mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, the Steinberg representation, or Steinberg module or Steinberg character, denoted by ''St'', is a particular

linear representation Representation theory is a branch of mathematics that studies abstract algebraic structures by ''representing'' their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures. In essenc ...

of a

reductive algebraic group In mathematics, a reductive group is a type of linear algebraic group over a field. One definition is that a connected linear algebraic group ''G'' over a perfect field is reductive if it has a representation with finite kernel which is a direc ...

over a

finite field In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subtr ...

local field In mathematics, a field ''K'' is called a (non-Archimedean) local field if it is complete with respect to a topology induced by a discrete valuation ''v'' and if its residue field ''k'' is finite. Equivalently, a local field is a locally compact t ...

, or a group with a

BN-pair In mathematics, a (''B'', ''N'') pair is a structure on groups of Lie type that allows one to give uniform proofs of many results, instead of giving a large number of case-by-case proofs. Roughly speaking, it shows that all such groups are similar ...

. It is analogous to the 1-dimensional

sign representation A sign is an object, quality, event, or entity whose presence or occurrence indicates the probable presence or occurrence of something else. A natural sign bears a causal relation to its object—for instance, thunder is a sign of storm, or me ...

ε of a

Coxeter Harold Scott MacDonald "Donald" Coxeter, (9 February 1907 – 31 March 2003) was a British and later also Canadian geometer. He is regarded as one of the greatest geometers of the 20th century. Biography Coxeter was born in Kensington to ...

Weyl group In mathematics, in particular the theory of Lie algebras, the Weyl group (named after Hermann Weyl) of a root system Φ is a subgroup of the isometry group of that root system. Specifically, it is the subgroup which is generated by reflections th ...

that takes all reflections to –1. For groups over finite fields, these representations were introduced by , first for the general linear groups, then for classical groups, and then for all

Chevalley group In mathematics, specifically in group theory, the phrase ''group of Lie type'' usually refers to finite groups that are closely related to the group of rational points of a reductive linear algebraic group with values in a finite field. The phras ...

s, with a construction that immediately generalized to the other groups of Lie type that were discovered soon after by Steinberg, Suzuki and Ree. Over a finite field of characteristic ''p'', the Steinberg representation has degree equal to the largest power of ''p'' dividing the order of the group. The Steinberg representation is the Alvis–Curtis dual of the trivial 1-dimensional representation. , , and defined analogous Steinberg representations (sometimes called special representations) for algebraic groups over

s. For the

general linear group In mathematics, the general linear group of degree ''n'' is the set of invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again invertible, ...

GL(2), the dimension of the

Jacquet module In mathematics, the Jacquet module is a module used in the study of automorphic representations. The Jacquet functor is the functor that sends a linear representation to its Jacquet module. They are both named after Hervé Jacquet. Definition The ...

of a special representation is always one.

The Steinberg representation of a finite group

*The character value of ''St'' on an element ''g'' equals, up to sign, the order of a

Sylow subgroup In mathematics, specifically in the field of finite group theory, the Sylow theorems are a collection of theorems named after the Norwegian mathematician Peter Ludwig Sylow that give detailed information about the number of subgroups of fixed ...

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 subgroup In the theory of algebraic groups, a Borel subgroup of an algebraic group ''G'' is a maximal Zariski closed and connected solvable algebraic subgroup. For example, in the general linear group ''GLn'' (''n x n'' invertible matrices), the subgroup ...

s containing a

Borel subgroup In the theory of algebraic groups, a Borel subgroup of an algebraic group ''G'' is a maximal Zariski closed and connected solvable algebraic subgroup. For example, in the general linear group ''GLn'' (''n x n'' invertible matrices), the subgroup ...

, of the representation induced from the identity representation of the parabolic subgroup. *The Steinberg representation is both regular and

unipotent In mathematics, a unipotent element ''r'' of a ring ''R'' is one such that ''r'' − 1 is a nilpotent element; in other words, (''r'' − 1)''n'' is zero for some ''n''. In particular, a square matrix ''M'' is a unipotent ...

, and is the only irreducible regular unipotent representation (for the given prime ''p''). *The Steinberg representation is used in the proof of

Haboush's theorem In mathematics Haboush's theorem, often still referred to as the Mumford conjecture, states that for any semisimple algebraic group ''G'' over a field ''K'', and for any linear representation ρ of ''G'' on a ''K''-vector space ''V'', given ''v''& ...

(the Mumford conjecture). 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. For symmetric groups (and other Coxeter groups) the sign representation is analogous to the Steinberg representation. Some of the sporadic simple groups act as doubly transitive permutation groups so have a BN-pair for which one can define a Steinberg representation, but for most of the sporadic groups there is no known analogue of it.

The Steinberg representation of a ''p''-adic group

, , and introduced Steinberg representations for algebraic groups over

s. showed that the different ways of defining Steinberg representations are equivalent. and showed how to realize the Steinberg representation in the cohomology group ''H''(''X'') of the

Bruhat–Tits building In mathematics, a building (also Tits building, named after Jacques Tits) is a combinatorial and geometric structure which simultaneously generalizes certain aspects of flag manifolds, finite projective planes, and Riemannian symmetric spaces. Bu ...

of the group.

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) * * * * * * * * *R. Steinberg, ''Collected Papers'', Amer. Math. Soc. (1997) 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 , mr=876960, doi=10.1090/S0273-0979-1987-15512-1, issue=2, doi-access=free Representation theory of algebraic groups Finite fields