# Category: Theorems in representation theory

Borel–Weil–Bott theorem
In mathematics, the Borel–Weil–Bott theorem is a basic result in the representation theory of Lie groups, showing how a family of representations can be obtained from holomorphic sections of certain c
Haboush's theorem
In mathematics Haboush's theorem, often still referred to as the Mumford conjecture, states that for any semisimple algebraic group G over a field K, and for any linear representation ρ of G on a K-ve
Engel's theorem
In representation theory, a branch of mathematics, Engel's theorem states that a finite-dimensional Lie algebra is a nilpotent Lie algebra if and only if for each , the adjoint map given by , is a nil
Lie–Kolchin theorem
In mathematics, the Lie–Kolchin theorem is a theorem in the representation theory of linear algebraic groups; Lie's theorem is the analog for linear Lie algebras. It states that if G is a connected an
Multiplicity-one theorem
In the mathematical theory of automorphic representations, a multiplicity-one theorem is a result about the representation theory of an adelic reductive algebraic group. The multiplicity in question i
Peter–Weyl theorem
In mathematics, the Peter–Weyl theorem is a basic result in the theory of harmonic analysis, applying to topological groups that are compact, but are not necessarily abelian. It was initially proved b
Brauer's three main theorems
Brauer's main theorems are three theorems in representation theory of finite groups linking the blocks of a finite group (in characteristic p) with those of its p-local subgroups, that is to say, the
Commutation theorem for traces
In mathematics, a commutation theorem for traces explicitly identifies the commutant of a specific von Neumann algebra acting on a Hilbert space in the presence of a trace. The first such result was p
Frobenius reciprocity
In mathematics, and in particular representation theory, Frobenius reciprocity is a theorem expressing a duality between the process of restricting and inducting. It can be used to leverage knowledge
Lafforgue's theorem
In mathematics, Lafforgue's theorem, due to Laurent Lafforgue, completes the Langlands program for general linear groups over algebraic function fields, by giving a correspondence between automorphic
Mautner's lemma
In mathematics, Mautner's lemma in representation theory states that if G is a topological group and π a unitary representation of G on a Hilbert space H, then for any x in G, which has conjugates yxy
Gabriel's theorem
In mathematics, Gabriel's theorem, proved by Pierre Gabriel, classifies the quivers of finite type in terms of Dynkin diagrams.
Wigner–Eckart theorem
The Wigner–Eckart theorem is a theorem of representation theory and quantum mechanics. It states that matrix elements of spherical tensor operators in the basis of angular momentum eigenstates can be
Brauer's theorem on induced characters
Brauer's theorem on induced characters, often known as Brauer's induction theorem, and named after Richard Brauer, is a basic result in the branch of mathematics known as character theory, within repr
Harish-Chandra's regularity theorem
In mathematics, Harish-Chandra's regularity theorem, introduced by Harish-Chandra, states that every invariant eigendistribution on a semisimple Lie group, and in particular every character of an irre
Maschke's theorem
In mathematics, Maschke's theorem, named after Heinrich Maschke, is a theorem in group representation theory that concerns the decomposition of representations of a finite group into irreducible piece