Littlewood–Richardson coefficients for reflection groups

Abstract

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 aijaji≥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 Γi(G).