Littlewood-Richardson coefficients for reflection groups

In this paper we explicitly compute all Littlewood-Richardson coefficients for semisimple and Kac-Moody groups G, that is, the structure constants (also known as the Schubert structure constants) of the cohomology algebra H*(G/P, C), 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 purely combinatorial and 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) >= 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 GIP and Bott Samelson varieties Gamma(i)(G). (C) 2015 Elsevier Inc. All rights reserved.