zbMATH — the first resource for mathematics

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)

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
Full Text: DOI