Related papers: Hurwitz numbers for reflection groups II: Paraboli…

The integral group rings $\mathbb{Z}G$ for finite groups $G$ are precisely those fusion rings whose basis elements have Frobenius-Perron dimension 1, and each is categorifiable in the sense that it arises as the Grothendieck ring of a…

An element of a Coxeter group $W$ is called fully commutative if any two of its reduced decompositions can be related by a series of transpositions of adjacent commuting generators. In the preprint "Fully commutative elements in finite and…

We introduce Coxeter-sortable elements of a Coxeter group W. For finite W, we give bijective proofs that Coxeter-sortable elements are equinumerous with clusters and with noncrossing partitions. We characterize Coxeter-sortable elements in…

We investigate representations of Coxeter groups into $\mathrm{GL}(n,\mathbb{R})$ as geometric reflection groups which are convex cocompact in the projective space $\mathbb{P}(\mathbb{R}^n)$. We characterize which Coxeter groups admit such…

Hurwitz numbers count branched covers of the Riemann sphere with specified ramification, or equivalently, transitive permutation factorizations in the symmetric group with specified cycle types. Monotone Hurwitz numbers count a restricted…

Let G be a reductive group over a non-archimedean local field F. Consider an arbitrary Bernstein block Rep(G)^s in the category of complex smooth G-representations. In earlier work the author showed that there exists an affine Hecke algebra…

Let $q$ be a positive integer. In our recent paper, we proved that the cardinality of the complement of an integral arrangement, after the modulo $q$ reduction, is a quasi-polynomial of $q$, which we call the characteristic…

We construct finite volume hyperbolic manifolds with large symmetry groups. The construction makes use of the presentations of finite Coxeter groups provided by Barot and Marsh and involves mutations of quivers and diagrams defined in the…

Fully commutative elements in types $B$ and $D$ are completely characterized and counted by Stembridge. Recently, Feinberg-Kim-Lee-Oh have extended the study of fully commutative elements from Coxeter groups to the complex setting, giving…

In this paper, we define and study a kind of Steinberg representation for linear algebraic groups of a particular kind, called groups of parahoric type, defined overa finite field; in particular, when G is the group of F-points of a…

We combinatorially characterize the number $\mathrm{cc}_2$ of conjugacy classes of involutions in any Coxeter group in terms of higher rank odd graphs. This notion naturally generalizes the concept of odd graphs, used previously to count…

By the work of Sela, for any free group $F$, the Coxeter group $W_ 3 = \mathbb{Z}/2\mathbb{Z} \ast \mathbb{Z}/2\mathbb{Z} \ast \mathbb{Z}/2\mathbb{Z}$ is elementarily equivalent to $W_3 \ast F$, and so Coxeter groups are not closed under…

We study the algebra of invariant representative functions over the N-fold Cartesian product of copies of a compact Lie group G modulo the action of conjugation by the diagonal subgroup. We construct a basis of invariant representative…

Let S be a finite graph and G be the corresponding free partially commutative group. In this paper we study subgroups generated by vertices of the graph S, which we call canonical parabolic subgroups. A natural extension of the definition…

In this work we introduce the notion of almost-symmetry for generalized numerical semigroups. In addition to the main properties occurring in this new class, we present several characterizations for its elements. In particular we show that…

We prove that even Coxeter groups, whose Coxeter diagrams contain no (4,4,2) triangles, are conjugacy separable. In particular, this applies to all right-angled Coxeter groups or word hyperbolic even Coxeter groups. For an arbitrary Coxeter…

We present procedures to calculate the set of Arf numerical semigroups with given genus, given conductor and given genus and conductor. We characterize the Kunz coordinates of an Arf numerical semigroup. We also describe Arf numerical…

We determine the maximal hyperbolic reflection groups associated to the quadratic forms $-3x_0^2 + x_1^2 + ... + x_n^2$, $n \ge 2$, and present the Coxeter schemes of their fundamental polyhedra. These groups exist in dimensions up to 13,…

We define a type B analogue of the category of finite sets with surjections, and we study the representation theory of this category. We show that the opposite category is quasi-Grobner, which implies that submodules of finitely generated…

An element of a Coxeter group $W$ is fully commutative if any two of its reduced decompositions are related by a series of transpositions of adjacent commuting generators. These elements were extensively studied by Stembridge, in particular…