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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.