Glossary of representation theory

This is a glossary of representation theory in mathematics.

See also list of representation theory topics and Category:Representation theory.

Notations: We write ${\mathbb {G}}_{m}=GL_{1}$ . Thus, for example, a one-representation (i.e., a character) of a group G is of the form $\chi :G\to \mathbb {G} _{m}$ .

A

adjoint: The adjoint representation of a Lie group G is the representation given by the adjoint action of G on the Lie algebra of G (an adjoint action is obtained, roughly, by differentiating a conjugation action.)
alternating: The alternating square of a representation V is a subrepresentation $\operatorname {Alt} ^{2}(V)$ of the second tensor power $V^{\otimes 2}=V\otimes V$ .

B

Borel–Weil–Bott theorem: Over an algebraically closed field of characteristic zero, the Borel–Weil–Bott theorem realizes an irreducible representation of a reductive algebraic group as the space of the global sections of a line bundle on a flag variety. (In the positive characteristic case, the construction only produces Weyl modules, which may not be irreducible.)

C

Casimir element: A Casimir element is a distinguished element of the center of the universal enveloping algebra of a Lie algebra.
category of representations: Representations and equivariant maps between them form a category of representations.
character: 1. A character is a one-dimensional representation.; 2. The character of a finite-dimensional representation π is the function $g\mapsto \operatorname {tr} \pi (g)$ . In other words, it is the composition $G{\overset {\pi }{\to }}GL(V){\overset {\operatorname {tr} }{\to }}\mathbb {G} _{m}$ .; 3. An irreducible character (resp. a trivial character) is the character of an irreducible representation (resp. a trivial representation).; 4. The character group of a group G is the group of all characters on G; namely, $\operatorname {Hom} (G,\mathbb {G} _{m})$ .; 5. The character ring is the group ring (over the integers) of the character group of G.; 6. A virtual character is an element of a character ring.; 7. A distributional character may be defined for an infinite-dimensional representation.; 8. An infinitesimal character.
class function: A class function f on a group G is a function such that $f(g)=f(hgh^{-1})$ ; it is a function on conjugacy classes.
coadjoint: A coadjoint representation is the dual representation of an adjoint representation.
complete: “completely reducible" is another term for "semisimple".
complex: A complex representation is a representation of G on a complex vector space. Many authors refer complex representations simply as representations.
cyclic: A cyclic G-module is a G-module generated by a single vector. For example, an irreducible representation is necessarily cyclic.

D

defined over: Given a field extension $K/F$ , a representation V of a group G over K is said to be defined over F if $V\simeq V_{0}\otimes _{F}K$ for some representation $V_{0}$ over F such that $g:V\to V$ is induced by $g:V_{0}\to V_{0}$ ; i.e., $g\cdot (v\otimes \lambda )=gv\otimes \lambda$ . Here, $V_{0}$ is called an F-form of V (and is not necessarily unique).
direct sum: The direct sum of representations V, W is a representation that is the direct sum $V\oplus W$ of the vector spaces together with the linear group action $\pi _{V\oplus W}(g)(v+w)=\pi _{V}(g)v+\pi _{W}(g)w$ .
dual: The dual representation (or the contragredient representation) of a representation V is a representation that is the dual vector space $V^{*}=\operatorname {Hom} (V,k)$ together with the linear group action that preserves the natural pairing $V^{*}\times V\to k$

E

equivariant: The term “G-equivariant” is another term for “G-linear”.
exterior: An exterior power of a representation V is a representation $\wedge ^{n}(V)$ with the group action induced by $V^{\otimes n}\to \wedge ^{n}(V)$ .

F

faithful

A faithful representation

\pi :G\to GL(V)

is a representation such that

\pi

is injective as a function.

Frobenius reciprocity

The Frobenius reciprocity states that for each representation

\sigma

of H and representation

\pi

of G there is a bijection

\operatorname {Hom} _{G}(\pi ,\operatorname {Ind} _{H}^{G}\sigma )=\operatorname {Hom} _{H}(\pi |_{H},\sigma )

that is natural in the sense that

\operatorname {Ind} _{H}^{G}

is the right adjoint functor to the restriction functor

\pi \mapsto \pi |_{H}

.

