View source for Steinberg module

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130530/s1305301.png" />, the group of all invertible <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130530/s1305302.png" />-matrices over the finite [[Field|field]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130530/s1305303.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130530/s1305304.png" /> elements and characteristic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130530/s1305305.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130530/s1305306.png" /> be the subgroup of all superdiagonal elements, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130530/s1305307.png" /> be the subgroup of elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130530/s1305308.png" /> whose diagonal entries are all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130530/s1305309.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130530/s13053010.png" /> be the subgroup of permutation matrices. In the [[Group algebra|group algebra]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130530/s13053011.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130530/s13053012.png" /> over any field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130530/s13053013.png" /> of characteristic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130530/s13053014.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130530/s13053015.png" />, the element

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130530/s13053016.png" /></td> </tr></table>

is an idempotent, called the Steinberg idempotent, and the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130530/s13053017.png" />-module that it generates in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130530/s13053018.png" /> by right multiplication is called the Steinberg module (see [[#References|[a8]]]) and is commonly denoted <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130530/s13053019.png" /> (as are all modules isomorphic to it). A similar construction holds for any [[Finite group|finite group]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130530/s13053020.png" /> of Lie type (and for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130530/s13053022.png" />-pair, which is an axiomatic generalization due to J. Tits) defined over a field of characteristic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130530/s13053023.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130530/s13053024.png" /> replaced by a [[Borel subgroup|Borel subgroup]] (which is a certain kind of solvable subgroup), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130530/s13053025.png" /> by a maximal unipotent subgroup (cf. [[Unipotent group|Unipotent group]]) of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130530/s13053026.png" /> (which is also a Sylow <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130530/s13053027.png" />-subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130530/s13053028.png" />; cf. also [[Sylow subgroup|Sylow subgroup]]; [[P-group|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130530/s13053029.png" />-group]]) and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130530/s13053030.png" /> by the corresponding [[Weyl group|Weyl group]]. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130530/s13053031.png" /> is always irreducible and it has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130530/s13053032.png" /> as a basis, so that its dimension is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130530/s13053033.png" /> (see [[#References|[a8]]]). Its character values are given as follows [[#References|[a3]]]. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130530/s13053034.png" /> has order prime to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130530/s13053035.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130530/s13053036.png" /> equals, up to a sign which can be determined, the order of a Sylow <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130530/s13053037.png" />-subgroup of the centralizer of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130530/s13053038.png" />; otherwise it equals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130530/s13053039.png" />.

In case the characteristic of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130530/s13053040.png" /> equals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130530/s13053041.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130530/s13053042.png" /> has the following further properties [[#References|[a5]]]. It is the only module (for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130530/s13053043.png" />) 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, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130530/s13053044.png" /> plays a prominent role in ongoing work in the still (2000) unresolved problem of determining all of the irreducible <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130530/s13053045.png" />-modules (with characteristic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130530/s13053046.png" /> still equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130530/s13053047.png" />), or equivalently, as it turns out, of determining all of the irreducible rational <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130530/s13053048.png" />-modules, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130530/s13053049.png" /> is the [[Algebraic group|algebraic group]] obtained from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130530/s13053050.png" /> by replacing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130530/s13053051.png" /> by its algebraic closure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130530/s13053052.png" />, i.e., where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130530/s13053053.png" /> is any simple affine algebraic group of characteristic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130530/s13053054.png" /> (see [[#References|[a6]]]). This equivalence comes from the fact that every irreducible <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130530/s13053055.png" />-module extends to a rational <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130530/s13053056.png" />-module. In particular, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130530/s13053057.png" /> extends to the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130530/s13053058.png" />-module with highest weight <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130530/s13053059.png" /> times the sum of the fundamental weights, which is accordingly also denoted <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130530/s13053060.png" />, or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130530/s13053061.png" /> since there is one such <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130530/s13053062.png" />-module for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130530/s13053063.png" />. These modules are ubiquitous in the module theory of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130530/s13053064.png" /> and figure prominently, for example, in the proofs of many cohomological vanishing theorems and in W. Haboush's proof of the [[Mumford hypothesis|Mumford hypothesis]] (see [[#References|[a4]]]).

