Steinberg representation
From Wikipedia, the free encyclopedia
In mathematics, the Steinberg representation, or Steinberg module, denoted by St, is a particular linear representation of a group of Lie type over a finite field of characteristic p, of degree equal to the largest power of p dividing the order of the group. These representations were discovered by Robert Steinberg (1957).
Most finite simple groups have exactly one Steinberg representation. A few have more than one because they are groups of Lie type in more than one way, and sporadic and most alternating groups have no Steinberg representation.
[edit] Properties
- The character value of St on an element g equals, up to sign, the order of a Sylow subgroup of the centralizer of g if g has order prime to p, and is zero if the order of g is divisible by p.
- The Steinberg representation is equal to an alternating sum over all parabolic subgroups containing a Borel subgroup, of the representation induced from the identity representation of the parabolic subgroup.
- The Steinberg representation is both regular and unipotent, and is the only irreducible regular unipotent representation (for the given prime p).
[edit] Applications
- The Steinberg representation is used in the proof of Haboush's theorem (the Mumford conjecture).
[edit] References
- Finite Groups of Lie Type: Conjugacy Classes and Complex Characters (Wiley Classics Library) by Roger W. Carter, John Wiley & Sons Inc; New Ed edition (August 1993) ISBN 0-471-94109-3
- Steinberg, Robert (2001), “Steinberg module”, in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Kluwer Academic Publishers, ISBN 978-1556080104
- Steinberg, R. (1957), “Prime power representations of finite linear groups II”, Canad. J. Math. 9: 347-351
- R. Steinberg, Collected Papers , Amer. Math. Soc. (1997) ISBN 0-8218-0576-2 pp. 580–586
- Humphreys, J.E. (1987), “The Steinberg representation”, Bull. Amer. Math. Soc. (N.S.) 16: 237–263, MR876960, <http://www.ams.org/bull/1987-16-02/S0273-0979-1987-15512-1/home.html>
