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 conjugacy class of elements of $W$, and gave a uniform proof of this claim for symmetric groups. In this note we outline an inductive approach to our conjecture. As an application of this method, we prove the inductive version of the conjecture for finite Coxeter groups of rank up to 2.
In recent papers we have refined a conjecture of Lehrer and Solomon expressing the character of a finite Coxeter group $W$ acting on the $p$th graded component of its Orlik-Solomon algebra as a sum of characters induced from linear characters of centralizers of elements of $W$. Our refined conjecture relates the character above to a component of a decomposition of the regular character of $W$ related to Solomon's descent algebra of $W$. The refined conjecture has been proved for symmetric and dihedral groups, as well as finite Coxeter groups of rank three and four. In this paper, the second in…
In recent papers we have refined a conjecture of Lehrer and Solomon expressing the characters of a finite Coxeter group $W$ afforded by the homogeneous components of its Orlik-Solomon algebra as sums of characters induced from linear characters of centralizers of elements of $W$. Our refined conjecture also relates the Orlik-Solomon characters above to the terms of a decomposition of the regular character of $W$ related to the descent algebra of $W$. A consequence of our conjecture is that both the regular character of $W$ and the character of the Orlik-Solomon algebra have parallel, graded de…
We refine a conjecture by Lehrer and Solomon on the structure of the Orlik-Solomon algebra of a finite Coxeter group $W$ and relate it to the descent algebra of $W$. As a result, we claim that both the group algebra of $W$, as well as the Orlik-Solomon algebra of $W$ can be decomposed into a sum of induced one-dimensional representations of element centralizers, one for each conjugacy class of elements of $W$. We give a uniform proof of the claim for symmetric groups. In addition, we prove that a relative version of the conjecture holds for every pair $(W, W_L)$, where $W$ is arbitrary and $W_…
In this paper we extend the computations in parts I and II of this series of papers and complete the proof of a conjecture of Lehrer and Solomon expressing the character of a finite Coxeter group W acting on the pth graded component of its Orlik-Solomon algebra as a sum of characters induced from linear characters of centralizers of elements of W for groups of rank seven and eight. For classical Coxeter groups, these characters are given using a formula that is expected to hold in all ranks.
The descent algebra $Σ(W)$ is a subalgebra of the group algebra $\Q W$ of a finite Coxeter group $W$, which supports a homomorphism with nilpotent kernel and commutative image in the character ring of $W$. Thus $Σ(W)$ is a basic algebra, and as such it has a presentation as a quiver with relations. Here we construct $Σ(W)$ as a quotient of a subalgebra of the path algebra of the Hasse diagram of the Boolean lattice of all subsets of $S$, the set of simple reflections in $W$. From this construction we obtain some general information about the quiver of $Σ(W)$ and an algorithm for the constructi…
Let W be a finite Coxeter group. In this paper, we show that the properties of the Solomon algebra homomorphism Phi (from the Solomom descent algebra to the algebra of class functions) are strongly related to enumerative results: certain joint statistics on W may be enumerated by the scalar products of appropriate characters.
The purpose of the present article is to define and study a new class of descent algebras, called twisted descent algebras. These algebras are associated to the Barratt-Joyal theory of twisted bialgebras in the same way than classical descent algebras are associated to classical bialgebras. The formal properties of twisted descent algebras seem particularly meaningful in view of applications to discrete probabilities, to the geometry of Coxeter groups and buildings, and to symmetric group combinatorics.
We study different problems related to the Solomon's descent algebra $Σ(W)$ 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 and if the longest element is central, then the Loewy length of $Σ(W)$ is equal to $\lceil |S| / 2 \rceil$.
Let $(W,S)$ be a finite Coxeter system. Tits defined an associative product on the set $Σ$ of simplices of the associated Coxeter complex. The corresponding semigroup algebra is the Solomon-Tits algebra of $W$. It contains the Solomon algebra of $W$ as the algebra of invariants with respect to the natural action of $W$ on $Σ$. For the symmetric group $S_n$, there is a 1-1 correspondence between $Σ$ and the set of all set compositions (or ordered set partitions) of $\{1,...,n\}$. The product on $Σ$ has a simple combinatorial description in terms of set compositions. We study in detail the repre…
We expand upon the notion of equivariant log concavity, and make equivariant log concavity conjectures for Orlik--Solomon algebras of matroids, Cordovil algebras of oriented matroids, and Orlik--Terao algebras of hyperplane arrangements. In the case of the Coxeter arrangement for the Lie algebra $\mathfrak{sl}_n$, we exploit the theory of representation stability to give computer assisted proofs of these conjectures in low degree.
