Vector Bundles and Hermitian Operators

Alexander A. Klyachko

ABSTRACT

We will 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 GLn(C);
  3. Structure of the moduli space of stable bundles on the projective plane P2.
1. Hermitian operators. We begin with the first subject. Let A: E® E be a Hermitian operator in a unitary space E of finite dimension n and let
l(A)  :   l1(A) ³ l2(A) ³ ... ³ ln(A)
be its spectrum. The problem is to find all restrictions on spectra l(A), l(B) and l(A+B). First of all, we have the trace identity

S
i 
li(A+B)  =
S
i 
li(A) +
S
i 
li(B)
and a number of classical inequalities, all of the form
(IJK)

S
k Î K 
lk(A+B)  £
S
i Î I 
li(A) +
S
j Î J 
lj(B)
for some triple of subsets I,J,K Ì {1,2,...,n} of the same cardinality.

Let us fix a decomposition n = p+q. We have a bijection between subsets I Ì {1,2,...,n} of cardinality p = |I| and Young diagrams s = sI in a rectangular box of dimension p (North) by q (East) given as follows. Let G = GI 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 Î I and to the East otherwise. The line G = GI cuts out from the box a Young diagram s = sI Ì p x q situated in its North-West angle. The diagram sI in the usual way corresponds to a Schubert cycle sI in a Chow ring of the Grassmannian.

1.1. Theorem. Consider a triple of subsets I,J,K Ì {1,2,...,n} such that the Schubert cycle sK is a component of sI·sJ. Then

  1. The inequality (IJK) holds.
  2. 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
a  :  a1 ³ a2 ³ ... ³ an,    ai Î Z,
and associate with it the following dominant weight of general linear group GLn(C)
wa  :   diag(x1,  x2,  ...,  xn)   ®   x1a1  x2a2  ...  xnan.
The weight wa in the usual way corresponds to an irreducible representation V(wa) of the group GLn(C) with highest weight wa.

We are iterested in the problem: which irreducible representations V(wg) are components of tensor product V(wa)ÄV(wb)? The result is that the inequalities (IJK) of Theorem 1.1 answer this question too. More precisely,
2.1. Theorem. The irreducible representation V(wNg) is a component of tensor product V(wNa) ÄV(wN b) for some positive N if and only if a,b and g are spectra of Hermitian operators A, B and C = A+B.

3. Vector bundles. It will be explained in the lecture that both previous theorems follow from the existence of Hermite-Einstein metrics on stable toric vector bundles on P2.

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Created by Alexandru I. Suciu, Saturday, Sept. 5, 1998
alexsuciu@neu.edu
http://www.math.neu.edu/bhmn/klyachko_abs.html