Representations of Groups

Introduction

A representation of a group is a module over its group algebra, and with that identification the entire structure theory of the preceding articles applies to group representations. This article develops the consequences: complete reducibility in the semisimple case, the character theory that computes multiplicities, the decomposition of the regular representation, and the tensor product decomposition known as the Clebsch–Gordan theory. It is a bridge entry. The group theory itself — subgroups, cosets, conjugacy classes, the class equation — is that of the companion article Groups and is not restated; the Lie-theoretic analogue for continuous groups is that of the companion article Lie Algebras and is cited where it is needed rather than developed. A representation is a module and is treated as such throughout: the objects are vector spaces with an action, the maps are intertwining operators, and no physical interpretation is attached to any of them.

The conventions are those of Modules over an Algebra and Simple and Semisimple Modules. The base ring is a commutative ring $R$ with identity; $F$ is a field; $G$ is a group; $F[G]$ is the group algebra, the free $F$-module on $G$ with the product extending the group multiplication. An $F[G]$-module is a left module, and $V$, $W$ denote representations. The group algebra is noncommutative exactly when $G$ is nonabelian, and the general theory of modules over a possibly noncommutative algebra therefore governs the subject. The balanced product, extension of scalars and Frobenius reciprocity are those of The Balanced Product over an Algebra and Change of Rings.

Representations and Group Algebras

Definition

Let $G$ be a group and let $V$ be a vector space over $F$. A representation of $G$ on $V$ is a group homomorphism

$$ \rho : G \longrightarrow GL_F(V)=GL(V), $$

the group of invertible $F$-linear endomorphisms of $V$. Equivalently, $V$ is a left $F[G]$-module, with the action of a formal sum $\sum_g a_g g$ given by

$$ \Bigl(\sum_g a_g g\Bigr)\cdot v=\sum_g a_g\,\rho(g)v. $$

The equivalence is exact: a group homomorphism $G\to GL(V)$ extends uniquely to a unital $F$-algebra homomorphism $F[G]\to\operatorname{End}_F(V)$, and conversely the restriction of an algebra homomorphism to the basis $G$ is a group homomorphism into the units. Hence

$$ \operatorname{Rep}_F(G) \cong \operatorname{Mod}(F[G]), $$

an equivalence of categories, the $F[G]$-linear maps being exactly the linear maps intertwining the group action. The dimension $\dim_F V$ is the degree of the representation, and a representation is irreducible if $V$ is a simple $F[G]$-module, completely reducible if $V$ is semisimple.

This identification is the reason the module theory of the preceding articles is the right language for representation theory. Submodules are the invariant subspaces; quotient modules are the quotients by invariant subspaces; direct sums are direct sums of representations; and the endomorphism ring $\operatorname{End}_{F[G]}(V)=\rho(G)'$ is the commutant, whose size measures the reducibility of $V$, by Automorphisms of Modules over an Algebra.

The regular representation

The left regular representation of $G$ is the action of $G$ on the group algebra $F[G]$ by left multiplication, equivalently the regular module ${}_{F[G]}F[G]$ of Modules over an Algebra. The right regular representation is right multiplication, and the two differ when $G$ is nonabelian: right multiplication gives $F[G]$-linear endomorphisms of the left regular module, by the theorem on the regular module, while left multiplication is the module action itself. The regular representation has degree $|G|$ when $G$ is finite, and its character is computed in §The Regular Representation.

Basic operations

The dual of a representation $V$ is the representation on $V^*=\operatorname{Hom}_F(V,F)$ with $(\rho^*(g)f)(v)=f(\rho(g^{-1})v)$; the inverse is what makes this a left action, since $V^*$ is naturally only a right $F[G]$-module. For a general algebra the dual of a left module is a left module over the opposite algebra, and it is the antipode $g\mapsto g^{-1}$ of the group algebra that converts it back into a left module. The direct sum $V\oplus W$ carries the diagonal action $g\cdot(v\oplus w)=gv\oplus gw$, and the tensor product $V\otimes_F W$ carries $g\cdot(v\otimes w)=gv\otimes gw$. The tensor product is a representation because the map

$$ \Delta : F[G] \longrightarrow F[G]\otimes_F F[G], \qquad \Delta(g)=g\otimes g, $$

is a unital algebra homomorphism; together with the counit $\varepsilon(g)=1$ and the antipode $S(g)=g^{-1}$ it makes $F[G]$ a Hopf algebra, and the tensor product of representations is the pullback along $\Delta$. The tensor product of two representations of an arbitrary algebra is not a representation — the diagonal map is not an algebra homomorphism in general — and the group algebra is the setting where it is.