The discrete group generated by reflections of the sphere, or Euclidean space, or hyperbolic space are said to be Coxeter groups of, respectively, spherical, or Euclidean, or hyperbolic type. The hyperbolic Coxeter groups are said to be (quasi-)Lannér if the tiles covering the space are of finite volume and all (resp. some of them) are compact. For any Coxeter group stratified by the length of its elements, the Poincaré series (a.k.a. growth function) is the generating function of the cardinalities of sets of elements of equal length. Solomon established that, for ANY Coxeter group, its Poinca…
We develop the homology theory of the algebra of a regular semigroup, which is a particularly nice case of a quasi-hereditary algebra in good characteristic. Directedness is characterized for these algebras, generalizing the case of semisimple algebras studied by Munn and Ponizovksy. We then apply homological methods to compute (modulo group theory) the quiver of a right regular band of groups, generalizing Saliola's results for a right regular band. Right regular bands of groups come up in the representation theory of wreath products with symmetric groups in much the same way that right regul…
Let $V$ be an $\ell$-dimensional Euclidean space. Let $G \subset O(V)$ be a finite irreducible orthogonal reflection group. Let ${\cal A}$ be the corresponding Coxeter arrangement. Let $S$ be the algebra of polynomial functions on $V.$ For $H \in {\cal A}$ choose $α_H \in V^*$ such that $H = {\rm ker}(α_H).$ For each nonnegative integer $m$, define the derivation module $\sD^{(m)}({\cal A}) = \{θ\in {\rm Der}_S | θ(α_H) \in S α^m_H\}$. The module is known to be a free $S$-module of rank $\ell$ by K. Saito (1975) for $m=1$ and L. Solomon-H. Terao (1998) for $m=2$. The main result of this paper …
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 Schur-Weyl dualities. It leads to a new class of representations of the infinite symmetric group, which have not appeared earlier. We describe these representations and, in particular, find their spectral types with respect to the Gelfand-Tsetlin algebra. The main example of such a representation acts in an incomplete infinite tensor product. As an important app…
If $x$ and $y$ are roots in the root system with respect to the standard (Tits) geometric realization of a Coxeter group $W$, we say that $x$ \emph{dominates} $y$ if for all $w\in W$, $wy$ is a negative root whenever $wx$ is a negative root. We call a positive root \emph{elementary} if it does not dominate any positive root other than itself. The set of all elementary roots is denoted by $\E$. It has been proved by B. Brink and R. B. Howlett (Math. Ann. \textbf{296} (1993), 179--190) that $\E$ is finite if (and only if) $W$ is a finite-rank Coxeter group. Amongst other things, this finiteness …
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 multiplicative structure on the Whitney homology of $\mathcal{L}$ in terms of the basis, and the resulting algebra has similarities to the Orlik-Solomon algebra. As an application, we obtain four chain complexes which compute the integral homology of the Milnor fibre of the reflection arrangement of $W$, the Milnor fibre of the discriminant of $W$, the hyperpl…
The descent algebra of a finite Coxeter group W is a subalgebra of the group algebra defined by Solomon. Descent algebras of symmetric groups have properties that are not shared by other Coxeter groups. For instance, the natural map from the descent algebra of a symmetric group to its character ring is a surjection with kernel equal the Jacobson radical. Thus, the descent algebra implicitly encodes information about the representations of the symmetric group, and a complete set of primitive idempotents in the character ring leads to a decomposition of the group algebra into a sum of right idea…
Let $W$ be a finite Coxeter group. We give an algebraic presentation of what we refer to as ``the non-crossing algebra'', which is associated to the hyperplane complement of $W$ and to the cohomology of its Milnor fibre. This is used to produce simpler and more general chain (and cochain) complexes which compute the integral homology and cohomology groups of the Milnor fibre $F$ of $W$. In the process we define a new, larger algebra $\widetilde{A}$, which seems to be ``dual'' to the Fomin-Kirillov algebra, and in low ranks is linearly isomorphic to it. There is also a mysterious connection bet…
The new approach to the theory of complex representrations of the finite symmetric groups which based on the notions of Coxeter generators., Gelfand-Zetlin algebras, Hecke algebra, Young-Jucys-Murphi generators and which hardly used inductive structure - is systematically developed. The appearence of Young diagrams, tables is naturally explained - the set of content vectors of all Young tables is the spectra of Gel'fand-Zetlin algebra. The first steps of the theory (the list of the irrediucible representatinos, branching rule, Young orthopgonal form, Murnagan-Nakayama rule etc.)are established…
