# Moment map, generalized Horn inequalities and quivers (Michèle Vergne, IMJ-PRG, Paris)

23.05.2018 16:30

Work in common with Velleda Baldoni and Michael Walter.

Let $G$ be a complex reductive group acting on a finite-dimensional complex vector space $\CH$. Let $\t^*_{\geq 0}$ be a choice of positive Weyl chamber. The moment map associated to a Hermitian metric on $\CH$ determines a polyhedral cone $\Cone(\CH)$ in $\t^*_{\geq 0}$. It is the cone generated by the $T$-weights of the polynomial functions on $\CH$ which are semi-invariant under the Borel subgroup.

We determine inductively the inequalities of the cone $\Cone(\CH)$ in the case of the linear representation of the group $G=\prod_x \GL(n_x)$ associated to a quiver and a dimension vector $n=(n_x)$. When the quiver has $s$ arrows $x_i\to y$, and the dimension vector is $n_{x_i}=n_y=n$ the problem is the Horn problem: if $A=\sum_{i=1}^s A_i$ is a $n\times n$ Hermitian matrix which is the sum of $s$ Hermitian matrices $A_i$, what are the relations between the eigenvalues of $A$ and $A_i$. In general, we determine inductively the inequalities of the Cone $Cone(\CH)$ in terms of filtered subrepresentations of the quiver $Q$. In particular, this gives yet another proof of Horn inequalities.

### Lieu

Bâtiment: Battelle

Séminaire "Groupes de Lie et espaces des modules"

### Organisé par

Section de mathématiques

### Intervenants

Michèle Vergne, IMJ-PRG, Paris

entrée libre

### Classement

Catégorie: Séminaire