Arkady Berenstein

Total positivity in Schubert varieties

Abstract. We extend the results of [2] on totally positive matrices to totally positive elements in arbitrary semisimple groups.

Tensor product multiplicities and convex polytopes in partition space

Westudy multiplicities in the decompositionof tensorproduct of two lire- ducible finite dimensionalmodules over a semisimple complex Lie algebra. A con- jectural expressionfor such multiplicity is given asthe numberof integralpoints of a certain convexpolytope. Wediscusssome specialcases,corollariesandconfirmations of the conjecture.

GEOMETRIC AND UNIPOTENT CRYSTALS

Let G be a split semisimple algebraic group over ℚ, g be the Lie algebra of G and U q (g) be the corresponding quantized enveloping algebra. Lusztig has introduced in [Lul] canonical bases for finite-dimensional U q(g)-modules. About the same time E as hi war a introduced in [Kl] crystal bases as a natural framework for parametrizing bases of finite-dimensional U q(g)-modules. It was shown in [Lu2] that Eashiwara’s crystal bases are the limits as q → 0 of Lusztig’s canonical bases. Later, in [K2] Kashiwara introduced a new combinatorial concept — crystals. Kashiwara’s crystals generalize the crystal bases and provide a natural framework for their study.

Canonical bases for the quantum group of type

This work was motivated by the following two problems from the classical representation theory. (Both problems make sense for an arbitrary complex semisimple Lie algebra but since we shall deal only with the Ar case, we formulate them in this generality). 1. Construct a “good” basis in every irreducible finite-dimensional slr+1-module Vλ, which “materializes” the Littlewood-Richardson rule. A precise formulation of this problem was given in [3]; we shall explain it in more detail a bit later. 2. Construct a basis in every polynomial representation of GLr+1, such that the maximal element w0 of the Weyl group Sr+1 (considered as an element of GLr+1) acts on this basis by a permutation (up to a sign), and explicitly compute this permutation. This problem is motivated by recent work by John Stembridge [10] and was brought to our attention by his talk at the Jerusalem Combinatorics Conference, May 1993.

A_r

The goal of the paper is to introduce and study symmetric and exterior algebras in certain braided monoidal categories such as the category O for quantum groups. We relate our braided symmetric algebras and braided exterior algebras with their classical counterparts.

and piecewise-linear combinatorics

We construct 2-dimensonal thick nondiscrete affine buildings associated with an arbitrary finite dihedral group.

BRAIDED SYMMETRIC AND EXTERIOR ALGEBRAS

This paper is a first attempt to generalize results of A. Berenstein, S. Fomin, and A. Zelevinsky on total positivity of matrices over commutative rings to matrices over noncommutative rings. The classical theory of total positivity studies matrices whose minors are all nonnegative. Motivated by a surprising connection discovered by Lusztig [10, 11] between total positivity of matrices and canonical bases for quantum groups, Berenstein, Fomin, and Zelevinsky, in a series of papers [1, 2, 3, 4], systematically investigated the problem of total positivity from a representation-theoretic point of view. In particular, they showed that a natural framework for the study of totally positive matrices is provided by the decomposition of a reductive group G into the disjoint union of double Bruhat cells G = BuB∩B−vB−, where B and B− are two opposite Borel subgroups in G, and u and v belong to the Weyl group W of G. According to [1, 3, 4] there, exist families of birational parametrizations of G, one for each reduced expression of the element (u, v) in the Coxeter group W ×W. Every such parametrization can be thought of as a system of local coordinates in G. Such coordinates are called the factorization parameters associated to the reduced expression of (u, v). The coordinates are obtained by expressing a generic element x ∈ G as an element of the maximal torus H = B ∩ B− multiplied by the product of elements of various

Affine buildings for dihedral groups

The goal of the paper is to introduce a version of Schubert calculus for each dihedral reflection group W. That is, to each “sufficiently rich” spherical building Y of type W we associate a certain cohomology theory

Noncommutative double Bruhat cells and their factorizations

H_{BK}^*(Y)

Stability inequalities and universal Schubert calculus of rank 2

and verify that, first, it depends only on W (i.e., all such buildings are “homotopy equivalent”), and second,