Introduction. Algebraic combinatorics is concerned with the interaction between combinatorics and such other branches of mathematics as commutative algebra, algebraic geometry, algebraic topology, and representation theory. Many of the major open problems of algebraic combinatorics are related to positivity questions, i.e., showing that certain integers are nonnegative. The significance of positivity to algebraic combinatorics stems from the fact that a nonnegative integer can have both a combinatorial and an algebraic interpretation. The archetypal algebraic interpretation of a nonnegative integer is as the dimension of a vector space. Thus to show that a certain integer m is nonnegative, it suces to nd a vector space Vm of dimension m. Similarly to show that m n, it suces to nd an injective map Vm ! V n or surjective map V n ! Vm . Of course the inequality m n is equivalent to the positivity statement n m 0, while the injectivity of the map ' : Vm ! V n is equivalent to the

Abstract. We describe a method of computing equivariant and ordinary intersection cohomology of certain varieties with actions of algebraic tori, in terms of structure of the zero- and one-dimensional orbits. The class of varieties to which our formula applies includes Schubert varieties in flag varieties and affine flag varieties. We also prove a monotonicity result on local intersection cohomology stalks. 1. Statement of the Main Results To a variety X with an appropriate torus action (§1.1), we will associate a moment graph (§1.2), a combinatorial object which reflects the structure of the 0 and 1-dimensional orbits. There is a canonical sheaf (§1.3) on the moment graph, combinatorially constructed from it (§1.4), which we denote by M. The main result (§1.5) uses the sheaf M to compute the local and global equivariant and ordinary intersection cohomology of X functorially. 1.1. Assumptions on the Variety X. We assume that X is an irreducible complex algebraic variety endowed with two structures: 1. An action of an algebraic torus T ∼ = (C ∗ ) d. We assume that (a) for every fixed point x ∈ X T there is a one-dimensional subtorus which is contracting near x, i.e. there is a homomorphism i: C ∗ → T and a Zariski open neighborhood U of x so that limα→0 i(α)y = x for all y ∈ U (this implies X T is finite), and (b) X has finitely many one-dimensional orbits 2. A T-invariant Whitney stratification by affine spaces. It follows that each stratum contains exactly one fixed point, since a contracting C ∗ action on an affine space must act linearly with respect to some coordinate system (see [2], Theorem 2.5). Let Cx denote the stratum containing the fixed point x, so X = ⋃ x∈XT Cx. Every one dimensional orbit L has exactly two distinct limit points: the T fixed

In 1979 Kazhdan and Lusztig defined, for every Coxeter group W, a family of polynomials, indexed by pairs of elements of W, which have become known as the Kazhdan-Lusztig polynomials of W, and which have proven to be of importance in several areas of mathematics. In this paper we show that the combinatorial concept of a special matching plays a fundamental role in the computation of these polynomials. Our results also imply, and generalize, the recent one in [12] on the combinatorial invariance of Kazhdan-Lusztig polynomials.

Europe", grant HPRN-CT-2001-00272. We write elements of S n in three ways, namely: . disjoint cycle form (e.g., # = (7, 5, 2)(1, 3)) ; . one-line notation (e.g., # = 3714265); (Meaning that #(1) = 3, #(2) = 7, etc...) . matrix (e.g. # = 6 6 6 6 6 6 6 6 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 7 7 7 7 7 7 7 7 ). There are two important subsets for this story: S = 2), (2, 3), . . . , (n 1, n)} T = j) : 1 i < j n}. Note that #(i, j) switches #(i) and #(j), while (i, j)# switches i and j, in the one-line notation of #. There are also two important statistics. For # S n let inv(#) = [n] : i < j, #(i) > #(j)}| (number inversions of #, or length of #, denoted l(#)) and D(#) = i + 1) S : #(i) > #(i + 1)}(# l(#(i, i + 1)) < l(#)) (descent set of # ). 1.2 Bruhat graph and Bruhat order There are two main combinatorial objects related to Kazhdan-Lusztig and R-polynomials. The Bruhat graph of S n is the directed graph B(S n )

Abstract. To any Coxeter system we associate an exact category and study the projective objects therein. We discuss an analog of the Kazhdan-Lusztig conjecture and show how it follows from a “genericity ” conjecture and how the latter follows from a “Hard Lefschetz ” conjecture. 1.

In 1979, Kazhdan and Lusztig defined a family Pu,v(q) of polynomials indexed by pairs of elements of a Coxeter group W that have proven to be fundamental objects of study in representation theory. At the same time, they can be defined combinatorially, and so have also been studied by combinatorialists. Although it is now known that Pu,v(q) depends only on the structure of the Bruhat interval [u, v] as an abstract poset, explicit formulas which exhibit this invariance are only known in general for intervals of length at most 4. In this paper we use a formula of Brenti to give an explicit formula for the quadratic coefficient of Pu,v(q) which is almost combinatorially invariant, and use this formula to give a combinatorially In [6], Kazhdan and Lusztig introduced polynomials Pu,v(q), the K azhdan-Lusztig polynomials, indexed by pairs of elements in a Coxeter group W. These polynomials have since become a fundamental object of study in representation theory, and due to their combinatorial definition have become of interest to combinatorialists as well. For more information

Abstract. We discuss the enumeration theory for flags in Eulerian partially ordered sets, emphasizing the two main geometric and algebraic examples, face posets of convex polytopes and regular CW-spheres, and Bruhat intervals in Coxeter groups. We review the two algebraic approaches to flag enumeration – one essentially as a quotient of the algebra of noncommutative symmetric functions and the other as a subalgebra of the algebra of quasisymmetric functions – and their relation via duality of Hopf algebras. One result is a direct expression for the Kazhdan-Lusztig polynomial of a Bruhat interval in terms of a new invariant, the complete cd-index. Finally, we summarize the theory of combinatorial Hopf algebras, which gives a unifying framework for the quasisymmetric generating functions developed here.