15 research outputs found
Path counting and random matrix theory
We establish three identities involving Dyck paths and alternating Motzkin
paths, whose proofs are based on variants of the same bijection. We interpret
these identities in terms of closed random walks on the halfline. We explain
how these identities arise from combinatorial interpretations of certain
properties of the -Hermite and -Laguerre ensembles of random
matrix theory. We conclude by presenting two other identities obtained in the
same way, for which finding combinatorial proofs is an open problem.Comment: 14 pages, 13 figures and diagrams; submitted to the Electronic
Journal of Combinatoric
A polynomiality property for Littlewood-Richardson coefficients
We present a polynomiality property of the Littlewood-Richardson coefficients
c_{\lambda\mu}^{\nu}. The coefficients are shown to be given by polynomials in
\lambda, \mu and \nu on the cones of the chamber complex of a vector partition
function. We give bounds on the degree of the polynomials depending on the
maximum allowed number of parts of the partitions \lambda, \mu and \nu. We
first express the Littlewood-Richardson coefficients as a vector partition
function. We then define a hyperplane arrangement from Steinberg's formula,
over whose regions the Littlewood-Richardson coefficients are given by
polynomials, and relate this arrangement to the chamber complex of the
partition function. As an easy consequence, we get a new proof of the fact that
c_{N\lambda N\mu}^{N\nu} is given by a polynomial in N, which partially
establishes the conjecture of King, Tollu and Toumazet that c_{N\lambda
N\mu}^{N\nu} is a polynomial in N with nonnegative rational coefficients.Comment: 14 page
A vector partition function for the multiplicities of sl_k(C)
We use Gelfand-Tsetlin diagrams to write down the weight multiplicity
function for the Lie algebra sl_k(C) (type A_{k-1}) as a single partition
function. This allows us to apply known results about partition functions to
derive interesting properties of the weight diagrams. We relate this
description to that of the Duistermaat-Heckman measure from symplectic
geometry, which gives a large-scale limit way to look at multiplicity diagrams.
We also provide an explanation for why the weight polynomials in the boundary
regions of the weight diagrams exhibit a number of linear factors. Using
symplectic geometry, we prove that the partition of the permutahedron into
domains of polynomiality of the Duistermaat-Heckman function is the same as
that for the weight multiplicity function, and give an elementary proof of this
for sl_4(C) (A_3).Comment: 34 pages, 11 figures and diagrams; submitted to Journal of Algebr
On positivity of Ehrhart polynomials
Ehrhart discovered that the function that counts the number of lattice points
in dilations of an integral polytope is a polynomial. We call the coefficients
of this polynomial Ehrhart coefficients, and say a polytope is Ehrhart positive
if all Ehrhart coefficients are positive (which is not true for all integral
polytopes). The main purpose of this article is to survey interesting families
of polytopes that are known to be Ehrhart positive and discuss the reasons from
which their Ehrhart positivity follows. We also include examples of polytopes
that have negative Ehrhart coefficients and polytopes that are conjectured to
be Ehrhart positive, as well as pose a few relevant questions.Comment: 40 pages, 7 figures. To appear in in Recent Trends in Algebraic
Combinatorics, a volume of the Association for Women in Mathematics Series,
Springer International Publishin
Signature quantization and representations of compact Lie groups
We discuss some applications of signature quantization to the representation theory of compact Lie groups. In particular, we prove signature analogues of the Kostant formula for weight multiplicities and the Steinberg formula for tensor product multiplicities. Using symmetric functions, we also find, for type A, analogues of the Weyl branching rule and the Gel'fand–Tsetlin theorem