Contemporary Mathematics

Volume 194, 1996

COMMUTING ACTIONS

- A Tale of Two Groups

GEORGIA BENKART

ABSTRACT.

When two groups or two algebras have commuting actions on

the same space, their representation theories and combinatorics are inti-

mately connected. In this article we will concentrate on three examples to

illustrate the power of commuting actions and will discuss generalizations

of these examples to other settings and various applications of them to such

subjects as the theory of symmetric functions and knot theory.

I. Three Examples

(1.1) Let V

=

pn where F is a field of characteristic zero, and identify the vectors

in V with t:D.e n x

1

matrices over F. The general linear group GLn of n x n

invertible matrices with entries in F acts on V by matrix multiplication and on

the k-fold tensor product

V®k

of

V

by the diagonal action,

g(v1 ® · · · ® vk) =

gvl ® ... ®

9Vk,

making them GLn-modules. The symmetric group

sk

also

acts on

y®k

as place permutations. Since these two actions commute, the space

EndaLn (V181k)

of transformations commuting with the action of the general linear

group on

y®k

contains a homomorphic image of the group algebra

FSk·

When

F

has characteristic zero, the group algebra

FSk

is a semisimple associative

algebra, and it decomposes into a sum,

FSk = EBAf-klA,

of simple ideals

JA

indexed by the partitions . of

k.

Let t'(.) denote the number of nonzero parts

of..

1991 Mathematics Subject Classification. Primary 17B10, 22E46, 81R50; Secondary 05E05,

05E10.

Partially supported by National Science Foundation Grant #DMS-9300523.

©

1996 American Mathematical Society

1

http://dx.doi.org/10.1090/conm/194/02387