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

Klyachko, Alexander A.

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)

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