Fedor Part - Crystal Bases for the Kronecker Problem

Kronecker coefficients arise as multiplicities in decompositions of GL(mn) irreducibles after restricting the GL(mn) action to the subgroup GL(m) x GL(n) embedded via tensor product. A long-standing open problem in algebraic combinatorics asks for a combinatorial rule for these coefficients akin to Littlewood-Richardson rule. Passing to quantum group Uq(gl(mn)) one gets a nice bases on representations called crystal bases which might be useful for the solution of the problem. However the standard quantum version Uq(gl(m)+gl(n)) of the subgroup is no longer Hopf subalgebra of Uq(gl(mn)). I will give an overview of attempts to overcome this and still define and use crystal bases, in particular, via considering nonstandard quantum universal enveloping and Hecke algebras.