About: Steinberg representation

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In mathematics, the Steinberg representation, or Steinberg module or Steinberg character, denoted by St, is a particular linear representation of a reductive algebraic group over a finite field or local field, or a group with a BN-pair. It is analogous to the 1-dimensional ε of a Coxeter or Weyl group that takes all reflections to –1. The Steinberg representation is the Alvis–Curtis dual of the trivial 1-dimensional representation.

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rdfs:label	Steinberg representation (en) Representação de Steinberg (pt)
rdfs:comment	In mathematics, the Steinberg representation, or Steinberg module or Steinberg character, denoted by St, is a particular linear representation of a reductive algebraic group over a finite field or local field, or a group with a BN-pair. It is analogous to the 1-dimensional ε of a Coxeter or Weyl group that takes all reflections to –1. The Steinberg representation is the Alvis–Curtis dual of the trivial 1-dimensional representation. (en) Em matemática, a representação de Steinberg ou módulo de Steinberg, denotado por St, é uma representação linear específica de um grupo redutivo algébrico sobre um corpo finito ou campo local. É análogo a representação de sinal unidimensional ε de um Coxeter ou que leva todas as reflexões para -1. (pt)
dcterms:subject	Representation theory of algebraic groups Finite fields
Wikipage page ID	5862365 (xsd:integer)
Wikipage revision ID	1032155046 (xsd:integer)
Link from a Wikipage to another Wikipage	Cambridge University Press Annals of Mathematics Representation theory of algebraic groups Inventiones Mathematicae Jacquet module Mathematics General linear group Haboush's theorem Local field American Mathematical Society Finite field Coxeter group Finite fields Chevalley group Sylow subgroup Topology (journal) Borel subgroup Canadian Journal of Mathematics Unipotent representation Weyl group BN-pair Transactions of the American Mathematical Society Linear representation Bruhat–Tits building Parabolic subgroup Alvis–Curtis dual Reductive algebraic group dbr:Sign_representation
Link from a Wikipage to an external page	http://www.ams.org/bull/1987-16-02/S0273-0979-1987-15512-1/home.html https://books.google.com/books%3Fid=5xIZAQAAIAAJ https://books.google.com/books%3Fid=cNvqfp80jpEC
sameAs	Steinberg representation Steinberg representation Steinberg representation Steinberg representation Steinberg representation
txt	yes (en)
dbp:wikiPageUsesTemplate	dbt:Springer dbt:Citation dbt:Harvtxt dbt:Reflist dbt:Isbn dbt:Su dbt:Harvs
authorlink	Robert Steinberg (en)
b	c (en)
first	Robert (en)
id	S/s130530 (en)
last	Steinberg (en)
p	l (en)
title	Steinberg module (en)
year	1951 (xsd:integer) 1956 (xsd:integer) 1957 (xsd:integer)
has abstract	In mathematics, the Steinberg representation, or Steinberg module or Steinberg character, denoted by St, is a particular linear representation of a reductive algebraic group over a finite field or local field, or a group with a BN-pair. It is analogous to the 1-dimensional ε of a Coxeter or Weyl group that takes all reflections to –1. For groups over finite fields, these representations were introduced by Robert Steinberg , first for the general linear groups, then for classical groups, and then for all Chevalley groups, 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 local fields. For the general linear group GL(2), the dimension of the Jacquet module of a special representation is always one. (en) Em matemática, a representação de Steinberg ou módulo de Steinberg, denotado por St, é uma representação linear específica de um grupo redutivo algébrico sobre um corpo finito ou campo local. É análogo a representação de sinal unidimensional ε de um Coxeter ou que leva todas as reflexões para -1. Para os grupos sobre corpos/campos finitos, estas representações foram introduzidas por Robert Steinberg, primeiro (1951) para os grupos lineares gerais, em seguida (1956), para os grupos clássicos, e depois (1957), para todos os grupos de Chevalley, com uma construção que, imediatamente generalizada para os outros grupos do tipo Lie que foram descobertos logo depois por Steinberg, Suzuki e Ree. Ao longo de um corpo finito de característica p, a representação Steinberg possui graduação igual ao maior poder de p dividindo a ordem do grupo. (pt)
gold:hypernym	Representation
prov:wasDerivedFrom	wikipedia-en:Steinberg_representation?oldid=1032155046&ns=0
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foaf:isPrimaryTopicOf	wikipedia-en:Steinberg_representation
is Link from a Wikipage to another Wikipage of	Jacquet module Robert Steinberg Glossary of representation theory Deligne–Lusztig theory Steinberg (disambiguation) Haboush's theorem List of Romanian inventors and discoverers Suzuki groups Unipotent representation Steinberg character Steinberg module Special representation
is Wikipage redirect of	Steinberg character Steinberg module Special representation
is Wikipage disambiguates of	Steinberg (disambiguation)
is foaf:primaryTopic of	wikipedia-en:Steinberg_representation

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