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Some representation-theoretic multiplicities, such as the Kostka and the Littlewood-Richardson coefficients, admit a combinatorial interpretation that places their computation in the complexity class #P. Whether this holds more generally is…
This paper presents a noncommutative theory of symmetric functions, based on the notion of quasi-determinant. We begin with a formal theory, corresponding to the case of symmetric functions in an infinite number of independent variables.…
We study the set of square-free parts of volume polynomials associated with four planar lattice polytopes. This is motivated by the problem of describing possible pairwise intersection numbers of four curves in $(\mathbb{C}^*)^2$ with…
We consider orthogonal polynomials on the unit circle with respect to a weight which is a quotient of $q$-gamma functions. We show that the Verblunsky coefficients of these polynomials satisfy discrete Painlev\'e equations, in a Lax form,…
We discuss three distinct topics of independent interest; one in enumerative combinatorics, one in symmetric function theory, and one in algebraic geometry. The topic in enumerative combinatorics concerns a q-analog of a generalization of…
A fundamental open problem in algebraic combinatorics is to find a positive combinatorial formula for Kronecker coefficients, which are multiplicities of the decomposition of the tensor product of two \S_r-irreducibles into irreducibles.…
In [J. Phys. A: Math. Theor. 45 (2012)], while looking for spin chains that admit perfect state transfer, Vinet and Zhedanov found an apparently new sequence of orthogonal polynomials, that they called para-Krawtchouk polynomials, defined…
Laurent polynomials related to the Hahn-Exton $q$-Bessel function, which are $q$-analogues of the Lommel polynomials, have been introduced by Koelink and Swarttouw. The explicit strong moment functional with respect to which the Laurent…
Let V be a vector bundle on a scheme X endowed with a nondegenerate symplectic or orthogonal form. Let G be a Grassmannian bundle parametrizing maximal isotropic subbundles of V. The main goal of the paper is to give formulas for the…
We study the module $C^r(\mathcal{P})$ of piecewise polynomial functions of smoothness $r$ on a pure $n$-dimensional polytopal complex $\mathcal{P}\subset\mathbb{R}^n$, via an analysis of certain subcomplexes $\mathcal{P}_W$ obtained from…
We explore the applications of Lorentzian polynomials to the fields of algebraic geometry, analytic geometry and convex geometry. In particular, we establish a series of intersection theoretic inequalities, which we call rKT property, with…
Let $\mathcal{P} \subseteq \mathbb{R}^{n}$ be a polytope whose vertices have rational coordinates. By a seminal result of E. Ehrhart, the number of integer lattice points in the $k$th dilate of $\mathcal{P}$ ($k$ a positive integer) is a…
New bispectral polynomials orthogonal on a quadratic bi-lattice are obtained from a truncation of Wilson polynomials. Recurrence relation and difference equation are provided. The recurrence coefficients can be encoded in a perturbed…
We prove the quasimodularity of generating functions for counting pillowcase covers, with and without Siegel-Veech weight. Similar to prior work on torus covers, the proof is based on analyzing decompositions of half-translation surfaces…
We introduce the notion of "quasi-symmetric" polynomials, which is a generalization of the notion of symmetry, and is particularly suited to the setting of polynomial rings over finite fields. The properties of this new class of functions…
We present a study of three families of Kronecker coefficients, which we describe in terms of reduced Kronecker coefficients. This study is grounded on the generating function of the coefficients, proved by a bijection between two…
Several results concerning pairs of polynomially convex sets whose union is not even rationally convex are given. It is shown that there is no restriction on how two spaces can be embedded in some $\C^N$ so as to be polynomially convex but…
E. Bayer-Fluckiger gave a necessary and sufficient condition for a polynomial to be realized as the characteristic polynomial of a semisimple isometry of an even unimodular lattice, by describing the local-global obstruction, and the author…
A common tool in the theory of numerical semigroups is to interpret a desired class of semigroups as the integer lattice points in a rational polyhedron in order to leverage computational and enumerative techniques from polyhedral geometry.…
In this paper we discuss a couple of observations related to polynomial convexity. More precisely, (i) We observe that the union of finitely many disjoint closed balls with centres in $\cup_{\theta\in[0,\pi/2]}e^{i\theta}V$ is polynomially…