Related papers: Random Subwords and Billiard Walks in Affine Weyl …

We introduce a natural way to construct a random subset of a homogeneous Cantor set $C$ in $[0,1]$ via random labelings of an infinite $M$-ary tree, where $M\geq 2$. The Cantor set $C$ is the attractor of an equicontractive iterated…

Let G be a semisimple algebraic group over an algebraically closed field of characteristic p>0, and let g be its Lie algebra. The crucial Lie algebra representations to understand are those associated with the reduced enveloping algebra…

Let g be a simple Lie algebra, with fixed Borel subalgebra b and with Weyl group W. Expanding on previous work of Fan and Stembridge in the simply laced case, this note aims to study the fully commutative elements of W, and their…

Let $\CaC\subset \Q^p$ be a rational cone. An affine semigroup $S\subset \CaC$ is a $\CaC$-semigroup whenever $(\CaC\setminus S)\cap \N^p$ has only a finite number of elements. In this work, we study the tree of $\CaC$-semigroups, give a…

We extend properties of the weak order on finite Coxeter groups to Weyl groupoids admitting a finite root system. In particular, we determine the topological structure of intervals with respect to weak order, and show that the set of…

Two notions of riffle shuffling on finite Coxeter groups are given: one using Solomon's descent algebra and another using random walk on chambers of hyperplane arrangements. These coincide for types $A$,$B$,$C$, $H_3$, and rank two groups.…

Let $S=\langle d_1,\dots,d_m\rangle$ be a numerical semigroup and $k[S]$ its semigroup ring. The Hilbert numerator of $k[S]$ determines normalized alternating syzygy power sums $K_p(S)$ encoding alternating power sums of syzygy degrees. Fel…

Let $W_0$ be a reflection subgroup of a finite complex reflection group $W$, and let $B_0$ and $B$ be their respective braid groups. In order to construct a Hecke algebra $\widetilde{H}_0$ for the normalizer $N_W(W_0)$, one first considers…

Let $\mathbb{P}$ denote the set of primes and $\mathcal{N}\subset \mathbb{N}$ be a set with arbitrary weights attached to its elements. Set $\mathfrak{p}_{\mathcal{N}}(n)$ to be the restricted partition function which counts partitions of…

First and second fundamental theorems are given for polynomial invariants of a class of pseudo-reflection groups (including the Weyl groups of type $B_n$), under the assumption that the order of the group is invertible in the base field.…

For affine special linear superalgebra $\widehat{sl}(m|n, \Pi)$ defined by an arbitrary system of simple roots $\Pi$ we define the affine super Yangian $Y_{\hbar}(\widehat{sl}(m|n, \Pi))$ as Hopf superalgebra which is a quantization of…

We study a deliberately simple, fully non-linguistic model of text: a sequence of independent draws from a finite alphabet of letters plus a single space symbol. A word is defined as a maximal block of non-space symbols. Within this…

Given $p \in (0,1)$, we let $Q_p= Q_p^d$ be the random subgraph of the $d$-dimensional hypercube $Q^d$ where edges are present independently with probability $p$. It is well known that, as $d \rightarrow \infty$, if $p>\frac12$ then with…

Let $\mathbb{F}_q[t]$ be the ring of polynomials over $\mathbb{F}_q$, the finite field of $q$ elements, and let $p$ be the characteristic of $\mathbb{F}_q$. We denote $\widetilde{G}_q(k)$ to be the least integer $t_0$ with the property that…

We investigate particle transport in the honeycomb billiard that consists of connected channels placed on the edges of a honeycomb structure. The spreading of particles is superdiffusive due to the existence of ballistic trajectories which…

We describe an approach, via Malle's permutation $\Psi$ on the set of irreducible characters $\text{Irr}(W)$, that gives a uniform derivation of the Chapuy-Stump formula for the enumeration of reflection factorizations of the Coxeter…

In a previous paper, the authors studied the radical filtration of a Weyl module $\Delta_\zeta(\lambda)$ for quantum enveloping algebras $U_\zeta(\overset\circ{\mathfrak g})$ associated to a finite dimensional complex semisimple Lie algebra…

Using maximal isotropic submodules in a quadratic module over Z_p, we prove the existence of a natural discrete probability distribution on the set of isomorphism classes of short exact sequences of co-finite type Z_p-modules, and then…

We consider the semisimple orbits of a Vinberg $\theta$-representation. First we take the complex numbers as base field. By a case by case analysis we show a technical result stating the equality of two sets of hyperplanes, one…

Given a finite-dimensional complex Lie algebra g equipped with a nondegenerate, symmetric, invariant bilinear form B, let V_k(g,B) denote the universal affine vertex algebra associated to g and B at level k. For any reductive group G of…