We study the asymptotic behaviour of the statistic (des+ides) which assigns to an element w of a finite Coxeter group W the number of descents of w plus the number of descents of its inverse. Our main result is a central limit theorem for the probability distributions associated to this statistic. This answers a question of Kahle-Stump and generalises work of Chatterjee-Diaconis, Özdemir and Röttger.
In 2018, Kahle and Stump raised the following problem: identify sequences of finite Coxeter groups $W_n$ for which the two-sided descent statistics on a uniform random element of $W_n$ is asymptotically normal. Recently, Brück and Röttger provided an almost-complete answer, assuming some regularity condition on the sequence $W_n$. In this note, we provide a shorter proof of their result, which does not require any regularity condition. The main new proof ingredient is the use of the second Wasserstein distance on probability distributions, based on the work of Mallows (1972).
A Coxeter group of classical type $A_n$, $B_n$ or $D_n$ contains a chain of subgroups of the same type. We show that intersections of conjugates of these subgroups are again of the same type, and make precise in which sense and to what extent this property is exclusive to the classical types of Coxeter groups. As the main tool for the proof we use Solomon's descent algebra. Using Stirling numbers, we express certain basis elements of the descent algebra as polynomials and derive explicit multiplication formulas for a commutative subalgebra of the descent algebra.
For any Coxeter system $(W,S)$ of rank $n$, we introduce an abstract boolean complex (simplicial poset) of dimension $2n-1$ that contains the Coxeter complex as a relative subcomplex. Faces are indexed by triples $(I,w,J)$, where $I$ and $J$ are subsets of the set $S$ of simple generators, and $w$ is a minimal length representative for the parabolic double coset $W_I w W_J$. There is exactly one maximal face for each element of the group $W$. The complex is shellable and thin, which implies the complex is a sphere for the finite Coxeter groups. In this case, a natural refinement of the $h$-pol…
We classify a class of complex representations of an arbitrary Coxeter group via characters of the integral homology of certain graphs. Such representations can be viewed as a generalization of the geometric representation and correspond to the second-highest 2-sided cell in the sense of Kazhdan-Lusztig. We also give a description of the cell representation provided by this 2-sided cell, and find out all its simple quotients for simply laced Coxeter system with no more than one circuit in the Coxeter graph.
We show that any lower Bruhat interval in a Coxeter group is a disjoint union of certain two-sided cosets as a consequence of Lifting Property and Subword Property. Furthermore, we describe these details in terms of Bruhat graphs, graded posets, and two-sided quotients altogether. For this purpose, we introduce some new ideas, quotient lower intervals and quotient Bruhat graphs.
We investigate Mahonian and Eulerian probability distributions given by inversions and descents in general finite Coxeter groups. We provide uniform formulas for the means and variances in terms of Coxeter group data in both cases. We also provide uniform formulas for the double-Eulerian probability distribution of the sum of descents and inverse descents. We finally establish necessary and sufficient conditions for general sequences of Coxeter groups of increasing rank under which Mahonian and Eulerian probability distributions satisfy central and local limit theorems.
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…
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 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…
The descent algebra of a finite Coxeter group $W$ is a basic algebra, and as such it has a presentation as quiver with relations. In recent work, we have developed a combinatorial framework which allows us to systematically compute such a quiver presentation for a Coxeter group of a given type. In this article, we use that framework to determine quiver presentations for the descent algebras of the Coxeter groups of exceptional or non-crystallographic type, i.e., of type $E_6$, $E_7$, $E_8$, $F_4$, $H_3$, $H_4$ or $I_2(m)$.
We develop combinatorics of parabolic double cosets in finite Coxeter groups as a follow-up of recent articles by Billey-Konvalinka-Petersen-Slofstra-Tenner and Petersen. (1) We construct a double coset system as a generalization of a two-sided analogue of a Coxeter complex and present its order structure with its local dimension function on certain connected components. As applications of double cosets to Bruhat graphs, we also prove: (2) every parabolic double coset is regular, (3) invariance of degree on Bruhat graph on lower intervals as an analogy of the one for Kazhdan-Lusztig polynomial…
The descent algebra of finite Coxeter groups is studied by many famous mathematicians like Bergeron, Brown, Howlett, or Reutenauer. Blessenohl, Hohlweg, and Schocker, for example, proved a symmetry property of the descent algebra, when it is linked to the representation theory of its Coxeter group. The interest is particularly showed for the descent algebra of symmetric group. Thibon determined the eigenvalues and their multiplicities of the action on the group algebra of symmetric group of the descent algebra element, which is the sum over all permutations weighted by q^maj. And even the auth…
The Eulerian numbers count permutations according to the number of descents. The two-sided Eulerian numbers count permutations according to number of descents and the number of descents in the inverse permutation. Here we derive some results for Eulerian and two-sided Eulerian numbers using an elementary "balls-in-boxes" approach. We also discuss an open conjecture of Ira Gessel about the two-sided Eulerian numbers.
