
School of Mathematics and Statistics
Andrew Mathas
University of Sydney
Irreducible Specht modules for symmetric groups
Friday 21st August, 121pm, Carslaw 273.
The ordinary irreducible representation of the symmetric group
of degree n are known as the Specht modules; they
are indexed by partitions of n.
James showed that every irreducible representation of
the symmetric group in characteristic p can be obtained
in a unique way as the head of the reduction mod p of some
Specht module.
In this talk we ask, and partially answer, the following question:
When does a Specht module remain irreducible when
reduced mod p?
It turns out that the answer is known in full only in the following
four cases:
 When the characteristic p is larger than n;
this is trivial since the group algebra is still semisimple.
 When the corresponding partition is pregular; this follows from
the CarterPayne theorem and a result of James and Murphy.
 When the corresponding partition has only two nonzero parts; in this
case, James has determined all of the rows of the decomposition matrix.
 When p=2; this is a recent result of James and the speaker.
