Kroenecker coefficients

Given two finite--dimensional vector spaces $V$ and $W$ on $\mathbb{C}$, the group $GL(V) \times GL(W)$ maps naturally to $GL(V \otimes W)$ (in coordinates, this is the ``Kronecker product'' of matrices). This gives to every $GL(V \otimes W)$--module a structure of $GL(V) \times GL(W)$--module. In particular, every irreducible representation of $GL(V \otimes W)$ decomposes as a sum of irreducible $GL(V) \times GL(W)$--modules. The multiplicities in these decompositions are the Kronecker coefficients. Finding a nice combinatorial description of the Kronecker coefficients, akin to the Littlewood--Richardson rule for the Littlewood--Richardson coefficients, is a longstanding open problem in algebraic combinatorics. Recently, Mulmuley has set this problem at the heart of his \emph{Geometric complexity theory}, a geoemtric approach to the algebraic analog of $P \neq NP$. I will present recent results obtained with Rosa Orellana and Emmanuel Briand about a particular family of Kronecker coefficients: the \emph{Kronecker coefficients indexed by three two--row shapes}. We obtained explicit formulas for these coefficients. These formulas are polynomial functions on $\mathbb{Z}^6$ (the indices of the considered Kronecker coefficients are points in this space), whose domains are the intersections $\sigma \cap C$ of the maximal cells $\sigma$ of a fan of $\mathbb{R}^6$ with the cosets $C$ of a full--rank sublattice of $\mathbb{Z}^6$. I will explain how these formulas were obtained, and how they allowed us to disprove a conjecture, due to Mulmuley, about the behaviour of the stretching functions associated to the Kronecker coefficients.