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 in about 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] References
- Steinberg module at the Encyclopaedia of Mathematics
- R. Steinberg, Prime power representations of finite linear groups II Canad. J. Math. , 9 (1957) pp. 347–351
- R. Steinberg, Collected Papers , Amer. Math. Soc. (1997) ISBN 0-8218-0576-2 pp. 580–586
- J.E. Humphreys, The Steinberg representation Bull. Amer. Math. Soc. (N.S.) , 16 (1987) pp. 237–263
- 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
