A combinatorial model for irreducible representations

Alexander Postnikov, MIT                                                                                
April 15, 2005

Seminar "Geometric Methods in Representation Theory"
University of North Carolina, Chapel Hill

Abstract:
                                                                                
     We  present  a  combinatorial  model  for  the  characters  of  the
irreducible  representations  of  semi-simple  Lie  algebras  and,  more
generally,  of  Kac-Moody algebras.   This  model  can  be viewed  as  a
discrete counterpart to the Littelmann path model.  We use this model to
give an explicit Chevalley-type formula  for the equivariant K-theory of
generalized flag varieties G/P.  The construction is given in terms of a
certain R-matrix that satisfies the  Yang-Baxter equation.  The model is
based on  enumeration of  saturated chains  in the  Bruhat order  on the
Weyl  group  and combinatorics  of  reduced  decompositions of  elements
in  the  associated  affine  Weyl  group.   The  model  implies  several
symmetries of  coefficients in  the equivariant  K-theory.  We  deduce a
combinatorial Littlewood-Richardson rule for decomposing tensor products
of irreducible representations and a  branching rule.  The talk is based
on a joint work with Cristian Lenart.