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Nilpotent matrices and the permuation group

David Vogan
Massachusetts Institute of Technology

Thursday, November 4, 2004
102 Bradley Hall, 4 pm
Tea 3:30 pm, Math Lounge

Abstract: The problem of classifying conjugacy classes of n x n matrices over a field k has two aspects. One is arithmetic: any degree-n extension field K of k can be embedded in n x n matrices, in a way that is canonical up to conjugacy. In terms of linear algebra, the corresponding matrices have their eigenvalues in K. For this reason, the arithmetic of k affects conjugacy classes. I will ignore this aspect almost entirely.

The second aspect is independent of the field; one could call it purely ``algebraic," if that word is divorced from arithmetic. There are non-zero n x n matrices all of whose eigenvalues are zero. These are the nilpotent matrices. Any nilpotent n x n matrix is conjugate to one in Jordan normal form, and in this way conjugacy classes of nilpotent matrices are in bijection with partitions of n.

More than a hundred years ago, Frobenius discovered that exactly the same set---partitions of n---parametrizes the irreducible representations of the symmetric group S_n. Since that time, there has been a tremendous amount of work aimed at using information about S_n and its representations (which are a part of combinatorics and finite mathematics) to study the group GL(n) and its representations (which are a part of algebraic geometry, arithmetic, and analysis).

I'll describe a classical example of this work, due to Green in 1955. The starting point is the analogies

{S_n acting on {1,2,...,n}} \leftrightarrow {GL(n) acting on projective space}

{S_n on k-element subsets of {1,2,...,n}} \leftrightarrow {GL(n) on the Grassmannian of k-planes}.

Green developed these analogies in the case of the general linear group over a finite field with q elements. He showed that the corresponding representations of GL(n) are ``q-analogues" of symmetric group representations, and that all of these representations decompose in exactly the same way.

I'll explain Green's work with an eye toward later extensions to the case of GL(n) over the real numbers and other infinite fields.

This talk should be entirely accessible to grad students and a lot of it works for undergrads who know about groups and homogeneous spaces.