From: George J McNinch 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 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 Newsgroups: sci.math.research Subject: Re: symplectic group Date: 30 Dec 1998 09:02:46 -0500 >>>>> "George" == George J McNinch 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.