Steinberg module
Let , the group of all invertible
-matrices over the finite field
with
elements and characteristic
, let
be the subgroup of all superdiagonal elements, let
be the subgroup of elements of
whose diagonal entries are all
, and let
be the subgroup of permutation matrices. In the group algebra
of
over any field
of characteristic
or
, the element
![]() |
is an idempotent, called the Steinberg idempotent, and the -module that it generates in
by right multiplication is called the Steinberg module (see [a8]) and is commonly denoted
(as are all modules isomorphic to it). A similar construction holds for any finite group
of Lie type (and for any
-pair, which is an axiomatic generalization due to J. Tits) defined over a field of characteristic
with
replaced by a Borel subgroup (which is a certain kind of solvable subgroup),
by a maximal unipotent subgroup (cf. Unipotent group) of
(which is also a Sylow
-subgroup of
; cf. also Sylow subgroup;
-group) and
by the corresponding Weyl group.
is always irreducible and it has
as a basis, so that its dimension is
(see [a8]). Its character values are given as follows [a3]. If
has order prime to
, then
equals, up to a sign which can be determined, the order of a Sylow
-subgroup of the centralizer of
; otherwise it equals
.
In case the characteristic of equals
,
has the following further properties [a5]. It is the only module (for
) 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,
plays a prominent role in ongoing work in the still (2000) unresolved problem of determining all of the irreducible
-modules (with characteristic
still equal to
), or equivalently, as it turns out, of determining all of the irreducible rational
-modules, where
is the algebraic group obtained from
by replacing
by its algebraic closure
, i.e., where
is any simple affine algebraic group of characteristic
(see [a6]). This equivalence comes from the fact that every irreducible
-module extends to a rational
-module. In particular,
extends to the
-module with highest weight
times the sum of the fundamental weights, which is accordingly also denoted
, or
since there is one such
-module for each
. These modules are ubiquitous in the module theory of
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 , with the characteristic of
now equal to
, are as follows. According to C.W. Curtis [a2]
![]() |
in which runs through the
(
equal to the rank of
) (parabolic) subgroups of
containing
,
is the
-module induced by the trivial
-module, and the
or
is used according as the rank
of
is even or odd. For
, for example, there is one
for each solution of
(
, each
); it consists of all of the elements of
that are superdiagonal in the corresponding block matrix form. A third construction, due to L. Solomon and Tits [a7], yields
as the top homology space
for the Tits simplicial complex or Tits building
of
, formed as follows: corresponding to each parabolic subgroup
there exists an
-simplex
in
, and
is a facet of
just when
contains
. These three constructions are, in fact, closely related to each other (see [a9]). In particular, the idempotent
used at the start can be identified with an
-sphere in the Tits building, the sum over
corresponding to a decomposition of the sphere into simplexes: in the usual action of
on
the reflecting hyperplanes divide
into
oriented spherical simplexes, each of which is a fundamental domain for
. Finally,
has a simple presentation (as a linear space). It is generated by the Borel subgroups of
subject only to the relations that for every parabolic subgroup of rank
the sum of the Borel subgroups that it contains is
.
There are also infinite-dimensional versions of the above constructions, usually for reductive Lie groups — real, complex or -adic — such as
. The
-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
and
), 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 .
References
[a1] | A. Borel, J-P. Serre, "Cohomologie d'immeubles et de groupes ![]() |
[a2] | C.W. Curtis, "The Steinberg character of a finite group with ![]() |
[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 |
[a4] | W. Haboush, "Reductive groups are geometrically reductive" Ann. of Math. , 102 (1975) pp. 67–83 |
[a5] | J.E. Humphreys, "The Steinberg representation" Bull. Amer. Math. Soc. (N.S.) , 16 (1987) pp. 237–263 |
[a6] | J.C. Jantzen, "Representations of algebraic groups" , Acad. Press (1987) |
[a7] | L. Solomon, "The Steinberg character of a finite group with ![]() |
[a8] | R. Steinberg, "Prime power representations of finite linear groups II" Canad. J. Math. , 9 (1957) pp. 347–351 |
[a9] | R. Steinberg, "Collected Papers" , Amer. Math. Soc. (1997) pp. 580–586 |
Steinberg module. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Steinberg_module&oldid=18782