Back to Departmental Colloquium: Fall 2004
Departmental Colloquium
Date: Thursday, Sep 30, 2004
Time: 4:15PM
Location: JWB 335
Distinguished Lecture Series Colloquium
Nilpotent Matrices and the Permutation Group
Title |
Nilpotent Matrices and the Permutation Group |
Abstract |
The problem of classifying n x n matrices over a field k has two aspects. One is arithemetic: any degree n extension field K of k can be embedded in nxn matrices, in a way canonical up to conjugacy. In terms of linear algebra, the corresponding matrices have their eigenvalues in K. For this reason, the arithemtic of k affects their conjugacy classes. I will ignore this aspect entirely. The second aspect is independent of field; one could call it purely “algebraic,” if that word is divorced from arithmetic. There are nonzero n x n matrices all of whose eigenvalues are zero. These are nilpotent matrices. Any nilpotent nxn matrix is conjugate to one in Jordan normal form, and in this way conjugacy classes of nilpotent matrices are in bijection with the partitions of n. More than a hundred years ago, Frobenius discovered exactly the same set (partitions of n ) parameterizes 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 GL(n) and its representations (which are part of algebraic geometry, arithmetic and analysis.) I will describe two examples of this work. The first, due to Green in 1955, shows how to use the symmetric group to understand certain geometrically natural representations of GL(n) over a finite field. He shows that these representations in GL(n) are " q -analogues" of certain natural symmetric group representations, and that all of these representations decompose in exactly the same way. The second example concerns the algebraic variety N of all n x n nilpotent matrices over an algebraically closed field. Fifteen years ago, Lusztig conjectured a very precise relationship between the GL(n) invariant coherent sheaves on N, and some combinatorics related to the symmetric group S n (and its semidirect product with Z n.) Lusztig’s conjecture has been proven by Bezrukavnikov and Achar. |
Distinguished Lecture Series
Number Theory
Representation Theory