We introduce a new statistic on the hyperoctahedral groups (Coxeter groups of type B), and give a conjectural formula for its signed distributions over arbitrary descent classes. The statistic is analogous to the classical Coxeter length function, and features a parity condition. For descent classes which are singletons the conjectured formula gives the Poincaré polynomials of the varieties of symmetric matrices of fixed rank. For several descent classes we prove the conjectural formula. For this we construct suitable "supporting sets" for the relevant generating functions. We prove cancellati…
Louis Solomon showed that the group algebra of the symmetric group $\mathfrak{S}_{n}$ has a subalgebra called the descent algebra, generated by sums of permutations with a given descent set. In fact, he showed that every Coxeter group has something that can be called a descent algebra. There is also a commutative, semisimple subalgebra of Solomon's descent algebra generated by sums of permutations with the same number of descents: an "Eulerian" descent algebra. For any Coxeter group that is also a Weyl group, Paola Cellini proved the existence of a different Eulerian subalgebra based on a modi…
If $A$ is a subset of the set of reflections of a finite Coxeter group $W$, we define a sub-${\mathbb{Z}}$-module ${\mathcal{D}}_A(W)$ of the group algebra ${\mathbb{Z}} W$. We provide examples where this submodule is a subalgebra. This family of subalgebras includes strictly the Solomon descent algebra and, if $W$ if of type $B$, the Mantaci-Reutenauer algebra.
We find a simple product formula for the characteristic polynomial of the permutations with a fixed descent set under the weak order. As a corollary we obtain a simple product formula for the characteristic polynomial of alternating permutations. We generalize these results to Coxeter groups. We also find a formula for the generating function for the characteristic polynomials of classical Coxeter groups, which is then related to affine Coxeter groups.
Introduced by Solomon in his 1976 paper, the descent algebra of a finite Coxeter group received significant attention over the past decades. As proved by Gessel, in the case of the symmetric group its structure constants give the comultiplication table for the fundamental basis of quasisymmetric functions. We show that this property actually implies several well known relations linked to the Robinson-Schensted-Knuth correspondence and some of its generalisations. We further use the theory of type B quasisymmetric functions introduced by Chow to provide analogue results when the Coxeter group i…
Associated to each irreducible crystallographic root system $Φ$, there is a certain cell complex structure on the torus obtained as the quotient of the ambient space by the coroot lattice of $Φ$. This is the Steinberg torus. A main goal of this paper is to exhibit a module structure on (the set of faces of) this complex over the (set of faces of the) Coxeter complex of $Φ$. The latter is a monoid under the Tits product of faces. The module structure is obtained from geometric considerations involving affine hyperplane arrangements. As a consequence, a module structure is obtained on the space …
We consider a generalization of the fundamental theorem of finitely generated abelian groups for some non-abelian groups, which is called OGS. First, we consider the dihedral group, which is a non-abelian extension of an abelian group by an involution. Then, we focus on a special case, where the abelian group is cyclic, which is the two-generated Coxeter group I{2}(m). We mention interesting connections between the reduced Coxeter presentation and a particular OGS canonical presentation, which we call the standard OGS canonical presentation. These connections motivate us to offer a generalizat…
For symmetric groups we show that the p-canonical basis can be extended to a cell datum for the Iwahori-Hecke algebra H and that the two-sided p-cell preorder coincides with the Kazhdan-Lusztig two-sided cell preorder. Moreover, we show that left (or right) p-cells inside the same two-sided p-cell for Hecke algebras of finite crystallographic Coxeter systems are incomparable (Property A).
We generalize some identities and q-identities previously known for the symmetric group to Coxeter groups of type B and D. The extended results include theorems of Foata and Schützenberger, Gessel, and Roselle on various distributions of inversion number, major index, and descent number. In order to show our results we provide caracterizations of the systems of minimal coset representatives of Coxeter groups of type B and D.
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…
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. Both notions have the same, simple eigenvalues. The hyperplane definition is especially natural and satisfies a positivity property when $W$ is crystallographic and the relevant parameter is a good prime. The hyperplane viewpoint suggests interesting connections with Lie theory and leads to a notion of riffle shuffling for arbitrary real hyperplane arrangeme…
We study the joint distribution of descents and sign for elements of the symmetric group and the hyperoctahedral group (Coxeter groups of types $A$ and $B$). For both groups, this has an application to riffle shuffling: for large decks of cards the sign is close to random after a single shuffle. In both groups, we derive generating functions for the Eulerian distribution refined according to sign, and use them to give two proofs of central limit theorems for positive and negative Eulerian numbers.
