Klyachko, Alexander A. Stable bundles, representation theory and Hermitian operators. (English) Zbl 0915.14010 Sel. Math., New Ser. 4, No. 3, 419-445 (1998). 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) Cited in 26 ReviewsCited in 121 Documents 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 Keywords:Hermitian operator; Hermite-Einstein metrics; tensor products; moduli space of toric vector bundles; representations of general linear group PDF BibTeX XML Cite \textit{A. A. Klyachko}, Sel. Math., New Ser. 4, No. 3, 419--445 (1998; Zbl 0915.14010) Full Text: DOI Link OpenURL