G

G-linear: A G-linear map $f:V\to W$ between representations is a linear transformation that commutes with the G-actions; i.e., $f\circ \pi _{V}(g)=\pi _{W}(g)\circ f$ for every g in G.
G-module: Another name for a representation. It allows for the module-theoretic terminology: e.g., trivial G-module, G-submodules, etc.
G-equivariant vector bundle: A G-equivariant vector bundle is a vector bundle $p:E\to X$ on a G-space X together with a G-action on E (say right) such that $g:p^{-1}(x)\to p^{-1}(xg)$ is a well-defined linear map.
good: A good filtration of a representation of a reductive group G is a filtration such that the quotients are isomorphic to $H^{0}(\lambda )=\Gamma (G/B,L_{\lambda })$ where $L_{\lambda }$ are the line bundles on the flag variety $G/B$ .

H

highest weight: 1. Given a complex semisimple Lie algebra ${\mathfrak {g}}$ , Cartan subalgebra ${\mathfrak {h}}$ and a choice of a positive Weyl chamber, the highest weight of a representation of ${\mathfrak {g}}$ is the weight of an ${\mathfrak {h}}$ -weight vector v such that $E_{\alpha }v=0$ for every positive root $\alpha$ (v is called the highest weight vector).; 2. The theorem of the highest weight states (1) two finite-dimensional irreducible representations of ${\mathfrak {g}}$ are isomorphic if and only if they have the same highest weight and (2) for each dominant integral $\lambda \in {\mathfrak {h}}^{*}$ , there is a finite-dimensional irreducible representation having $\lambda$ as its highest weight.
Hom: The Hom representation $\operatorname {Hom} (V,W)$ of representations V, W is a representation with the group action obtained by the vector space identification $\operatorname {Hom} (V,W)=V^{*}\otimes W$ .

I

indecomposable

An indecomposable representation is a representation that is not a direct sum of at least two proper subrepresebtations.

induction

1. Given a representation

(\sigma ,W)

of a subgroup H of a group G, the induced representation

\operatorname {Ind} _{H}^{G}(\sigma )=\{f:G\to W|f(hg)=\sigma (h)f(g)\}

is a representation of G that is induced on the H-linear functions

f:G\to W

; cf. #Frobenius reciprocity.

2. Depending on applications, it is common to impose further conditions on the functions

f:G\to W

; for example, if the functions are required to be compactly supported, then the resulting induction is called the compact induction.

intertwining

The term "intertwining operator" is an old name for a G-linear map between representations.

irreducible

An irreducible representation is a representation whose only subrepresentations are zero and itself. The term "irreducible" is synonymous with "simple".

isomorphism

An isomorphism between representations of a group G is an invertible G-linear map between the representations.

isotypic

Given a representation V and a simple representation W (subrepresebtation or otherwise), the isotypic component of V of type W is the direct sum of all subrepresentations of V that are isomorphic to W. For example, let A be a ring and G a group acting on it as automorphisms. If A is semisimple as a G-module, then the ring of invariants

A^G

is the isotypic component of A of trivial type.

L

lattice: 1. The root lattice is the free abelian group generated by the roots.; 2. The weight lattice is the group of all linear functionals $\chi \in {\mathfrak {h}}^{*}$ on a Cartan subalgebra ${\mathfrak {h}}$ that are integral: $\chi (H_{\alpha })$ is an integer for every root $\alpha$ .

M

Maschke's theorem: Maschke's theorem states that a finite-dimensional representation over a field F of a finite group G is a semisimple representation if the characteristic of F does not divide the order of G.
Mackey theory: The Mackey theory may be thought of a tool to answer the question: given a representation W of a subgroup H of a group G, when is the induced representation $\operatorname {Ind} _{H}^{G}W$ an irreducible representation of G?^[1]
matrix coefficient: A matrix coefficient of a representation $\pi :G\to GL(V)$ is a linear combination of functions on G of the form $g\mapsto \langle \pi (g)v,\alpha \rangle$ for v in V and $\alpha$ in the dual space $V^{*}$ . Note the notion makes sense for any group: if G is a topological group and $\pi$ is continuous, then a matrix matrix coefficient would be a continuous function on G. If G and $\pi$ are algebraic, it would be a regular function on G.