We study the quiver of the descent algebra of a finite Coxeter group W. The results include a derivation of the quiver of the descent algebra of types A and B. Our approach is to study the descent algebra as an algebra constructed from the reflection arrangement associated to W.
The open intervals in the Bruhat order on twisted involutions in a Coxeter group are shown to be PL spheres. This implies results conjectured by F. Incitti and sharpens the known fact that these posets are Gorenstein* over Z_2. We also introduce a Boolean cell complex which is an analogue for twisted involutions of the Coxeter complex. Several classical Coxeter complex properties are shared by our complex. When the group is finite, it is a shellable sphere, shelling orders being given by the linear extensions of the weak order on twisted involutions. Furthermore, the h-polynomial of the comple…
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…
Given an essential semilattice congruence $\equiv$ on the left weak order of a Coxeter group $W$, we define the Coxeter stack-sorting operator ${\bf S}_\equiv:W\to W$ by ${\bf S}_\equiv(w)=w\left(π_\downarrow^\equiv(w)\right)^{-1}$, where $π_\downarrow^\equiv(w)$ is the unique minimal element of the congruence class of $\equiv$ containing $w$. When $\equiv$ is the sylvester congruence on the symmetric group $S_n$, the operator ${\bf S}_\equiv$ is West's stack-sorting map. When $\equiv$ is the descent congruence on $S_n$, the operator ${\bf S}_\equiv$ is the pop-stack-sorting map. We establish …
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.
Central Limit Theorems are known for the Eulerian statistic "descent" (or "excedance") in the symmetric group $\SSS_n$. Recently, Fulman, Kim, Lee and Petersen gave a Central Limit Theorem for "descent" over the alternating group $\AAA_n$ and also gave a Carlitz identity in $\AAA_n$ using descents. In this paper, we give a Central Limit Theorem in $\AAA_n$ involving excedances. We extend these to the positive elements in type B and type D Coxeter groups. Boroweic and Młotkowski enumerated type B descents over $\DD_n$, the type D Coxeter group and gave similar results. We refine their results f…
It is known that Euler numbers, defined as the Taylor coefficients of the tangent and secant functions, count alternating permutations in the symmetric group. Springer defined a generalization of these numbers for each finite Coxeter group by considering the largest descent class, and computed the value in each case of the classification. We consider here another generalization of Euler numbers for finite Coxeter groups, building on Stanley's result about the number of orbits of maximal chains of set partitions. We present a method to compute these integers and obtain the value in each case of…
In theory of Coxeter groups, bigrassmannian elements are well known as elements which have precisely one left descent and precisely one right descent. In this article, we prove formulas on enumeration of bigrassmannian permutations weakly below a permutation in Bruhat order in the symmetric groups. For the proof, we use equivalent characterizations of bigrassmannian permutations by Lascoux-Schutzenberger and Reading.
Various partial orders related to the structures of dual canonical monoids are investigated. It is shown that the nilpotent variety of a dual canonical monoid is equidimensional; its dimension is found. It is shown in type A that certain intervals of the Putcha poset of a dual canonical monoid are isomorphic to the Renner monoids of matrices. The notion of a two-sided weak order on a normal reductive monoid is introduced. A criterion, in terms of type maps, for the covering relations in a two-sided weak order to have degree 2 is found. It is shown that, for the unique equivariant divisor of a …
Solomon's descent algebra is used to define a family of signed measures M(W,x) for a finite Coxeter group W and non-zero x. The measures corresponding to W of types A and B are known to arise from the theory of card shuffling and to be related to the Poincare-Birkhoff-Witt theorem and splittings of Hochschild homology. Formulas for these measures are obtained in special cases. The eigenvalues of the associated Markov chains are computed. By elementary algebraic group theory, choosing a random semisimple orbit on a Lie algebra corresponding to a finite group of Lie type G^F induces a measure on…
This paper develops techniques to study the number of descents in random permutations via martingales. We relax an assumption in the Berry-Esseen theorem of Bolthausen (1982) to extend the theorem's scope to martingale differences of time-dependent variances. This extension leads to a new proof of the fact that the number of descents in random permutations is asymptotically normal with an error bound of order $1/\sqrt{n}.$ The same techniques are shown to be applicable to other descent and descent-related statistics as they satisfy certain recurrence relation conditions. These statistics inclu…
The Loewy length of the descent algebra of type D_{2m+1} is shown to be m+2, for m \geq 2, by providing an upper bound that agrees with the lower bound in \cite{BonnafePfeiffer2006}. The bound is obtained by showing that the length of the longest path in the quiver of the descent algebra of D_{2m+1} is at most m+1. To achieve this bound, the geometric approach to the descent algebra is used, in which the descent algebra of a finite Coxeter group is identified with an algebra associated to the reflection arrangement of the group.
