Related papers: The module structure of the Solomon-Tits algebra o…

Motivated by work of Kac and Lusztig, we define a root system and a Weyl groupoid for a large class of semisimple Yetter-Drinfeld modules over an arbitrary Hopf algebra. The obtained combinatorial structure fits perfectly into an existing…

The faces of the braid arrangement form a monoid. The associated monoid algebra -- the face algebra -- is well-studied, especially in relation to card shuffling and other Markov chains. In this paper, we explore the action of the symmetric…

We attach to every Coxeter system (W,S) an extension C_W of the corresponding Iwahori-Hecke algebra. We construct a 1-parameter family of (generically surjective) morphisms from the group algebra of the corresponding Artin group onto C_W.…

Combinatorial Hopf algebras give a linear algebraic structure to infinite families of combinatorial objects, a technique further enriched by the categorification of these structure via the representation theory of families of algebras. This…

We study different problems related to the Solomon's descent algebra $\Sigma(W)$ of a finite Coxeter group $(W,S)$: positive elements, morphisms between descent algebras, Loewy length... One of the main result is that, if $W$ is irreducible…

Associated to each irreducible crystallographic root system $\Phi$, there is a certain cell complex structure on the torus obtained as the quotient of the ambient space by the coroot lattice of $\Phi$. This is the Steinberg torus. A main…

Let $\mathcal{L}$ be the noncrossing partition lattice associated to a finite Coxeter group $W$. In this paper we construct explicit bases for the top homology groups of intervals and rank-selected subposets of $\mathcal{L}$. We define a…

The Solomon-Tits theorem says that the poset of proper non-trivial subspaces of a finite-dimensional vector space has realisation equivalent to a wedge of spheres. In this paper we prove a variant of this result for collections of geodesic…

We give a combinatorial description (including explicit differential-form bases) for the cohomology groups of the space of n distinct nonzero complex numbers, with coefficients in rank-one local systems which are of finite monodromy around…

In this paper we present the construction of explicit quasi-isomorphisms that compute the cyclic homology and periodic cyclic homology of crossed-product algebras associated with (discrete) group actions. In the first part we deal with…

In this paper we continue our study of the geometric properties of full symmetric Toda systems from \cite{CSS14,CSS17,CSS19}. Namely we describe here a simple geometric construction of a commutative family of vector fields on compact…

The symmetric group $\mathfrak S_{n+1}$ of degree $n+1$ admits an $n$-dimensional irreducible $\mathbf Q \mathfrak S_n$-module $V$ corresponding to the hook partition $(2,1^{n-1})$. By the work of Craig and Plesken we know that there are…

Let $\text{GL}(n) = \text{GL}(n, {\mathbb C})$ denote the complex general linear group and let $G \subset \text{GL}(n)$ be one of the classical complex subgroups $\text{O}(n)$, $\text{SO}(n)$, and $\text{Sp}(2k)$ (in the case $n = 2k$). We…

Let us fix a positive integer $\nu>1$. For each positive integer $n>1$, we consider a normal supercharacter theory $\mathcal{S}_n$ of $G_n$, where $G_n$ is the direct-product of $n-1$ copies of the cyclic group of order $\nu$. Then we endow…

These notes -- originating from a one-semester class by their second author at the University of Minnesota -- survey some of the most important Hopf algebras appearing in combinatorics. After introducing coalgebras, bialgebras and Hopf…

Let $(W, S)$ be a Coxeter system equipped with a fixed automorphism $\ast$ of order $\leq 2$ which preserves $S$. Lusztig (and with Vogan in some special cases) have shown that the space spanned by set of "twisted" involutions was naturally…

We introduce the Hopf algebra of quasi-symmetric functions with semigroup exponents generalizing the Hopf algebra QSym of quasi-symmetric functions. As a special case we obtain the Hopf algebra WCQSym of weak composition quasi-symmetric…

A (semi)brick over an algebra $A$ is a module $S$ such that the endomorphism ring $\operatorname{\mathsf{End}}_A(S)$ is a (product of) division algebra. For each Dynkin diagram $\Delta$, there is a bijection from the Coxeter group $W$ of…

The properties of Hopf star operations and twisted Hopf stars operations on quantum groups are discussed in relation with the theory of representations (star representations). Invariant Hermitian sesquilinear forms (scalar products) on…

For a space X acted by a finite group $\G$, the product space $X^n$ affords a natural action of the wreath product $\Gn$. In this paper we study the K-groups $K_{\tG_n}(X^n)$ of $\Gn$-equivariant Clifford supermodules on $X^n$. We show that…