University of Sydney

    School of Mathematics and Statistics

    Algebra Seminar

    Andrew Mathas
    University of Sydney

    Irreducible Specht modules for symmetric groups

    Friday 21st August, 12-1pm, 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:
    1. When the characteristic p is larger than n; this is trivial since the group algebra is still semisimple.
    2. When the corresponding partition is p-regular; this follows from the Carter-Payne theorem and a result of James and Murphy.
    3. When the corresponding partition has only two non-zero parts; in this case, James has determined all of the rows of the decomposition matrix.
    4. When p=2; this is a recent result of James and the speaker.