Complete Reducibility

Maschke's theorem

Theorem (Maschke). Let $G$ be a finite group and $F$ a field with $\operatorname{char} F \nmid |G|$. Then $F[G]$ is semisimple, and every representation of $G$ over $F$ is a direct sum of irreducible representations.

Proof. Let $V$ be a representation and let $W \subseteq V$ be an invariant subspace. Choose any $F$-linear projection $\pi: V \to W$ and average it over the group,

$$ \bar\pi=\frac{1}{|G|}\sum_{g \in G}\rho(g)\,\pi\,\rho(g)^{-1}, $$

the factor $|G|^{-1}$ existing because $\operatorname{char} F \nmid |G|$. Then $\bar\pi$ is $F$-linear and $G$-equivariant, since conjugation by any $\rho(h)$ permutes the terms of the sum, and its restriction to $W$ is the identity, because $W$ is invariant and $\pi$ fixes $W$ pointwise. Hence $\bar\pi$ is an $F[G]$-linear projection of $V$ onto $W$, so that $V=W\oplus\ker\bar\pi$; every submodule of every module is therefore a direct summand, and every representation is completely reducible by Simple and Semisimple Modules. Applying this to the regular module gives that $F[G]$ is semisimple.

The proof is the module-theoretic form of the classical averaging argument: the group supplies an idempotent $\frac{1}{|G|}\sum_g g$ that is central and projects onto the invariants. When $\operatorname{char} F$ divides $|G|$, the averaging fails, the group algebra has a nonzero radical, and the theory becomes modular; the simple modules are then the simple modules of $F[G]/J(F[G])$, and the general theory of Change of Rings applies with $I=J(F[G])$.

Structure of the group algebra

Assume $G$ finite and $\operatorname{char} F \nmid |G|$. By the Wedderburn–Artin theorem of Simple and Semisimple Modules,

$$ F[G] \cong \prod_{i=1}^{r} M_{n_i}(D_i), $$

where $D_i$ is a finite-dimensional division algebra over $F$, the simple $F[G]$-modules are $S_1,\dots,S_r$ up to isomorphism, and $n_i=\dim_{D_i}S_i$. The number $r$ is the number of isomorphism classes of simple $F[G]$-modules, and $\operatorname{End}_{F[G]}(S_i)\cong D_i^{\mathrm{op}}$. Taking dimensions,

$$ |G|=\dim_F F[G]=\sum_{i=1}^{r} n_i^2 \dim_F D_i. $$

Over an algebraically closed field $F$ of characteristic zero, every $D_i=F$, and the formula becomes

$$ |G|=\sum_{i=1}^{r} d_i^2, \qquad d_i=\dim_F S_i. $$

The integer $r$ is then the number of conjugacy classes of $G$, by the character theory of §Characters. Over a general field the division algebras $D_i$ need not be fields, and their presence is exactly the failure of the representation theory to be governed by the base field alone; for $F=\mathbb{R}$ the division algebra $\mathbb{H}$ occurs for the quaternion group, as in Morita Equivalence.

Characters

Definition and elementary properties

Assume for this section that $F=\mathbb{C}$, so that the Wedderburn factors are matrix algebras over $\mathbb{C}$ and the simple modules are absolutely irreducible. For a finite-dimensional representation $\rho: G\to GL(V)$, the character is the function

$$ \chi_V : G \longrightarrow \mathbb{C}, \qquad \chi_V(g)=\operatorname{Tr}\rho(g). $$

It depends only on the isomorphism class of $V$, because the trace is invariant under conjugation. The following are immediate from the linear algebra of the trace.

$\chi_V(1)=\dim_{\mathbb{C}}V$.
$\chi_V(hgh^{-1})=\chi_V(g)$: the character is a class function.
$\chi_V(g^{-1})=\overline{\chi_V(g)}$, because $\rho(g^{-1})=\rho(g)^{-1}$ has eigenvalues the inverses of the eigenvalues of $\rho(g)$, which are roots of unity.
$\chi_{V\oplus W}=\chi_V+\chi_W$ and $\chi_{V\otimes W}=\chi_V\chi_W$, the second since $\operatorname{Tr}(A\otimes B)=\operatorname{Tr}A\operatorname{Tr}B$.
$\chi_{V^*}(g)=\chi_V(g^{-1})=\overline{\chi_V(g)}$ for the dual.

