From: George J McNinch <McNinch.1@nd.edu>
Newsgroups: sci.math.research
Subject: Re: symplectic group
Date: 29 Dec 1998 10:48:07 -0500
Keywords: Brauer characters of symplectic groups in characteristic  p

>>>>> "Siman" == Siman Wong <siman@math.mit.edu> writes:

    Siman> Let p = prime F_p = finite field with p elements

    Siman> Question: Where can I look up the Brauer characters of the
    Siman> symplectic group Sp(2n, F_p) in characteristic p?

The simple representations in characteristic p of Sp(2n,F_p) may be
obtained by regarding Sp(2n,F_p) as a subgroup of the simple algebraic
group (of type C_n) Sp(2n,k), where k is an algebraic closure of
F_p. The simple modules of Sp(2n,F_p) are the precisely the
restrictions to Sp(2n,F_p) of the "restricted" simple rational
representations for this algebraic group; there are p^n such simple
representations, and in general their _dimensions_ are not even known.

Lusztig's conjectures give the formal characters in terms of
Kazhdan-Lusztig polynomials associated with the affine Weyl group of
type C_n; these conjectures are known to hold for p>>0 but no specific
value of p is known to be sufficient (results of H. Andersen,
J. Janzten, and W. Soergel; see their Asterisque volume of, I believe,
1994).

    Siman> I am particularly interested in the degree n characters of
    Siman> Sp(2n, F_p).

Any representation of dimension n of Sp(2n,F_p) is trivial; the
smallest non-trivial simple representation is the "natural" symplectic
representation of dimension 2n. There is only once representation of
that dimension; I think the next smallest has dimension n(n-1)/2 - e
where e=1 if (n,p)=1 and e=2 otherwise.

Best,
George McNinch
==============================================================================

From: George J McNinch <McNinch.1@nd.edu>
Newsgroups: sci.math.research
Subject: Re: symplectic group
Date: 30 Dec 1998 09:02:46 -0500

>>>>> "George" == George J McNinch <McNinch.1@nd.edu> writes:

    George> Any representation of dimension n of Sp(2n,F_p) is
    George> trivial; the smallest non-trivial simple representation is
    George> the "natural" symplectic representation of dimension
    George> 2n. There is only once representation of that dimension; I
    George> think the next smallest has dimension n(n-1)/2 - e where
    George> e=1 if (n,p)=1 and e=2 otherwise.

Oops; that "next smallest" dimension should be instead:

  n(2n -1) - e   with e as indicated above.

(This simple module comes from the exterior square of the natural
module, so its dimension is roughly "2n choose 2"; e is determined by
the number of trivial composition factors of the exterior square).

If n=2 and p=2 I loks like I fibbed that there is only one simple
representation of Sp(2n,F_p) of dimension 4; for n>2 or p>2 that
assertion is correct, though.