The usual combinatorial model for the 0-Hecke algebra of the symmetric group is to consider the algebra (or monoid) generated by the bubble sort operators. This construction generalizes to any finite Coxeter group W. The authors previously introduced the Hecke group algebra, constructed as the algebra generated simultaneously by the bubble sort and antisort operators, and described its representation theory. In this paper, we consider instead the monoid generated by these operators. We prove that it has |W| simple and projective modules. In order to construct a combinatorial model for the simp…
Gessel conjectured that the two-sided Eulerian polynomial, recording the common distribution of the descent number of a permutation and that of its inverse, has non-negative integer coefficients when expanded in terms of the gamma basis. This conjecture has been proved recently by Lin. We conjecture that an analogous statement holds for simple permutations, and use the substitution decomposition tree of a permutation (by repeated inflation) to show that this would imply the Gessel-Lin result. We provide supporting evidence for this stronger conjecture.
Let $W\ltimes L$ be an irreducible affine Weyl group with Coxeter complex $Σ$, where $W$ denotes the associated finite Weyl group and $L$ the translation subgroup. The Steinberg torus is the Boolean cell complex obtained by taking the quotient of $Σ$ by the lattice $L$. We show that the ordinary and flag $h$-polynomials of the Steinberg torus (with the empty face deleted) are generating functions over $W$ for a descent-like statistic first studied by Cellini. We also show that the ordinary $h$-polynomial has a nonnegative $γ$-vector, and hence, symmetric and unimodal coefficients. In the class…
The distribution of Coxeter descents and block number over the set of fully commutative elements in the hyperoctahedral group $B_n$, $\FC(B_n)$, is studied in this paper. We prove that the associated Chow quasi-symmetric generating function is equal to a non-negative sum of products of two Schur functions. The proof involves a decomposition of $\FC(B_n)$ into a disjoint union of two-sided Barbash-Vogan combinatorial cells, a type $B$ extension of Rubey's descent preserving involution on $321$-avoiding permutations and a detailed study of the intersection of $\FC(B_n)$ with $S_n$-cosets which y…
We prove that the group algebra of the hyperoctahedral group contains a subalgebra corresponding to the flag descent number of Adin, Brenti, and Roichman. This algebra is in fact the span of the basis elements of the type A and type B Eulerian descent algebras. We describe a set of orthogonal idempotents which spans the flag descent algebra and prove that it contains the type A Eulerian descent algebra as a two-sided ideal. Using a new colored analogue of Stanley's $P$-partitions, we prove the existence of a colored Eulerian descent algebra which is a subalgebra of the Mantaci-Reutenauer algeb…
For any finite Coxeter group W, we introduce two new objects: its cutting poset and its biHecke monoid. The cutting poset, constructed using a generalization of the notion of blocks in permutation matrices, almost forms a lattice on W. The construction of the biHecke monoid relies on the usual combinatorial model for the 0-Hecke algebra H_0(W), that is, for the symmetric group, the algebra (or monoid) generated by the elementary bubble sort operators. The authors previously introduced the Hecke group algebra, constructed as the algebra generated simultaneously by the bubble sort and antisort o…
We generalize two bijections due to Garsia and Gessel to compute the generating functions of the two vector statistics $(\des_G, \maj,\ell_G, \col)$ and $(\des_G, \ides_G, \maj, \imaj, \col, \icol)$ over the wreath product of a symmetric group by a cyclic group. Here $\des_G$, $\ell_G$, $\maj$, $\col$, $\ides_G$, $\imaj_G$, and $\icol$ denote the number of descents, length, major index, color weight, inverse descents, inverse major index, and inverse color weight, respectively. Our main formulas generalize and unify several known identities due to Brenti, Carlitz, Chow-Gessel, Garsia-Gessel, a…
This is the second of two papers where we study polytopes arising from affine Coxeter arrangements. Our results include a formula for their volumes, and also compatible definitions of hypersimplices, descent numbers and major index for all Weyl groups. We give a q-analogue of Weyl's formula for the order of the Weyl group. For A_n, C_n and D_4, we give a Grobner basis which induces the triangulation of alcoved polytopes.
