<presented by
Tuesday, September 21, 4:00 p.m.
William Fulton
University of Michigan, Ann Arbor
Eigenvalues, Invariant Factors, Highest Weights, and Schubert Calculus
Wednesday, September 22, 5:00 p.m.
Thursday, September 23, 4:00 p.m.
Objects in widely differing areas of mathematics can be classified or described by n-tuples of real numbers:
- eigenvalues of real symmetric (or complex Hermitian) n by n matrices;
- invariant factors of n by n matrices with nonvanishing determinant over a discrete valuation ring (or isomorphism classes of torsion modules with at most n generators over such a ring; for the p-adic integers these are finite abelian p-groups);
- highest weights of finite-dimensional irreducible holomorphic representations of GL(n,C);
- Schubert classes in the cohomology of the Grassmann variety of n-planes in a vector space of larger dimension.
In each case there are natural questions about these invariants:
- What can be the eigenvalues of A, B, and C = A+B?
- What can be the invariant factors of matrices A, B, and C = AB (or the isomorphism types of a torsion module C, a subgroup B, and the quotient group A)?
- What are the highest weights of irreducible representations appearing in the tensor product of two irreducible representations?
- What Schubert classes appear in the product of two Schubert classes?
Some answers to, and relations among these questions had been known before, but recent work of Klyachko, Totaro, Knutson, Tao, Belkale, Woodward, and others have led to new and complete solutions. In each case the answers can in fact be given by induction on the integer n; for the eigenvalue problem this settles an old conjecture of A. Horn. There are related results about the singular values of sums and products of arbitrary matrices.
In these lectures we will explain these problems and discuss their histories, including the recent theorems and some of their relations to other areas of mathematics. The problems are elementary, so most of the lectures should be accessible to graduate students or advanced mathematics majors. Some of the the key proofs involve more advanced notions such as geometric invariant theory from algebraic geometry, but these will appear only in the third lecture.