Steinberg representation
Encyclopedia
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
. It is analogous to the 1-dimensional sign representation ε of a Coxeter or Weyl group 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 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 field
s.
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, and sporadic and most alternating groups have no Steinberg representation.
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
of the group.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, 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
Finite field
In abstract algebra, a finite field or Galois field is a field that contains a finite number of elements. Finite fields are important in number theory, algebraic geometry, Galois theory, cryptography, and coding theory...
or local field
Local field
In mathematics, a local field is a special type of field that is a locally compact topological field with respect to a non-discrete topology.Given such a field, an absolute value can be defined on it. There are two basic types of local field: those in which the absolute value is archimedean and...
. It is analogous to the 1-dimensional sign representation ε of a Coxeter or Weyl group 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 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 field
Local field
In mathematics, a local field is a special type of field that is a locally compact topological field with respect to a non-discrete topology.Given such a field, an absolute value can be defined on it. There are two basic types of local field: those in which the absolute value is archimedean and...
s.
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 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 subgroups containing a Borel subgroupBorel subgroupIn 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 group GLn ,...
, of the representation induced from the identity representation of the parabolic subgroup. - The Steinberg representation is both regular and unipotentUnipotent representationIn mathematics, a unipotent representation of a reductive group is a representation that has some similarities with unipotent conjugacy classes of groups....
, 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 (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, and sporadic and most alternating groups have no Steinberg representation.
The Steinberg representation of a p-adic group
, , and introduced Steinberg representations for algebraic groups over local fieldLocal field
In mathematics, a local field is a special type of field that is a locally compact topological field with respect to a non-discrete topology.Given such a field, an absolute value can be defined on it. There are two basic types of local field: those in which the absolute value is archimedean and...
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
Bruhat–Tits building
In mathematics, a building is a combinatorial and geometric structure which simultaneously generalizes certain aspects of flag manifolds, finite projective planes, and Riemannian symmetric spaces...
of the group.