## Stable bundles, representation theory and Hermitian operators.(English)Zbl 0915.14010

Let $$E$$ be a unitary vector space of dimension $$n$$ and let $$A:E\to E$$ be a Hermitian operator. The spectrum of $$A$$ is a set of $$n$$ real numbers $\lambda(A)=\bigl\{ \lambda_1(A)\geq\lambda_2(A)\geq \cdots\geq\lambda_n(A) \bigr\}.$ If $$B:E\to E$$ is another Hermitian operator one wants to understand the relation between the spectra $$\lambda(A)$$, $$\lambda(B)$$ and $$\lambda(A+B)$$. It turns out that all the relations are of the form $\sum_{i\in K} \lambda_i(A+B) \leq \sum_{i\in I} \lambda_i(A) + \sum_{i\in J} \lambda_i(B)$ for certain subsets $$I,J,K\subset\{1,\dots,n\}$$ and the author gives an explicit description of those triples of subsets.
Now let $$\alpha=\{ \alpha_1\geq\alpha_2\geq\cdots\geq \alpha_n \}$$ be a set of non-negative integers and denote by $$V(\alpha)$$ the irreducible representation of $$GL_n({\mathbb{C}})$$ with highest weight $$\alpha$$. The author shows that for any triple of highest weights $$(\alpha,\beta,\gamma)$$, the representation $$V(m\gamma)$$ is contained in $$V(m\alpha)\otimes V(m\beta)$$ for some positive integer $$m$$, if and only if there exists Hermitian operators $$A$$ and $$B$$ such that $$\alpha=\lambda(A)$$, $$\beta=\lambda(B)$$ and $$\gamma=\lambda(A+B)$$.
The relation between the tensor product decomposition of irreducible $$GL_n({\mathbb{C}})$$-modules and the spectra of Hermitian operators comes from the analysis of stability of toric vector bundles over the projective plane $${\mathbb{P}}^2({\mathbb{C}})$$ and the paper also contains some results concerning the moduli space of such bundles.
Reviewer: St.Helmke (Kyoto)

### MSC:

 14F05 Sheaves, derived categories of sheaves, etc. (MSC2010) 20G05 Representation theory for linear algebraic groups 14M25 Toric varieties, Newton polyhedra, Okounkov bodies 15A03 Vector spaces, linear dependence, rank, lineability
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