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Vector bundles and Hermitian operators Alexander A. Klyachko   Abstract. We'll deal with three apparently disjoint problems:

1.

The spectrum of a sum of Hermitian operators,

2.

Components of tensor product of irreducible representations of the group $\mathrm{GL}_n(\mathbb C)$,

3.

Structure of the moduli space of stable bundles on the projective plane $\mathbb P^2$.

1. Hermitian operators   We begin with the first subject. Let $A:E\rightarrow E$ be a Hermitian operator in a unitary space E of finite dimension n and let $$\lambda(A)\;:\;\lambda_1(A)\ge \lambda_2(A)\ge\cdots\ge\lambda_n(A)$$ be its spectrum. The problem is to find all restrictions on spectra $\lambda(A),\lambda(B)$ and $\lambda(A+B)$. First of all we have trace identity $$\sum_i\lambda_i(A+B)=\sum_i\lambda_i(A)+\sum_i\lambda_i(B)$$ and a number of classical inequalities, all of the form

$$\sum_{k\in K}\lambda_k(A+B)\le\sum_{i\in I}\lambda_i(A)+\sum_{j\inJ}\lambda_j(B) \tag IJK$$ (IJK)

for some triple of subsets $I,J,K\subset \{1,2,\ldots,n\}$ of the same cardinality.

Let us fix a decomposition n=p+q. We have a bijection between subsets $I\subset \{1,2,\ldots,n\}$ of cardinality p=|I| and Young diagrams $\sigma=\sigma_I$ in a rectangular box of dimension p (North) by q (East) given as follows. Let $\Gamma=\Gamma_I$ be a polygonal line with unit edges that runs from the South-West corner of the box to the East-North corner with the i-th edge running to the North for $i\in I$ and to the East otherwise. The line $\Gamma=\Gamma_I$ cuts out from the box a Young diagram $\sigma=\sigma_I\subset p\times q$ situated in its North-West angle. The diagram $\sigma_I$ in the usual way corresponds to a Schubert cycle s_I in a Chow ring of the Grassmannian.

Theorem 1.1. Consider a triple of subsets $I,J,K\subset \{1,2,\ldots,n\}$ such that the Schubert cycle s_K is a component of $s_I\cdot s_J$. Then

i)

The inequality (IJK) holds.

ii)

In union with the trace identity, this inequalities form a complete set of restrictions on spectra of A,B and A+B.

2. Tensor Products

Let us consider an integer spectrum $$\alpha\;:\; a_1\ge a_2\ge\cdots \ge a_n,\;\;a_i\in \mathbb Z,$$ and associate with it the following dominant weight of general linear group $GL_n(\mathbb C)$ $$\omega^\alpha:\mathrm{diag}(x_1.x_2,\dots,x_n)\mapstox_1^{a_1}x_2^{a_2}\cdots x_n^{a_n}.$$ The weight $\omega^\alpha$ in the usual way corresponds to an irreducible representation $V(\omega^\alpha)$ of the group $GL_n(\mathbb C)$ with highest weight $\omega^\alpha$.

 We are iterested in the problem: which irreducible representations $V(\omega^\gamma)$ are components of tensor product $V(\omega^\alpha)\otimes V(\omega^\beta)$? The result is that the inequalities (IJK) of Theorem 1.1 answer this question too. More precisely,

Theorem 2.1. The irreducible representation $V(\omega^{N\gamma})$ is a component of tensor product $V(\omega^{N\alpha})\otimes V(\omega^{N\beta})$ for some positive N if and only if $\alpha,\beta$ and $\gamma$ are spectra of Hermitian operators A,B and C=A+B.

3. Vector bundles It will be explained in the lecture that both of the previouse theorems follows from existence of Hermite-Einstein metric on stable toric vector bundles on $\mathbb P^2$.

9/8/1998