The characters of the irreducible representations are the irreducible characters, and they are the simple characters of the group algebra.

The orthogonality relations

On the space of class functions $G\to\mathbb{C}$ define the inner product

$$ \langle \chi, \psi \rangle_G=\frac{1}{|G|}\sum_{g \in G}\chi(g)\overline{\psi(g)}. $$

Theorem (First orthogonality). Let $\chi_i,\chi_j$ be the characters of two irreducible complex representations. Then

$$ \langle \chi_i, \chi_j \rangle_G=\delta_{ij}. $$

Proof. For irreducible $V_i,V_j$, the space $\operatorname{Hom}_{\mathbb{C}}(V_j,V_i)$ carries the action $g\cdot T=\rho_i(g)T\rho_j(g)^{-1}$, and averaging over $G$ with the factor $|G|^{-1}$ projects onto the invariants, which are $\operatorname{Hom}_G(V_j,V_i)$ by Schur's lemma. Computing the trace of the averaging operator and using Schur's lemma for the case $i=j$ gives the result; this is the standard unitary trick, and it is the statement that the matrix coefficients of the irreducible representations form an orthogonal basis of the space of functions on $G$, with the factorization

$$ \sum_{g \in G}\rho_i(g)\otimes\rho_j(g^{-1})=\frac{|G|}{d_i}\delta_{ij}\,P_i, $$

where for $i=j$ the operator $P_i$ is the flip $v\otimes w\mapsto w\otimes v$ of $V_i\otimes V_i$, whose trace is $d_i=\dim V_i$. Taking traces yields the displayed orthogonality.

Theorem (Second orthogonality). Let $\mathcal{C}$ be a conjugacy class of $G$ and let $g \in \mathcal{C}$. Then for representatives $g,h$ of classes,

$$ \sum_{i=1}^{r}\chi_i(g)\overline{\chi_i(h)}=\begin{cases} \dfrac{|G|}{|\mathcal{C}|}=|C_G(g)|, & g \sim h,\\[4pt] 0, & g \nsim h,\end{cases} $$

where $C_G(g)$ is the centralizer of $g$ and $|\mathcal{C}|=|G|/|C_G(g)|$ is the size of the class.

The two orthogonality relations together say that the character table — the matrix $\chi_i(\mathcal{C})$, indexed by irreducible characters and conjugacy classes — has orthogonal rows and orthogonal columns: weighting the column of a class $\mathcal{C}$ by $|\mathcal{C}|$, the rows are orthogonal with squared norm $|G|$, and the columns are orthogonal with squared norm $|G|/|\mathcal{C}|$; equivalently, the matrix $\bigl(\sqrt{|\mathcal{C}|/|G|}\,\chi_i(\mathcal{C})\bigr)$ is unitary. Consequently:

$r$, the number of irreducible complex characters, equals the number of conjugacy classes of $G$;
the class functions form an inner product space with orthonormal basis the irreducible characters;
every character is a non-negative integral linear combination of the irreducible characters.

Multiplicities

Theorem. Let $V$ be a finite-dimensional complex representation of the finite group $G$, and let $V_i$ run over the irreducible representations. Then the multiplicity of $V_i$ in a decomposition of $V$ as a direct sum of irreducibles is

$$ m_i=\langle \chi_V,\chi_i \rangle_G=\frac{1}{|G|}\sum_{g \in G}\chi_V(g)\overline{\chi_i(g)}, $$

and $V\cong\bigoplus_i V_i^{\oplus m_i}$ with uniquely determined multiplicities. In particular $V$ is irreducible if and only if $\langle\chi_V,\chi_V\rangle_G=1$, and $\langle\chi_V,\chi_V\rangle_G=\sum_i m_i^2$, so that the character determines the representation.

