Steinberg representation

Steinberg representation

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 harvs|txt=yes|authorlink=Robert Steinberg|first=Robert |last=Steinberg|year=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.

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").

Applications

*The Steinberg representation is used in the proof of Haboush's theorem (the Mumford conjecture).

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
*springer|id=S/s130530|title= Steinberg module|first=Robert |last=Steinberg
*citation|first=R. |last=Steinberg|title=Prime power representations of finite linear groups II |journal=Canad. J. Math. |volume= 9 |year=1957|pages=347-351
*R. Steinberg, "Collected Papers" , Amer. Math. Soc. (1997) ISBN 0-8218-0576-2 pp. 580–586
*citation|first=J.E.|last= Humphreys|title=The Steinberg representation|journal= Bull. Amer. Math. Soc. (N.S.) |volume= 16 |year=1987|pages=237–263
url=http://www.ams.org/bull/1987-16-02/S0273-0979-1987-15512-1/home.html
id=MR|876960


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