Abstract. For a finite Coxeter group W and a Coxeter element c of W, the c-Cambrian fan is a coarsening of the fan defined by the reflecting hyperplanes of W. Its maximal cones are naturally indexed by the c-sortable elements of W. The main result of this paper is that the known bijection clc between c-sortable elements and c-clusters induces a combinatorial isomorphism of fans. In particular, the c-Cambrian fan is combinatorially isomorphic to the normal fan of the generalized associahedron for W. The rays of the c-Cambrian fan are generated by certain vectors in the W-orbit of the fundamental weights, while the rays of the c-cluster fan are generated by certain roots. For particular (“bipartite”) choices of c, we show that the c-Cambrian fan is linearly isomorphic to the c-cluster fan. We characterize, in terms of the combinatorics of clusters, the partial order induced, via the map clc, on c-clusters by the c-Cambrian lattice. We give a simple bijection from c-clusters to c-noncrossing partitions that respects the refined (Narayana) enumeration. We relate the Cambrian fan to well known objects in the theory of cluster algebras, providing a geometric

In the first part of this paper the projective dimension of the structural modules in the BGG category O is studied. This dimension is computed for simple, standard and costandard modules. For tilting and injective modules an explicit conjecture relating the result to Lusztig’s a-function is formulated (and proved for type A). The second part deals with the extension algebra of Verma modules. It is shown that this algebra is in a natural way Z 2-graded and that it has two Z-graded Koszul subalgebras. The dimension of the space Ext 1 into the projective Verma module is determined. In the last part several new classes of Koszul modules and modules, represented by linear complexes of tilting modules, are constructed.

Abstract. We characterise the permutations π such that the elements in the closed lower Bruhat interval [id, π] of the symmetric group correspond to nontaking rook configurations on a skew Ferrers board. It turns out that these are exactly the permutations π such that [id, π] corresponds to a flag manifold defined by inclusions, studied by Gasharov and Reiner. Our characterisation connects the Poincaré polynomials (rank-generating function) of Bruhat intervals with q-rook polynomials, and we are able to compute the Poincaré polynomial of some particularly interesting intervals in the finite Weyl groups An and Bn. The expressions involve q-Stirling numbers of the second kind. As a by-product of our method, we present a new Stirling number identity connected to both Bruhat intervals and the poly-Bernoulli numbers defined by Kaneko. 1.

Abstract. We introduce a notion of “freely braided element ” for simply laced Coxeter groups. We show that an arbitrary group element w has at most 2 N(w) commutation classes of reduced expressions, where N(w) is a certain statistic defined in terms of the positive roots made negative by w. This bound is achieved if w is freely braided. In the type A setting, we show that the bound is achieved only for freely braided w. 1.

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.

. We depict the weight diagrams of basic and adjoint representations of complex simple Lie algebras/algebraic groups and describe some of their uses. Introduction In this paper we collect the weight diagrams of basic representations of complex simple Lie algebras as well as of those adjoint representations, which are not basic. These pictures arise in a number of contexts, but their main significance stems from the fact that they allow to visualize calculations with root systems, Weyl 1991 Mathematics Subject Classification. 17B10, 17B20, 17B25, 20G05, 20G15, 20G35, 20G40, 20G45, secondary 06A07, 20C33, 20D06. Key words and phrases. Semisimple Lie algebra, Chevalley group, Weyl modules, root systems, Weyl group, basic representation, microweight, adjoint representation, weight diagram, Bruhat order. The authors gratefully acknowledge the support of the Sonderforschungsbereich 343 and of the Alexander von Humboldt-Stiftung and the hospitality of the Universit at Bielefeld during t...

Abstract. We prove that the order complex of a geometric lattice has a convex ear decomposition. As a consequence, if ∆(L) is the order complex of a rank (r+1) geometric lattice L, then the for all i ≤ r/2 the h-vector of ∆(L) satisfies, hi−1 ≤ hi and hi ≤ hr−i. We also obtain several inequalities for the flag h-vector of ∆(L) by analyzing the weak Bruhat order of the symmetric group. As an application, we obtain a zonotopal cd-analogue of the Dowling-Wilson characterization of geometric lattices which minimize Whitney numbers of the second kind. In addition, we are able to give a combinatorial flag h-vector proof of hi−1 ≤ hi when i ≤ 2 5 (r + 7 2). 1.

We present a unified theory for permutation models of all the infinite families of finite and a#ne Weyl groups, including interpretations of the length function and the weak order. We also give new combinatorial proofs of Bott's formula (in the refined version of Macdonald) for the Poincareseriesofthese a#ne Weyl groups. 1991 Mathematics Subject Classification. primary 20B35; secondary 05A15. 1 Introduction The aim of this paper is to present a unified theory for permutation representations of the finite Weyl groups A n-1 , B n , C n , D n , and the a#ne Weyl groups # A n-1 , # B n , # C n , # D n . Our starting point is the symmetric group S n , the group of permutations of [1,...,n]. If S n is presented as the group generated by adjacent transpositions, it is isomorphic to the Weyl group A n-1 , and we obtain well-known interpretations of several Coxeter group concepts in permutation language: 1. The Coxeter generators are the adjacent transpositions. 2. Reflections corresp...

Abstract. Let W be an arbitrary Coxeter group, possibly of infinite rank. We describe a decomposition of the centralizer ZW(WI) of an arbitrary parabolic subgroup WI into the center of WI, a Coxeter group and a subgroup defined by a 2-cell complex. Only information about finite parabolic subgroups is required in an explicit computation. An analogous result on the normalizer NW(WI) is also mentioned. 1.

We give upper and lower bounds for the Kazhdan-Lusztig polynomials of any Coxeter group W . If W is finite we prove that, for any k 0, the k-th coefficient of the Kazhdan-Lusztig polynomial of two elements u, v of W is bounded from above and below by a polynomial (which depends only on k) in l(v) \Gamma l(u). In particular, this implies the validity of Lascoux-Schutzenberger's conjecture for all sufficiently long intervals, and gives supporting evidence in favor of the Dyer-Lusztig conjecture.