Proof. By direct orthogonality of the simple modules in a semisimple category, the multiplicity of $V_i$ in a semisimple $V$ is $\dim_{\mathbb{C}}\operatorname{Hom}_G(V_i,V)$, and the averaging projection computes this dimension as the inner product of characters. The last clause is the orthonormality $\langle\chi_i,\chi_j\rangle_G=\delta_{ij}$ together with additivity of the inner product in each variable.

This is the computational heart of the subject: entire representation-theoretic questions are reduced to inner products of class functions. Two irreducible representations are isomorphic exactly when their characters are equal, and the decomposition of any representation is determined by its character.

The Regular Representation

Let $G$ be finite and let $F=\mathbb{C}$. The character of the left regular representation is

$$ \chi_{\mathrm{reg}}(g)=\begin{cases} |G|, & g=1,\\ 0, & g \neq 1,\end{cases} $$

because the matrix of left multiplication by $g$ in the basis $G$ is a permutation matrix with no fixed point for $g\neq1$, and the number of fixed points is $|G|\delta_{g,1}$. Its decomposition follows by computing multiplicities:

$$ \langle \chi_{\mathrm{reg}}, \chi_i \rangle_G=\frac{1}{|G|}|G|\,\overline{\chi_i(1)}=d_i, $$

so that

$$ F[G] \cong \bigoplus_{i=1}^{r} V_i^{\oplus d_i}, \qquad \chi_{\mathrm{reg}}=\sum_{i=1}^{r} d_i\chi_i. $$

Evaluating at $1$ gives the dimension identity $\sum_i d_i^2=|G|$ of §Complete Reducibility. The regular representation therefore contains every irreducible exactly as many times as its degree, and it is the representation-theoretic embodiment of the group algebra; its endomorphism ring is $(F[G])^{\mathrm{op}}\cong F[G]$, by the theorem on the regular module of Modules over an Algebra, and the Wedderburn decomposition of the group algebra is read off from the isotypic decomposition of the regular module.

Tensor Products and Clebsch–Gordan

Let $V_i\otimes V_j$ be the tensor product with the diagonal action of §Basic operations. Since the tensor product character is $\chi_i\chi_j$, the multiplicities in its decomposition are

$$ V_i \otimes V_j \cong \bigoplus_{k=1}^{r}\bigl(V_k\bigr)^{\oplus c_{ij}^{k}}, \qquad c_{ij}^{k}=\langle \chi_i\chi_j,\chi_k\rangle_G=\frac{1}{|G|}\sum_{g \in G}\chi_i(g)\chi_j(g)\overline{\chi_k(g)}. $$

The integers $c_{ij}^{k}$ are the Clebsch–Gordan coefficients, or multiplicities, of the tensor product. They are non-negative integers, and they are symmetric in the first two indices, $c_{ij}^{k}=c_{ji}^{k}$, because $V_i\otimes V_j\cong V_j\otimes V_i$. They are not symmetric under an arbitrary permutation of $i,j,k$: a coefficient is the dimension of a space of invariants,

$$ c_{ij}^{k}=\dim_{\mathbb{C}}\bigl(V_i\otimes V_j\otimes V_k^*\bigr)^{G}, $$

and permuting the labels moves the dual from one tensor factor to another, so the general symmetry is $c_{ij}^{k}=c_{i^*j^*}^{k^*}$, obtained by dualising all three labels, because $\dim_{\mathbb{C}}W^{G}=\dim_{\mathbb{C}}(W^*)^{G}$. When every irreducible representation is self-dual, equivalently when every irreducible character is real-valued, the dual may be dropped and the coefficient is then symmetric under every permutation of $i,j,k$, as it is for $S_3$ below; for $G=\mathbb{Z}/3\mathbb{Z}$, whose nontrivial characters are not real-valued, $c_{11}^{2}=1$ while $c_{12}^{1}=0$, so the symmetry fails as soon as an irreducible representation non-isomorphic to its dual occurs. The coefficients define the multiplication of the representation ring or character ring $R(G)$, the free abelian group on the irreducible characters with product induced by the tensor product. The identity element of $R(G)$ is the class of the trivial one-dimensional representation, and $\chi_i\chi_j=\sum_k c_{ij}^k\chi_k$ is the structure-constant formula.

