Perfect models for finite Coxeter groups

Weights for symmetric and general linear groups

Variants of some of the Brauer-Fowler theorems

Let K be a commutative noetherian ring. It is proved that a representation of a finite group on a K-module of finite length or on a K-module of finite exponent has a finite basis for its identities. In particular, this implies an earlier result of Nguyen Hung Shon and the author stating that every representation of a finite group over a field is finitely based. The problem whether every representation of a finite group over a commutative noetherian ring is finitely based still remains open.

Let W be a finite Coxeter group and L be a weight function on W in the sense of Lusztig. We have recently introduced a pre-order relation ⪯ L on the set of irreducible characters of W which extends Lusztig’s definition of “families” and which, conjecturally, corresponds to the ordering given by Kazhdan–Lusztig cells. Here, we give an explicit description of ⪯ L for W of type B n and any L. (All other cases are known from previous work.) This crucially relies on some new combinatorial constructions around Lusztig’s “symbols”. Combined with previous work, we deduce general compatibility results between ⪯ L and Lusztig’s a-function, valid for any W,L.

Let W W be an infinite irreducible Coxeter group with ( s 1 , … , s n ) (s_1, \ldots , s_n) the simple generators. We give a short proof that the word s 1 s 2 ⋯ s n s 1 s 2 ⋯ s_1 s_2 \cdots s_n s_1 s_2 \cdots s n ⋯ s 1 s 2 ⋯ s n s_n \cdots s_1 s_2 \cdots s_n is reduced for any number of repetitions of s 1 s 2 ⋯ s n s_1 s_2 \cdots s_n . This result was proved for simply laced, crystallographic groups by Kleiner and Pelley using methods from the theory of quiver representations. Our proof uses only basic facts about Coxeter groups and the geometry of root systems. We also prove that, in finite Coxeter groups, there is a reduced word for w 0 w_0 which is obtained from the semi-infinite word s 1 s 2 ⋯ s n s 1 s 2 ⋯ s n ⋯ s_1 s_2 \cdots s_n s_1 s_2 \cdots s_n \cdots by interchanging commuting elements and taking a prefix.

In this note we give a new proof of the Macdonald-Mehta-Opdam integral identity for finite Coxeter groups (in the equal parameter case). This identity was conjectured by Macdonald and proved by Opdam in \cite{O1,O2} using the theory of multivariable Bessel functions, but in non-crystallographic cases the proof relied on a computer calculation by F. Garvan. Our proof is somewhat more elementary (in particular, it does not use multivariable Bessel functions), and uniform (does not refer to the classification of finite Coxeter groups). \footnote{We expect that this proof can be generalized to the case of non-equal parameters. Indeed, many of the steps of our proof, including key Proposition \ref{l2}, generalize without effort to this setting.

The notion of globally irreducible representations of finite groups was introduced by B.H. Gross, in order to explain new series of Euclidean lattices discovered recently by N. Elkies and T. Shioda using Mordell–Weil lattices of elliptic curves. It has been observed by R. Gow and Gross that irreducible Weil representations of certain finite classical groups lead to globally irreducible representations. In this paper we classify all globally irreducible representations coming from Weil representations of finite classical groups.

The (ordinary) representation theory of the symmetric group is fascinating and has rich connections to combinatorics, including the Frobenius correspondence to the self-dual graded Hopf algebra of symmetric functions. The $0$-Hecke algebra (type $A$) is a deformation of the group algebra of the symmetric group, and its representation theory has an analogous correspondence to the dual graded Hopf algebras of quasisymmetric functions and noncommutative symmetric functions. Macdonald used the hook length formula for the number of standard Young tableaux of a fixed shape to determine how many irreducible representations of the symmetric group have dimensions indivisible by a prime $p$. In this paper, we study the dimensions of the projective indecomposable modules of the $0$-Hecke algebra modulo $p$; such a module is indexed by a composition and its dimension is given by a ribbon number, i.e., the cardinality of a descent class. Applying a result of Dickson on the congruence of multinomial coefficients, we count how many ribbon numbers belong to each congruence class modulo $p$ and extend the result to other finite Coxeter groups.

