Steinberg module
Let , the group of all invertible ( n \times n )-matrices over the finite field \mathbf{F} _ { q } with q elements and characteristic p, let B be the subgroup of all superdiagonal elements, let U be the subgroup of elements of B whose diagonal entries are all 1, and let W be the subgroup of permutation matrices. In the group algebra k [ G ] of G over any field k of characteristic 0 or p, the element
\begin{equation*} e = \frac { | U | } { | G | } \left( \sum _ { b \in B } b \right) \left( \sum _ { w \in W } \operatorname { sign } ( w ) w \right) \end{equation*}
is an idempotent, called the Steinberg idempotent, and the G-module that it generates in k [ G ] by right multiplication is called the Steinberg module (see [a8]) and is commonly denoted \operatorname{St} (as are all modules isomorphic to it). A similar construction holds for any finite group G of Lie type (and for any B N-pair, which is an axiomatic generalization due to J. Tits) defined over a field of characteristic p with B replaced by a Borel subgroup (which is a certain kind of solvable subgroup), U by a maximal unipotent subgroup (cf. Unipotent group) of B (which is also a Sylow p-subgroup of G; cf. also Sylow subgroup; p-group) and W by the corresponding Weyl group. \operatorname{St} is always irreducible and it has \{ e u : u \in U \} as a basis, so that its dimension is | U | (see [a8]). Its character values are given as follows [a3]. If x \in G has order prime to p, then \chi ( x ) equals, up to a sign which can be determined, the order of a Sylow p-subgroup of the centralizer of x; otherwise it equals 0.
In case the characteristic of k equals p, \operatorname{St} has the following further properties [a5]. It is the only module (for G) which is both irreducible and projective. As an irreducible module it is the largest (in dimension), and as a projective module it is the smallest since it is a tensor factor of every projective module. It follows that it is also self-dual and that every projective module is also injective and vice versa. Because of these remarkable properties, \operatorname{St} plays a prominent role in ongoing work in the still (2000) unresolved problem of determining all of the irreducible G-modules (with characteristic k still equal to p), or equivalently, as it turns out, of determining all of the irreducible rational \overline { G }-modules, where \overline { G } is the algebraic group obtained from G by replacing \mathbf{F} _ { q } by its algebraic closure \mathbf{F} _ { q }, i.e., where \overline { G } is any simple affine algebraic group of characteristic p (see [a6]). This equivalence comes from the fact that every irreducible G-module extends to a rational \overline { G }-module. In particular, \operatorname{St} extends to the \overline { G }-module with highest weight q - 1 times the sum of the fundamental weights, which is accordingly also denoted \operatorname{St}, or \operatorname{St} _ { q } since there is one such \overline { G }-module for each q = p , p ^ { 2 } , p ^ { 3 } , . .. These modules are ubiquitous in the module theory of \overline { G } and figure prominently, for example, in the proofs of many cohomological vanishing theorems and in W. Haboush's proof of the Mumford hypothesis (see [a4]).
Back in the finite case, some other constructions of \operatorname{St}, with the characteristic of k now equal to 0, are as follows. According to C.W. Curtis [a2]
\begin{equation*} \text{St} = \sum _ { P } \pm 1 _ { P } ^ { G }, \end{equation*}
in which P runs through the 2 ^ { r } (r equal to the rank of G) (parabolic) subgroups of G containing B, 1 ^{ G } _ { P } is the G-module induced by the trivial P-module, and the + or - is used according as the rank r _ { P } of P is even or odd. For G = \operatorname{GL} _ { n } ( \mathbf{F} _ { q } ), for example, there is one P for each solution of n = a _ { 1 } + \ldots + a _ { s } (1 \leq s \leq n, each a _ { i } \geq 1); it consists of all of the elements of G that are superdiagonal in the corresponding block matrix form. A third construction, due to L. Solomon and Tits [a7], yields \operatorname{St} as the top homology space H _ { r - 1 } ( C ) for the Tits simplicial complex or Tits building C of G, formed as follows: corresponding to each parabolic subgroup P there exists an ( r - r _ { P } - 1 )-simplex S _ { P } in C, and S _ { P } is a facet of S _ { Q } just when P contains Q. These three constructions are, in fact, closely related to each other (see [a9]). In particular, the idempotent e used at the start can be identified with an ( r - 1 )-sphere in the Tits building, the sum over W corresponding to a decomposition of the sphere into simplexes: in the usual action of W on S ^ { r - 1 } \subset \mathbf{R} ^ { r } the reflecting hyperplanes divide S ^ { r - 1} into | W | oriented spherical simplexes, each of which is a fundamental domain for W. Finally, \operatorname{St} has a simple presentation (as a linear space). It is generated by the Borel subgroups of G subject only to the relations that for every parabolic subgroup of rank 1 the sum of the Borel subgroups that it contains is 0.
There are also infinite-dimensional versions of the above constructions, usually for reductive Lie groups — real, complex or p-adic — such as \operatorname { GL} _ { n }. The p-adic case most closely resembles the finite case. There, the affine Weyl group and a certain compact-open subgroup, called an Iwahori subgroup, come into play (in place of W and B), and the three constructions agree. In [a1] several types of buildings, Curtis' formula and the Steinberg idempotent, in the guise of a homology cycle, all appear. In the infinite case the constructed object is sometimes called the Steinberg representation, sometimes the special representation.
References [a5] and [a9] are essays on \operatorname{St}.
References
[a1] | A. Borel, J-P. Serre, "Cohomologie d'immeubles et de groupes S-arithmétiques" Topology , 15 (1976) pp. 211–232 MR447474 |
[a2] | C.W. Curtis, "The Steinberg character of a finite group with B N-pair" J. Algebra , 4 (1966) pp. 433–441 MR201524 |
[a3] | C.W. Curtis, G.I. Lehrer, J. Tits, "Spherical buildings and the character of the Steinberg representation" Invent. Math. , 58 (1980) pp. 201–220 MR0571572 Zbl 0435.20024 |
[a4] | W. Haboush, "Reductive groups are geometrically reductive" Ann. of Math. , 102 (1975) pp. 67–83 MR0382294 Zbl 0316.14016 |
[a5] | J.E. Humphreys, "The Steinberg representation" Bull. Amer. Math. Soc. (N.S.) , 16 (1987) pp. 237–263 MR0876960 Zbl 0627.20024 |
[a6] | J.C. Jantzen, "Representations of algebraic groups" , Acad. Press (1987) MR0899071 Zbl 0654.20039 |
[a7] | L. Solomon, "The Steinberg character of a finite group with B N-pair" , Theory of Finite Groups (Harvard Symp.) , Benjamin (1969) pp. 213–221 |
[a8] | R. Steinberg, "Prime power representations of finite linear groups II" Canad. J. Math. , 9 (1957) pp. 347–351 MR0087659 Zbl 0079.25601 |
[a9] | R. Steinberg, "Collected Papers" , Amer. Math. Soc. (1997) pp. 580–586 Zbl 0878.20002 |
Steinberg module. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Steinberg_module&oldid=54681