Example. For $G=S_3$ the irreducible complex characters are

$S_3$	$1$	transpositions $(3)$	$3$-cycles $(2)$
$\chi_{\mathrm{triv}}$	$1$	$1$	$1$
$\chi_{\mathrm{sign}}$	$1$	$-1$	$1$
$\chi_{\mathrm{std}}$	$2$	$0$	$-1$

with degrees $1,1,2$ and $\sum_i d_i^2=1+1+4=6=|S_3|$. The orthogonality relations hold, and the tensor square of the two-dimensional representation $\chi_{\mathrm{std}}$ satisfies

$$ \langle \chi_{\mathrm{std}}^2,\chi_{\mathrm{triv}}\rangle=\langle \chi_{\mathrm{std}}^2,\chi_{\mathrm{sign}}\rangle=\langle \chi_{\mathrm{std}}^2,\chi_{\mathrm{std}}\rangle=1, $$

so that

$$ V_{\mathrm{std}} \otimes V_{\mathrm{std}} \cong V_{\mathrm{triv}} \oplus V_{\mathrm{sign}} \oplus V_{\mathrm{std}}, $$

while $V_{\mathrm{std}}\otimes V_{\mathrm{sign}}\cong V_{\mathrm{std}}$. All these multiplicities were recomputed from the character table above by the formula for $c_{ij}^k$.

The Lie-theoretic analogue. For a compact Lie group the same formalism applies with the group algebra replaced by the algebra of representative functions and the finite sums by integrals; the classical Clebsch–Gordan rule for the representations of $\mathrm{SL}(2,\mathbb{C})$,

$$ V_m \otimes V_n \cong \bigoplus_{k=|m-n|}^{m+n} V_k, \qquad \text{all multiplicities one,} $$

is the oldest instance. The Lie theory is developed in the companion article Lie Algebras and is not repeated here; it is mentioned only to locate the finite-group tensor decomposition as the discrete case of the same construction.

Induction, Restriction and Frobenius Reciprocity

For a subgroup $H\leq G$, restriction and induction are the change-of-rings functors along the inclusion $F[H]\hookrightarrow F[G]$ of Change of Rings:

$$ \operatorname{Res}_H^G W = W \text{ as an } F[H]\text{-module}, \qquad \operatorname{Ind}_H^G V=F[G]\otimes_{F[H]} V. $$

The first adjunction of that article is Frobenius reciprocity:

$$ \operatorname{Hom}_{F[G]}(\operatorname{Ind}_H^G V, W) \cong \operatorname{Hom}_{F[H]}(V, \operatorname{Res}_H^G W), $$

and taking dimensions gives the multiplicity form: the multiplicity of an irreducible $G$-representation $W$ in $\operatorname{Ind}_H^G V$ equals the multiplicity of $V$ in $\operatorname{Res}_H^G W$. For finite $G$ the coinduced module $\operatorname{Hom}_{F[H]}(F[G],W)$ is isomorphic to $\operatorname{Ind}_H^G W$, so induction may be computed either way, and

$$ \dim_F \operatorname{Ind}_H^G V=[G:H]\dim_F V. $$

Induction is the standard source of irreducible representations of $G$ from those of $H$, and the character of an induced representation is given by the class-function formula

$$ \chi_{\operatorname{Ind}_H^G V}(g)=\frac{1}{|H|}\sum_{x \in G,\, x^{-1}gx \in H}\chi_V(x^{-1}gx), $$

which is standard and is used to compute character tables of groups from those of their subgroups.

Examples

(a) Abelian groups. If $G$ is abelian and $F$ is algebraically closed of characteristic not dividing $|G|$, every simple $F[G]$-module is one-dimensional, because $F[G]$ is commutative and its Wedderburn factors are fields. Over $\mathbb{C}$, the irreducible characters are the group homomorphisms $G\to\mathbb{C}^{\times}$; there are $|G|$ of them, forming the character group $\widehat{G}$. For $G=\mathbb{Z}/n\mathbb{Z}$, $\widehat{G}\cong\mathbb{Z}/n\mathbb{Z}$ generated by $\zeta(g)=\exp(2\pi i g/n)$, and the orthogonality relations are the finite Fourier orthogonality: $\frac{1}{n}\sum_{g}\zeta^{kg}\overline{\zeta^{lg}}=\delta_{kl}$. The group algebra is $\mathbb{C}[G]\cong\mathbb{C}^n$.