P

Peter–Weyl: The Peter–Weyl theorem states that the linear span of the matrix coefficients on a compact group G is dense in $L^{2}(G)$ .
permutation: Given a group G, a G-set X and V the vector space of functions from X to a fixed field, a permutation representation $\pi$ of G on V is a representation given by the induced action of G on V; i.e., $(\pi (g)v)(x)=v(g^{-1}x)$ . For example, if X is a finite set and V is viewed as a vector space with a basis parameteized by X, then the symmetric group $G=\operatorname {Sym} (X)$ permutates the elements of the basis and its linear extension is precisely the permutation representation.
projective: A projective representation of a group G is a group homomorphism $\pi :G\to PGL(V)=GL(V)/\mathbb {G} _{m}$ . Since $PGL(V)=\operatorname {Aut} (\mathbb {P} (V))$ , a projective representation is precisely a group action of G on $\mathbb {P} (V)$ as automorphisms.
proper: A proper subrepresentation of a representation V is a subrepresentstion that is not V.

Q

quotient: Given a representation V and a subrepresentation $W\subset V$ , the quotient representation is the representation $(\pi _{V/W},V/W)$ given by $\pi _{V/W}(g):V/W\to V/W,\,v+W\mapsto gv+W$ .
quaternionic: A quaternionic representation of a group G is a complex representation equipped with a G-invariant quaternionic structure.

R

rational: A representation V is rational if each vector v in V is contained in some finite-dimensional subrepresentation (depending on v.)
real: 1. A real representation of a vector space is a representation on a real vector space.; 2. A real character is a character $\chi$ of a group G such that $\chi (g)\in \mathbb {R}$ for all g in G.^[2]
regular: 1. A regular representation of a finite group G is the induced representation of G on the group algebra over a field of G.; 2. A regular representation of a linear algebraic group G is the induced representation on the coordinate ring of G. See also: representation on coordinate rings.
representation: 1. A linear representation of a group G is a group homomorphism $\pi :G\to GL(V)$ from G to the general linear group $GL(V)$ . Depending on the group G, the homomorphism $\pi$ is often implicitly required to be a morphishm in a category to which G belongs; e.g., if G is a topological group, then $\pi$ must be continuous. The adjective “linear” is often omitted.; 2. Equivalently, a linear representation is a group action of G on a vector space V that is linear: the action $\sigma :G\times V\to V$ such that for each g in G, $\sigma _{g}:V\to V,v\mapsto \sigma (g,v)$ is a linear transformation.; 3. A virtual representation is an element of the Grothendieck ring of the category of representations.

S

Schur: 1. Issai Schur; 2. Schur's lemma states that a G-linear map between irreducible representations must be either bijective or zero.; 3. The Schur orthogonality relations on a compact group says the characters of non-isomorphic irreducible representations are orthogonal to each other.; 4. The Schur functor $V\mapsto S^{\lambda }(V)$ constructs representations such as symmetric powers or exterior powers according to a partition $\lambda$ . The characters of $S^{\lambda }(V)$ are Schur polynomials.; 5. The Schur–Weyl duality computes the irreducible representations occurring in tensor powers of $GL(V)$ -modules.
simple: Another term for "irreducible".
smooth: A smooth representation of a locally profinite group G is a complex representation such that, for each v in V, there is some compact open subgroup K of G that fixes v; i.e., $g\cdot v=v$ for every g in K.
subrepresentation: A subrepresentation of a representation $(\pi ,V)$ of G is a vector subspace W of V such that $\pi (g):W\to W$ is well-defined for each g in G.
symmetric: 1. A symmetric power of a representation V is a representation $\operatorname {Sym} ^{n}(V)$ with the group action induced by $V^{\otimes n}\to \operatorname {Sym} ^{n}(V)$ .; 2. In particular, the symmetric square of a representation V is a representation $\operatorname {Sym} ^{2}(V)$ with the group action induced by $V^{\otimes 2}\to \operatorname {Sym} ^{2}(V)$ .
system of imprimitivity: A concept in the Mackey theory. See system of imprimitivity.

T

Tannakian duality: The Tannakian duality is roughly an idea that a group can be recovered from all of its representations.
tensor: A tensor representation is roughly a representation obtained from tensor products (of certain representations).
tensor product: The tensor product of representations V, W is the representation that is the tensor product of vector spaces $V\otimes W$ together with the linear group action $\pi _{V\otimes W}(g)(v\otimes w)=\pi _{V}(g)v\otimes \pi _{W}(g)w$ .
trivial: 1. A trivial representation of a group G is a representation π such that π(g) is the identity for every g in G.; 2. A trivial character of a group G is a character that is trivial as a representation.