Back in the finite case, some other constructions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130530/s13053065.png" />, with the characteristic of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130530/s13053066.png" /> now equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130530/s13053067.png" />, are as follows. According to C.W. Curtis [[#References|[a2]]]

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130530/s13053068.png" /></td> </tr></table>

in which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130530/s13053069.png" /> runs through the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130530/s13053070.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130530/s13053071.png" /> equal to the rank of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130530/s13053072.png" />) (parabolic) subgroups of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130530/s13053073.png" /> containing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130530/s13053074.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130530/s13053075.png" /> is the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130530/s13053076.png" />-module induced by the trivial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130530/s13053077.png" />-module, and the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130530/s13053078.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130530/s13053079.png" /> is used according as the rank <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130530/s13053080.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130530/s13053081.png" /> is even or odd. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130530/s13053082.png" />, for example, there is one <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130530/s13053083.png" /> for each solution of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130530/s13053084.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130530/s13053085.png" />, each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130530/s13053086.png" />); it consists of all of the elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130530/s13053087.png" /> that are superdiagonal in the corresponding block matrix form. A third construction, due to L. Solomon and Tits [[#References|[a7]]], yields <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130530/s13053088.png" /> as the top homology space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130530/s13053089.png" /> for the Tits simplicial complex or [[Tits building|Tits building]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130530/s13053090.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130530/s13053091.png" />, formed as follows: corresponding to each parabolic subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130530/s13053092.png" /> there exists an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130530/s13053093.png" />-simplex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130530/s13053094.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130530/s13053095.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130530/s13053096.png" /> is a facet of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130530/s13053097.png" /> just when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130530/s13053098.png" /> contains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130530/s13053099.png" />. These three constructions are, in fact, closely related to each other (see [[#References|[a9]]]). In particular, the idempotent <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130530/s130530100.png" /> used at the start can be identified with an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130530/s130530101.png" />-sphere in the Tits building, the sum over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130530/s130530102.png" /> corresponding to a decomposition of the sphere into simplexes: in the usual action of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130530/s130530103.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130530/s130530104.png" /> the reflecting hyperplanes divide <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130530/s130530105.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130530/s130530106.png" /> oriented spherical simplexes, each of which is a [[Fundamental domain|fundamental domain]] for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130530/s130530107.png" />. Finally, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130530/s130530108.png" /> has a simple presentation (as a linear space). It is generated by the Borel subgroups of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130530/s130530109.png" /> subject only to the relations that for every parabolic subgroup of rank <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130530/s130530110.png" /> the sum of the Borel subgroups that it contains is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130530/s130530111.png" />.

There are also infinite-dimensional versions of the above constructions, usually for reductive Lie groups — real, complex or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130530/s130530112.png" />-adic — such as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130530/s130530113.png" />. The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130530/s130530114.png" />-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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130530/s130530115.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130530/s130530116.png" />), and the three constructions agree. In [[#References|[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 [[#References|[a5]]] and [[#References|[a9]]] are essays on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130530/s130530117.png" />.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A. Borel,   J-P. Serre,   "Cohomologie d'immeubles et de groupes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130530/s130530118.png" />-arithmétiques"  ''Topology'' , '''15'''  (1976)  pp. 211–232</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  C.W. Curtis,   "The Steinberg character of a finite group with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130530/s130530119.png" />-pair"  ''J. Algebra'' , '''4'''  (1966)  pp. 433–441</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  C.W. Curtis,   G.I. Lehrer,   J. Tits,   "Spherical buildings and the character of the Steinberg representation"  ''Invent. Math.'' , '''58'''  (1980)  pp. 201–220</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  W. Haboush,   "Reductive groups are geometrically reductive"  ''Ann. of Math.'' , '''102'''  (1975)  pp. 67–83</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  J.E. Humphreys,   "The Steinberg representation"  ''Bull. Amer. Math. Soc. (N.S.)'' , '''16'''  (1987)  pp. 237–263</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  J.C. Jantzen,   "Representations of algebraic groups" , Acad. Press  (1987)</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  L. Solomon,   "The Steinberg character of a finite group with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130530/s130530120.png" />-pair" , ''Theory of Finite Groups (Harvard Symp.)'' , Benjamin  (1969)  pp. 213–221</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  R. Steinberg,   "Prime power representations of finite linear groups II"  ''Canad. J. Math.'' , '''9'''  (1957)  pp. 347–351</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top">  R. Steinberg,   "Collected Papers" , Amer. Math. Soc.  (1997)  pp. 580–586</TD></TR></table>