Littlewood-Richardson coefficients for reflection groups

Details Littlewood-Richardson coefficients for reflection groups Littlewood-Richardson coefficients for reflection groups

2010-12-08

arxiv.org/abs/1012.1714v4

Mathematics

Representation Theory

51 pages, AMSLaTeX, new results added (e.g., Theorem 2.9), references added, typos corrected

Scientific paper

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).

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