U

unitary: A unitary representation of a group G is a representation π such that π(g) is a unitary operator for every g in G.

V

Verma module: Given a complex semisimple Lie algebra ${\mathfrak {g}}$ , a Cartan subalgebra ${\mathfrak {h}}$ and a choice of a positive Weyl chamber, the Verma module $M_{\chi }$ associated to a linear functional $\chi :{\mathfrak {h}}\to \mathbb {C}$ is the quotient of the enveloping algebra $U({\mathfrak {g}})$ by the left ideal generated by $E_{\alpha }$ for all positive roots $\alpha$ as well as $H-\chi (H)1$ for all $H\in {\mathfrak {h}}$ .^[3]

W

weight: 1. The term "weight" is another name for a character.; 2. The weight subspace of a representation V of a weight $\chi :G\to \mathbb {G} _{m}$ is the subspace $V_{\chi }=\{v\in V|g\cdot v=\chi (g)v\}$ that has positive dimension.; 3. Similarly, for a linear functional $\chi :{\mathfrak {h}}\to \mathbb {C}$ of a complex Lie algebra ${\mathfrak {h}}$ , $\chi$ is a weight of an ${\mathfrak {h}}$ -module V if $V_{\chi }=\{v\in V|H\cdot v=\chi (H)v\}$ has positive dimension; cf. #highest weight.
Weyl: 1. Hermann Weyl; 2. The Weyl character formula expresses the character of an irreducible representations of a complex semisimple Lie algebra in terms of highest weights.; 3. Weyl module.; 4. A Weyl filtration is a filtration of a representation of a reductive group such that the quotients are isomorphic to Weyl modules.

Y

Young: 1. Alfred Young; 2. The Young symmetrizer is the G-linear endomorphism $c_{\lambda }:V^{\otimes n}\to V^{\otimes n}$ of a tensor power of a G-module V defined according to a given partition $\lambda$ . By definition, the Schur functor of a representation V assigns to V the image of $c_{\lambda }$ .

Z

zero: A zero representation is a zero-dimensional representation. Note: while a zero representation is a trivial representation, a trivial representation need not be zero (since “trivial” mean G acts trivially.)

Notes

↑ https://www.dpmms.cam.ac.uk/~nd332/Mackey.pdf
↑ James, Gordon Douglas (2001). Representations and characters of groups. Liebeck, Martin W. 1954- (2nd ed.). Cambridge, UK: Cambridge University Press. ISBN 052100392X. OCLC 52220683.
↑ Editorial note: this is the definition in (Humphreys 1972, § 20.3.) as well as (Gaitsgory 2005, § 1.2.) and differs from the original by $\rho =$ half the sum of the positive roots.

References

Bushnell, Colin J.; Henniart, Guy (2006), The local Langlands conjecture for GL(2), Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 335, Berlin, New York: Springer-Verlag, doi:10.1007/3-540-31511-X, ISBN 978-3-540-31486-8, MR 2234120
Fulton, William; Harris, Joe (1991). Representation theory. A first course. Graduate Texts in Mathematics, Readings in Mathematics. 129. New York: Springer-Verlag. doi:10.1007/978-1-4612-0979-9. ISBN 978-0-387-97495-8. MR 1153249. OCLC 246650103.
D. Gaitsgory, Geometric Representation theory, Math 267y, Fall 2005
Humphreys, James E. (1972). Introduction to Lie Algebras and Representation Theory. Graduate Texts in Mathematics. 9. New York: Springer-Verlag. ISBN 0-387-90053-5.
Knapp, Anthony W. (2001), Representation theory of semisimple groups. An overview based on examples., Princeton Landmarks in Mathematics, Princeton University Press, ISBN 0-691-09089-0 (elementary treatment for SL(2,C))
Claudio Procesi (2007) Lie Groups: an approach through invariants and representation, Springer, ISBN 9780387260402.

External links

https://math.stanford.edu/~bump/

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[1] ttps://www.dpmms.cam.ac.uk/~nd332/Mackey.pdf

[2] James, Gordon Douglas (2001). Representations and characters of groups. Liebeck, Martin W. 1954- (2nd ed.). Cambridge, UK: Cambridge University Press. ISBN 052100392X. OCLC 52220683.

[3] Editorial note: this is the definition in (Humphreys 1972, § 20.3.) as well as (Gaitsgory 2005, § 1.2.) and differs from the original by $\rho =$ half the sum of the positive roots.