Vanishing ranges for the mod p cohomology of alternating subgroups of Coxeter groups

In a finite Coxeter group W and with two given conjugacy classes of parabolic subgroups [X] and [Y] , we count those parabolic subgroups of W in [Y] that are full support, while simultaneously being simple extensions (i.e., extensions by a single reflection) of some standard parabolic subgroup of W in [X] . The enumeration is given by a product formula that depends only on the two parabolic types. Our derivation is case-free and combines a geometric interpretation of the “full support” property with a double-counting argument involving Crapo's beta invariant. As a corollary, this approach gives the first case-free proof of Chapoton's formula for the number of reflections of full support in a real reflection group W .

Cambrian lattices

Reflection groups of geodesic spaces and Coxeter groups

This thesis presents recent developments concerning two open problems related to algebra and discrete geometry. In 1934, Harold S. M. Coxeter introduced Coxeter groups as abstractions of groups generated by reflections in a vector space [Cox34]. The present work lays out how a study of geometric and combinatorial properties of Coxeter groups contributed to the comprehension of the two open problems. Furthermore, this thesis presents some progress in proving the first problem, and an application of the approach introduced in this thesis for the second problem. Open Problem (Lattice for infinite Coxeter groups, Dyer [Dye11]). Is there, for each infinite Coxeter group, a complete ortholattice that contains the weak order? In Chapter 2, we study the asymptotical behaviour of roots of infinite Coxeter groups, which is part of a joint work with Christophe Hohlweg and Vivien Ripoll [HLR13]. In particular, we show that the directions of roots tend to the isotropic cone of the geometric representation of the root system. Moreover, using this framework, this thesis presents a proof that there is a complete ortholattice structure enclosing the weak order of infinite Coxeter groups of rank at most 3. Open Problem (Existence of multiassociahedra, Jonsson [Jon05]). Let k be an integer such that k >= 1. Is there a polytope whose boundary complex corresponds to the simplicial complex of sets of diagonals of a convex polygon not containing k + 1 mutually crossing diagonals? In Chapter 3 we introduce, for any finite Coxeter group and any nonnegative integer k, a spherical subword complex called multi-cluster complex. This family generalizes the concept of multitriangulations of type A and B to arbitrary finite Coxeter groups. For k = 1, this simplicial complex coincides with the finite cluster complex of the given type. We study combinatorial and geometric properties of multi-cluster complexes. In particular, we show that every spherical subword complex is the link of a face of a multi-cluster complex. This work was realized jointly with Cesar Ceballos and Christian Stump [CLS13]. Finally, this approach allows us to exhibit formulas counting the number of common vertices of permutahedra and generalized associahedra for arbitrary finite Coxeter groups and Coxeter elements.

Involutory reflection groups and their models

We classify surjective lattice homomorphisms $W\to W'$ between the weak orders on finite Coxeter groups. Equivalently, we classify lattice congruences $\Theta$ on $W$ such that the quotient $W/\Theta$ is isomorphic to $W'$. Surprisingly, surjective homomorphisms exist quite generally: They exist if and only if the diagram of $W'$ is obtained from the diagram of $W$ by deleting vertices, deleting edges, and/or decreasing edge labels. A surjective homomorphism $W\to W'$ is determined by its restrictions to rank-two standard parabolic subgroups of $W$. Despite seeming natural in the setting of Coxeter groups, this determination in rank two is nontrivial. Indeed, from the combinatorial lattice theory point of view, all of these classification results should appear unlikely a priori. As an application of the classification of surjective homomorphisms between weak orders, we also obtain a classification of surjective homomorphisms between Cambrian lattices and a general construction of refinement relations between Cambrian fans.