $α$-pfaffians and study the positivity. Further we define point processes determined by the $α$-pfaffian. Also we provide a linear algebraic proof of the explicit pfaffian expression for the correlation function of the shifted Schur measure.

math.CO↗

arXiv2004

Correlation functions of the shifted Schur measure

The shifted Schur measure introduced by Tracy and Widom is a measure on the set of all strict partitions, which is defined by Schur $Q$-functions. The main aim of this paper is to calculate the correlation function of this measure, which is given by a pfaffian. As an application, we prove that a limit distribution of $λ_j$'s with respect to a shifted version of the Plancherel measure for symmetric groups is identical with the corresponding distribution of the original Plancherel measure. Further we give expressions of the mean value and the variance of the size of a partition with respect to the measure defined by Hall-Littlewood functions.

math.CO↗

arXiv2003

A Scaling Limit for t-Schur Measures

We introduce a generalization of the Schur measure. And we give a combinatorial interpretation to this generalized measure and prove that a scaling limit for this measure is given by the Tracy-Widom distribution.

math.PR↗