Related papers: Sylvester Waves in the Coxeter Groups

We consider the uniqueness of solution (i.e., nonsingularity) of systems of $r$ generalized Sylvester and $\star$-Sylvester equations with $n\times n$ coefficients. After several reductions, we show that it is sufficient to analyze periodic…

We extend the classical Schur-Weyl duality between representations of the groups $SL(n,\C)$ and $\sN$ to the case of $SL(n,\C)$ and the infinite symmetric group $\sinf$. Our construction is based on a "dynamic," or inductive, scheme of…

Let W be a Coxeter group and L be a weight function on W. Following Lusztig, we have a corresponding decomposition of W into left cells, which have important applications in representation theory. We study the case where $W$ is an affine…

We show that all groups in a very large class of Coxeter groups are locally quasiconvex and have uniform membership problem solvable in quadratic time. If a group in the class satisfies a further hypothesis it is subgroup separable and…

We give Euler-like recursive formulas for the $t$-colored partition function when $t=2$ or $t=3,$ as well as for all $t$-regular partition functions. In particular, we derive an infinite family of ``triangular number" recurrences for the…

We prove maximal Schauder regularity for solutions to elliptic systems and Cauchy problems, in the space $C_b(\mathbb{R}^d;\mathbb{R}^m)$ of bounded and continuous functions, associated to a class of nonautonomous weakly coupled…

We study the restriction of the absolute order on a Coxeter group $W$ to an interval $[1,w]_T$, where $w\in W$ is an involution. We characterize and classify those involutions $w$ for which $[1,w]_T$ is a lattice, using the notion of…

Given a Coxeter system $(W,S)$ and a positive real multiparameter $\bq$, we study the "weighted $L^2$-cohomology groups," of a certain simplicial complex $\Sigma$ associated to $(W,S)$. These cohomology groups are Hilbert spaces, as well as…

In this paper, we give a class of reflection rigid Coxeter systems. Let $(W,S)$ be a Coxeter system. Suppose that (1) for each $s,t\in S$ such that $m(s,t)$ is odd, $\{s,t\}$ is a maximal spherical subset of $S$, (2) there does not exist a…

We compute the number of irreducible linear representations of self-similar branch groups, by expressing these numbers as the co\"efficients a_n of a Dirichlet series sum a_n n^{-s}. We show that this Dirichlet series has a positive…

We identify vertex operator algebras (VOAs) of a class of Argyres-Douglas (AD) matters with two types of non-abelian flavor symmetries. They are the $W$ algebra defined using nilpotent orbit with partition $[q^m,1^s]$. Gauging above AD…

We consider a certain left action by the monoid $SL_2(\mathbf{N}_0)$ on the set of divisor pairs $\mathcal{D}_f := \{ (m, n) \in \mathbf{N}_0 \times \mathbf{N}_0 : m \lvert f(n) \}$ where $f \in \mathbf{Z}[x]$ is a polynomial with integer…

Using techniques from integrable systems, we obtain a number of exact results for random partitions. In particular, we prove a simple formula for correlation functions of what we call the Schur measure on partitions (which is a far reaching…

In a recent paper we claimed that both the group algebra of a finite Coxeter group $W$ as well as the Orlik-Solomon algebra of $W$ can be decomposed into a sum of induced one-dimensional representations of centralizers, one for each…

For a given sequence $\mathbf{\alpha} = [\alpha_1,\alpha_2,\dots,\alpha_{N+1}]$ of $N+1$ positive integers, we consider the combinatorial function $E(\mathbf{\alpha})(t)$ that counts the nonnegative integer solutions of the equation…

We revisit second-order-in-time space-time discretizations of the linear and semilinear wave equations by establishing precise equivalences with first-order-in-time formulations. Focusing on schemes using continuous piecewise-polynomial…

Let $(W,S)$ be a Coxeter system, let $\varphi$ be a weight function on $S$ and let ${\mathrm{Cact}}\_W$ denote the associated {\it cactus group}. Following an idea of I. Losev, we construct an action of ${\mathrm{Cact}}\_W \times…

We consider random normal matrix and planar symplectic ensembles, which can be interpreted as two-dimensional Coulomb gases having determinantal and Pfaffian structures, respectively. For general radially symmetric potentials, we derive the…

We introduce two entire functions $f_{A_{1/2\infty}}$and $f_{D_{1/2\infty}}$ in two variables. Both of them have only two critical values 0 and 1, and the associated maps $\C^2 to \C$ define topologically locally trivial fibrations over…

The Sylvester's denumerant \( d(t; \boldsymbol{a}) \) is a quantity that counts the number of nonnegative integer solutions to the equation \( \sum_{i=1}^{N} a_i x_i = t \), where \( \boldsymbol{a} = (a_1, \dots, a_N) \) is a sequence of…