Littlewood-Richardson coefficients for reflection groups

TL;DR

This paper provides explicit formulas for Littlewood-Richardson coefficients in the cohomology of semisimple and Kac-Moody groups, revealing conditions for their nonnegativity and extending results to equivariant cohomology and Bott-Samelson varieties.

Contribution

It introduces a new explicit formula for Littlewood-Richardson coefficients based on the Cartan matrix and Weyl group, with conditions for nonnegativity and extensions to equivariant cohomology.

Findings

Explicit formula for Littlewood-Richardson coefficients in terms of Cartan matrix and Weyl group.

Nonnegativity of coefficients when Cartan matrix entries satisfy certain conditions.

Extension of results to T-equivariant cohomology and Bott-Samelson varieties.

Abstract

In this paper we explicitly compute all Littlewood-Richardson coefficients for semisimple or Kac-Moody groups G, that is, the structure coefficients of the cohomology algebra H^*(G/P), where P is a parabolic subgroup of G. These coefficients are of importance in enumerative geometry, algebraic combinatorics and representation theory. Our formula for the Littlewood-Richardson coefficients is given in terms of the Cartan matrix and the Weyl group of G. However, if some off-diagonal entries of the Cartan matrix are 0 or -1, the formula may contain negative summands. On the other hand, if the Cartan matrix satisfies $a_{ij}a_{ji}\ge 4$ for all $i,j$, then each summand in our formula is nonnegative that implies nonnegativity of all Littlewood-Richardson coefficients. We extend this and other results to the structure coefficients of the T-equivariant cohomology of flag varieties G/P and Bott-Samelson varieties Gamma_\ii(G).