John R. Stembridge

Nonintersecting paths, pfaffians, and plane partitions

Gessel and Viennot have developed a powerful technique for enumerating various classes of plane partitions. The purpose of this article is to show by similar means that one may use pfaffiants to enumerate configurations of nonintersecting paths in which the initial and/or terminal vertices of the paths are allowed to vary over specified regions of the digraph. This leads to the possibility of enumerating classes of plane partitions in which the shape is allowed to vary, whereas the previous applications of Gessel and Viennot were largely confined to plane partitions of a given shape

Shifted tableaux and the projective representations of symmetric groups

Les groupes de representation de S n . Classes de conjugaison. Algebres de Clifford et la representation spin de base. Produits induits pour des representations spin. Une application caracteristique pour des caracteres de spin. Tableaux decales et Q-fonctions de Schur. Les caracteres de spin irreductibles de S~ n et A n . Une analogue decalee de la regle LR. Produits tensoriels interieurs

On the Fully Commutative Elements of Coxeter Groups

Let W be a Coxeter group. We define an element w ∈ W to be fully commutative if any reduced expression for w can be obtained from any other by means of braid relations that only involve commuting generators. We give several combinatorial characterizations of this property, classify the Coxeter groups with finitely many fully commutative elements, and classify the parabolic quotients whose members are all fully commutative. As applications of the latter, we classify all parabolic quotients with the property that (1) the Bruhat ordering is a lattice, (2) the Bruhat ordering is a distributive lattice, (3) the weak ordering is a distributive lattice, and (4) the weak ordering and Bruhat ordering coincide.

Rational tableaux and the tensor algebra of gl n

Abstract A generalization of the usual column-strict tableaux (equivalent to a construction of R. C. King) is presented as a natural combinatorial tool for the study of finite dimensional representations of GL n ( C ). These objects are called rational tableaux since they play the same role for rational representations of GL n as ordinary tableaux do for polynomial representations. A generalization of Schensteds insertion algorithm is given for rational tableaux, and is used to count the. multiplicities of the irreducible GL n -modules in the tensor algebra of GL n . The problem of counting multiplicities when the k th tensor power gl n ⊗ k is decomposed into modules which are simultaneously irreducible with respect to GL n and the symmetric group S k is also considered. The existence of an insertion algorithm which describes this decomposition is proved. A generalization of border strip tableaux, in which both addition and deletion of border strips is allowed, is used to describe the characters associated with this decomposition. For large n , these generalized border strip tableaux have a simple structure which allows derivation of identities due to Hanlon and Stanley involving the (large n ) decomposition of gl n ⊗ k .

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

Abstract Let χ be a character of the symmetric group Ln. The immanant of an n × n matrix A = [aij] with respect to χ is Σw ϵ Sn χ(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.

A local characterization of simply-laced crystals

We provide a simple list of axioms that characterize the crystal graphs of integrable highest weight modules for simply-laced quantum Kac-Moody algebras.

Some combinatorial aspects of reduced words in finite Coxeter groups

We analyze the structure of reduced expressions in the Coxeter groups An, Bn and Dn. Several special classes of elements are singled out for their connections with symmetric functions or the theory of P -partitions. Membership in these special classes is characterized in a variety of ways, including forbidden patterns, forbidden subwords, and by the form of canonically chosen reduced words.

A Maple package for symmetric functions

Abstract We describe the man features of a package of Maple programs for manipulating symmetric polynomials and related structures. Among the highlights of the package are (1) a collection of procedures for converting between polynomial expressions involving several fundamental bases, and (2) a general mechanism that allows the user to easily add new bases to the existing collection. The latter facilitates computations involving numerous important families of symmetric functions, including Schur functions, Zonal polynomials, Jack symmetric functions, Hall-Littlewood functions, and the two-parameter symmetric functions of Macdonald.

Counting points on varieties over finite fields related to a conjecture of Kontsevich

We describe a characteristic-free algorithm for “reducing” an algebraic variety defined by the vanishing of a set of integer polynomials. In very special cases, the algorithm can be used to decide whether the number of points on a variety, as the ground field varies over finite fields, is a polynomial function of the size of the field. The algorithm is then used to investigate a conjecture of Kontsevich regarding the number of points on a variety associated with the set of spanning trees of any graph. We also prove several theorems describing properties of a (hypothetical) minimal counterexample to the conjecture, and produce counterexamples to some related conjectures.

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 An, Bn, Dn, En,Fn, Hn and I2(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.