S Viswanath: Saturation for refined Littlewood-Richardson coefficients-1

The Littlewood-Richardson (LR) coefficients are the multiplicities of irreducible representations occurring in the tensor product of two irreducible polynomial representations of GL_n. To each permutation 'w' in S_n, we associate a 'w-refinement' of the LR coefficients. These correspond to multiplicities in the so-called Kostant-Kumar submodules of the tensor product, or equivalently of multiplicities in "excellent filtrations" of Demazure modules. We prove a saturation theorem for these w-refinements when 'w' is 312-avoiding or 231-avoiding, by adapting the proof via hives of the classical saturation conjecture due to Knutson-Tao. This is a report of work-in-progress with Mrigendra Singh Kushwaha and KN Raghavan. This talk will span two seminar days (Oct 1 and 8). In the first part, we describe the setting of the problem and the result. In the second part, we recall the key steps in the Knutson-Tao proof of the saturation conjecture via hives and indicate how it can be adapted to our case.

[rachidnajib]

Hello, i have to solve this following problem, E is finite linear space over finite field, prove that the number of invariant subspace by nilpotent operator is polynomial p(q) where q=card of the field, and p(0)=n+1, n=dimE, can you give me a reference or hint to solve this question. Thank you