Séance Séminaire

Séminaire Algèbre Géométrie Algébrique Topologie Algébrique

Théorie des invariants géométriques et problème de valeurs propres.

Let ${\mathfrak su}(n)$ denote the algebra of ${\rm SU}(n)$, and $ {\mathfrak t}=\{(\lambda_1,\cdots,\lambda_n)\in{\mathbb R}^n\ |\ \sum\lambda_i=0\}$, its Cartan subalgebra. Let $ {\mathfrak t}_+=\{(\lambda_1,\cdots,\lambda_n)\in{\mathbb R}^n\ | \lambda_i\geq \lambda_{i+1}, i=1,\cdots,n-1\} $ be a choice of closed positive Weyl chamber. For any matrix $A\in{\mathfrak su}(n)$ let $ \lambda(A)=(\lambda_1(A),\cdots,\lambda_n(A))\in{\mathfrak t}_+ $ be the spectrum of the Hermitian matrix $-iA$. Let $\Delta(l)\subset {(\mathfrak t}_+)^l$ denote the set $ \Delta(l)=\{(\lambda(A_1),\cdots,\lambda(A_l))\ |\ A_1,\cdots,A_l\in {\mathfrak su}(n),\,A_1+\cdots+A_l=0\} $. The set $\Delta(l)$ has several interesting interpretations. Firstly, it is the moment polytope of the cotangent bundle $T^*{\rm SU}(n)^{l-1}$ endowed with a natural ${\rm SU}(n)^l$-action. In particular, it is a convex polyedral cone defined by finitely many linear inequalities. The cone $\Delta(l)$ may also be described in terms of the tensor products of representations of ${\rm SU}(n)$ or ${\rm SL}(n)$. It may also be described in terms of Geometric Invariant Theory (GIT) for the diagonal action of ${\rm SL}(n)$ on the product ${\mathcal F}^l$ of $l$ copies of the variety of complete flags of ${\mathbb C}^n$. The question to determine explicitely the inequalities fullfilled by the points of $\Delta(l)$ began with H.Weyl in 1912 and has a rich history. Recently, Belkale and Kummar have proposed a list of inequalities which characterize the cone $\Delta(l)$ (and its genralisations for the others semi-simple groups) paramitrized by a condition expressed in terms of a new product on the cohomology group of the (generalized) flag varieties. Here, we assert that the list of Belkale and Kummar is minimal. The proof is made using the point of view of the GIT.