Saturated Horn inequalities for submodules and C_0 operators

par Wing Suet Li (Georgia Institute of Technology)

mardi 17 mai 2016, 14:00 → 15:00 Europe/Paris

Fokko du Cloux (Université Claude Bernard Lyon 1 - Campus de la Doua, Bâtiment Braconnier)

Consider three partitions of integers $\alpha=(\alpha_1\ge\alpha_2\ge\dots\ge\alpha_n\ge 0)$, $\beta=(\beta_1\ge\beta_2\ge\dots\ge \beta_n\ge 0)$, and $\gamma=(\gamma_1\ge\gamma_2\ge\dots\ge\gamma_n\ge 0)$. Such a triple of partitions $(\alpha,\beta,\gamma)$ that satisfies the so-called Horn inequalities, a set of inequalities conjectured by A. Horn in 1960 and later the conjecture was proved by the work of Klyachko and Knutson-Tao, describes the eigenvalues of the sum of $n$ by $n$ Hermitian matrices, i.e., Hermitian matrices $A$, $B$, and $C=A+B$ with eigenvalues $\alpha$, $\beta$, and $\gamma$ respectively. Such triple also describes the Jordan decompositions of a nilpotent matrix $T$, $T$ resticted to an invariant subspace $\cal{M}$, and $T$ compressed to $\cal{M}^{\perp}$. More precisely, $T$ is similar to $J(\gamma):=J_{\gamma_1}\oplus J_{\gamma_2}\oplus\cdots\oplus J_{\gamma_n}$, and $T|\cal{M}$ is similar to $J(\alpha)$ and $T_{{\cal{M}^{\perp}}}$ is similar to $J(\beta)$. (Here $J_k$ denotes the Jordan cell of size $k$ with $0$ on the diagonal.) This result for nilpotent matrices also has an analogue for operators in the class of $C_0$. In this talk I will explain, through the intersection of certain Schubert varieties, why the same combinatorics solves the eigenvalue and the Jordan form problems. I will also describe the additional information that we can obtain whenever a Horn inequality saturates.