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
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