Related papers: On the averages of characteristic polynomials from…
We discuss a conjecture which says that the automorphism group of the Weyl algebra in characteristic zero is canonically isomorphic to the automorphism group of the corresponding Poisson algebra of classical polynomial symbols. Several…
In this chapter are given necessary and sufficient conditions for the regularity of solutions of the functional equation appearing in the theory of classical orthogonal polynomials. In addition, we also present the functional Rodrigues…
The theory of polynomials orthogonal with respect to one inner product is classical. We discuss the extension of this theory to multiple inner products. Examples include the Lam\'e and Heine-Stieltjes polynomials.
One interesting combinatorial feature of classical determinantal varieties is that the character of their coordinate rings give a natural truncation of the Cauchy identity in the theory of symmetric functions. Natural generalizations of…
With a main tool is signed graphs, we give a full description of the characteristic quasi-polynomials of ideals of classical root systems ($ABCD$) with respect to the integer and root lattices. As a result, we obtain a full description of…
Character polynomials are used to study the restriction of a polynomial representation of a general linear group to its subgroup of permutation matrices. A simple formula is obtained for computing inner products of class functions given by…
We characterize characteristic polynomials of elements in a central simple algebra. We also give an account for the theory of rational canonical forms for separable linear transformations over a central division algebra, and a description…
By using combinatorics, we give a new proof for the recurrence relations of the characteristic polynomial coefficients, and then we obtain an explicit expression for the generic term of the coefficient sequence, which yields the trace…
A theorem due to Tokuyama expresses Schur polynomials in terms of Gelfand-Tsetlin patterns, providing a deformation of the Weyl character formula and two other classical results, Stanley's formula for the Schur $q$-polynomials and Gelfand's…
We study the averages of ratios of characteristic polynomials over circular $\beta$-ensembles, where $\beta$ is a positive real number. Using Jack polynomial theory, we obtain three expressions for ratio averages. Two of them are given as…
We consider discretisations of the Macdonald--Mehta integrals from the theory of finite reflection groups. For the classical groups, $\mathrm{A}_{r-1}$, $\mathrm{B}_r$ and $\mathrm{D}_r$, we provide closed-form evaluations in those cases…
We define and study odd and even analogues of the major index statistics for the classical Weyl groups. More precisely, we show that the generating functions of these statistics, twisted by the one-dimensional characters of the…
In this (partly expository) paper we show, using ideas from the theory of total positivity, how a number of properties of a semisimple group over the complex numbers can be presented purely in terms of the Weyl group. We also describe some…
We prove a family of identities, expressing generating functions of powers of characteristic polynomials of permutations, as finite or infinite products. These generalize formulae first obtained in a study of the geometry/topology of…
The decomposition of representations of compact classical Lie groups into representations of finite subgroups is discussed. A Mathematica package is presented that can be used to compute these branching rules using the Weyl character…
We introduce the definition of the typical irreducible modules of the generalized quantum groups, and prove the Weyl-Kac-type formulas of their characters. As a by-product, we obtain the Weyl-Kac-type character formulas of the typical…
The skew Schubert polynomials are those which are indexed by skew elements of the Weyl group, in the sense of arXiv:0812.0639. We obtain tableau formulas for the double versions of these polynomials in all four classical Lie types, where…
A Littlewood polynomial is a single-variable polynomial all of whose coefficients lie in $\{ \pm 1\}$. We establish the leading term asymptotics of the number of reciprocal or skew-reciprocal Littlewood polynomials with square discriminant.…
We revisit factorizations of classical characters under various specializations, some old and some new. We first show that all characters of classical families of groups twisted by odd powers of an even primitive root of unity factorize…
We give formulae relating the value of an irreducible character of a classical group at a matrix to entries of powers of the matrix. This yields a far-reaching generalization of a result of J. L. Cisneros-Molina concerning the $GL_2$ case.