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The goal of my talk (based on a recent joint paper with Edward Richmond) is to compute the Littlewood-Richardson (LR) 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 LR coefficients is purely combinatorial and is given in terms of the Cartan matrix and the Weyl group of G. In particular, our formula gives a combinatorial proof of positivity of the LR coefficients in the cases when off-diagonal Cartan matrix entries are less than or equal to -2.