A Weighted Enumeration of Maximal Chains in the Bruhat Order

Given a finite Weyl group W with root system Φ, assign the weight α ∈ Φ to each covering pair in the Bruhat order related by the reflection corresponding to α. Extending this multiplicatively to chains, we prove that the sum of the weights of all maximal chains in the Bruhat order has an explicit product formula, and prove a similar result for a weighted sum over maximal chains in the Bruhat ordering of any parabolic quotient of W . Several variations and open problems are discussed.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/46157/1/10801_2004_Article_403286.pd

Multiplicity-Free Products of Schur Functions

We classify all multiplicity-free products of Schur functions and all multiplicity-free products of characters of SL ( n , C ).Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/41839/1/26-5-2-113_10050113.pd

The Enumeration of Fully Commutative Elements of Coxeter Groups

A Coxeter group element w is fully commutative if any reduced expression for w can be obtained from any other via the interchange of commuting generators. For example, in the symmetric group of degree n, the number of fully commutative elements is the nth Catalan number. The Coxeter groups with finitely many fully commutative elements can be arranged into seven infinite families A n , B n , D n , E n ,F n , H n and I 2 (m). For each family, we provide explicit generating functions for the number of fully commutative elements and the number of fully commutative involutions; in each case, the generating function is algebraic.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/46294/1/10801_2004_Article_157578.pd

Quasi-Minuscule Quotients and Reduced Words for Reflections

We study the reduced expressions for reflections in Coxeter groups, with particular emphasis on finite Weyl groups. For example, the number of reduced expressions for any reflection can be expressed as the sum of the squares of the number of reduced expressions for certain elements naturally associated to the reflection. In the case of the longest reflection in a Weyl group, we use a theorem of Dale Peterson to provide an explicit formula for the number of reduced expressions. We also show that the reduced expressions for any Weyl group reflection are in bijection with the linear extensions of a natural partial ordering of a subset of the positive roots or co-roots.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/46149/1/10801_2004_Article_333190.pd

A Construction of H 4 without Miracles

We present a new construction of the root system H 4 .Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/42422/1/454-22-3-425_22n3p425.pd

Affine descents and the Steinberg torus

Abstract. Let W ⋉ 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 classical cases, we also provide expansions, identities, and generating functions for the h-polynomials of Steinberg tori. Résumé. Nous considérons un groupe de Weyl affine irréductible W ⋉ L avec complexe de Coxeter Σ, où W désigne le groupe de Weyl fini associé et L le sous-groupe des translations. Le tore de Steinberg est le complexe cellulaire Booléen obtenu comme le quotient de Σ par L. Nous montrons que les h-polynômes, ordinaires et de drapeaux, du tore de Steinberg (sans la face vide) sont des fonctions génératrices sur W pour une statistique de type descente, étudiée en premier lieu par Cellini. Nous montrons également qu’un h-polynôme ordinaire possède un γ-vecteur positif, et par conséquent, a des coéfficients symétriques et unimodaux. Dans les cas classiques, nous donnons également des développements, des identités et des fonctions génératrices pour les h-polynômes des tores de Steinberg

On immanants of Jacobi-Trudi matrices and permutations with restricted position

Let [chi] be a character of the symmetric group Ln. The immanant of an n x n matrix A = [aij] with respect to [chi] is [Sigma]w [epsilon] Sn [chi](w) a1,w(1) ... an,w(n). Goulden and Jackson conjectured, and Greene recently proved, that immanants of Jacobi-Trudi matrices are polynomials with nonnegative integer coefficients. This led one of us (Stembridge) to formulate a series of conjectures involving immanants, some of which amount to stronger versions of the original Goulden-Jackson conjecture. In this paper, we prove some special cases of one of the stronger conjectures. One of the special cases we prove develops from a generalization of the theory of permutations with restricted position which takes into account the cycle structure of the permutations. We also present two refinements of the immanant conjectures, as well as a related conjecture on the number of ways to partition a partially ordered set into chains.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/30946/1/0000617.pd

Affine descents and the Steinberg torus

Let $W \ltimes L$ be an irreducible affine Weyl group with Coxeter complex $\Sigma$, 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 $\Sigma$ 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 $\gamma$-vector, and hence, symmetric and unimodal coefficients. In the classical cases, we also provide expansions, identities, and generating functions for the $h$-polynomials of Steinberg tori