ebriand@us.esWe consider classical structural constants in the representation theory of the general linear groups GL(n,C): Littlewood-Richardson coefficients and Kronecker coefficients. They describe the decomposition into irreducible of, respectively: the tensor products of irreducibles; and the restrictions from GL(mn,C) to GL(m,C)× GL(n,C). It is known that they are given by piecewise quasipolynomial (polynomial with periodic coefficients) formulas. The domains of quasipolynomiality are the maximal cells of a fan (``chamber complex'') subdividing a cone.
These chamber complexes are quite mysterious, For instance, very little is known about their size and structure in general.
I will present explicit calculations on cases small enough to allow it. The observations based on the outcomes of these calculations lead to determining new symmetries of some of these coefficients, and suggest interesting relations between piecewise-quasipolynomiality and representation stability.
This talk is based on several works in collaboration with Mercedes Rosas and Stefan Trandafir.