Andrei Zelevinsky

Cluster algebras I: Foundations

In an attempt to create an algebraic framework for dual canonical bases and total positivity in semisimple groups, we initiate the study of a new class of commutative algebras.

Cluster algebras II: Finite type classification

This paper continues the study of cluster algebras initiated in math.RT/0104151. Its main result is the complete classification of the cluster algebras of finite type, i.e., those with finitely many clusters. This classification turns out to be identical to the Cartan-Killing classification of semisimple Lie algebras and finite root systems, which is intriguing since in most cases, the symmetry exhibited by the Cartan-Killing type of a cluster algebra is not at all apparent from its geometric origin. The combinatorial structure behind a cluster algebra of finite type is captured by its cluster complex. We identify this complex as the normal fan of a generalized associahedron introduced and studied in hep-th/0111053 and math.CO/0202004. Another essential combinatorial ingredient of our arguments is a new characterization of the Dynkin diagrams.

Double Bruhat cells and total positivity

We study intersections of opposite Bruhat cells in a semisimple complex Lie group, and associated totally nonnegative varieties.

Generalized associahedra via quiver representations

We provide a quiver-theoretic interpretation of certain smooth complete simplicial fans associated to arbitrary finite root systems in recent work of S. Fomin and A. Zelevinsky. The main properties of these fans then become easy consequences of the known facts about tilting modules due to K. Bongartz, D. Happel and C. M. Ringel.

Total positivity: Tests and parametrizations

An introduction to total positivity (TP), with the emphasis on efficient TP criteria and parametrizations of TP matrices. Intended for general mathematical audience.

Total positivity in Schubert varieties

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

Triple Multiplicities for sl (r + 1) and the Spectrum of the Exterior Algebra of the Adjoint Representation

A new combinatorial expression is given for the dimension of the space of invariants in the tensor product of three irreducible finite dimensional sl(r + 1)-modules (we call this dimension the triple multiplicity). This expression exhibits a lot of symmetries that are not clear from the classical expression given by the Littlewood–Richardson rule. In our approach the triple multiplicity is given as the number of integral points of the section of a certain “universal” polyhedral convex cone by a plane determined by three highest weights. This allows us to study triple multiplicities using ideas from linear programming. As an application of this method, we prove a conjecture of B. Kostant that describes all irreducible constituents of the exterior algebra of the adjoint sl(r + 1)-module.

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.

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 paper is motivated by an analogy between cluster algebras and Kac-Moody algebras: both theories share the same classification of finite type objects by familiar Cartan-Killing types. However, the underlying combinatorics beyond the two classifications is different: roughly speaking, Kac-Moody algebras are associated with (symmetrizable) Cartan matrices, while cluster algebras correspond to skew-symmetrizable matrices. We study an interplay between the two classes of matrices, in particular, establishing a new criterion for deciding whether a given skew-symmetrizable matrix gives rise to a cluster algebra of finite type.