(b) Finite groups with a nonabelian group algebra. For $G=S_3$ the character table of §Tensor Products and Clebsch–Gordan gives $\mathbb{C}[S_3]\cong\mathbb{C}\oplus\mathbb{C}\oplus M_2(\mathbb{C})$, and for the quaternion group $Q_8$,

$$ \mathbb{C}[Q_8] \cong \mathbb{C}^4 \oplus M_2(\mathbb{C}), $$

with five irreducible characters of degrees $1,1,1,1,2$ and $\sum_i d_i^2=8=|Q_8|$. Over $\mathbb{R}$ the two-dimensional factor is replaced by the division algebra $\mathbb{H}$, as in Morita Equivalence; the contrast between $\mathbb{C}$ and $\mathbb{R}$ here is not pursued further.

(c) The regular representation of a finite cyclic group. For $G=\mathbb{Z}/n\mathbb{Z}$ the regular representation is $\mathbb{C}[G]$ acting on itself; its character is $n$ at the identity and $0$ elsewhere, and its decomposition $\bigoplus_\zeta V_\zeta$ has each one-dimensional character with multiplicity $1$, matching $\chi_{\mathrm{reg}}=\sum_\zeta\zeta$.

(d) Semidirect products. For a semidirect product $G=N\rtimes H$ with $N$ normal, the irreducible representations are constructed by Clifford theory from the orbits of $H$ on the irreducibles of $N$ and the projective representations of the inertia groups; the group algebra is Morita equivalent to a crossed product of $F[N]$ by $H$, and the simple modules and their endomorphism division rings are computed from that equivalence. This is the standard application of Morita Equivalence to group representations and is treated in the sources cited below.

Summary

A representation of a group $G$ over a field $F$ is a group homomorphism $G\to GL_F(V)$, equivalently a left module over the group algebra $F[G]$, and the two notions form equivalent categories. Submodules, quotients, direct sums, duality and tensor products of representations are the corresponding module constructions, the tensor product existing because $F[G]$ is a Hopf algebra with comultiplication $\Delta(g)=g\otimes g$. When $G$ is finite and $\operatorname{char} F$ does not divide $|G|$, Maschke's theorem makes $F[G]$ semisimple and every representation completely reducible; the Wedderburn–Artin theorem then writes $F[G]\cong\prod_i M_{n_i}(D_i)$, with $|G|=\sum_i n_i^2\dim_F D_i$, and over an algebraically closed field of characteristic zero this is $|G|=\sum_i d_i^2$ with the number of factors equal to the number of conjugacy classes.

Over $\mathbb{C}$, the character $\chi_V(g)=\operatorname{Tr}\rho(g)$ is a class function, additive on direct sums, multiplicative on tensor products, and the irreducible characters are orthonormal under $\langle\chi,\psi\rangle_G=|G|^{-1}\sum_g\chi(g)\overline{\psi(g)}$; the second orthogonality relation gives orthogonality down the columns of the character table, so that the number of irreducible characters equals the number of conjugacy classes, and the multiplicity of an irreducible in any representation is an inner product of characters. The regular representation has character $|G|\delta_{g,1}$ and decomposes as $\bigoplus_i V_i^{\oplus d_i}$, so its dimension identity is $\sum_i d_i^2=|G|$. Tensor products decompose with the Clebsch–Gordan multiplicities $c_{ij}^k=\langle\chi_i\chi_j,\chi_k\rangle$, which give the representation ring $R(G)$ its multiplication; for $S_3$ the computation $V_{\mathrm{std}}\otimes V_{\mathrm{std}}\cong V_{\mathrm{triv}}\oplus V_{\mathrm{sign}}\oplus V_{\mathrm{std}}$ illustrates the general formula, and the Lie-theoretic Clebsch–Gordan rule is the continuous analogue, treated in the companion Lie theory article. Induction and restriction are extension and restriction of scalars along $F[H]\hookrightarrow F[G]$ and satisfy Frobenius reciprocity. Every result is a statement about modules over the group algebra, and none of them requires a physical interpretation.

Summary of Notation

Symbol	Meaning
$R$, $F$	base commutative ring, base field
$G$, $H$	groups; $G$ finite in the character theory
$F[G]$	group algebra
$V$, $W$, $V_i$	representations, i.e. $F[G]$-modules
$\rho : G \to GL_F(V)$	a representation
$\operatorname{Rep}_F(G)$	category of representations, $\cong\operatorname{Mod}(F[G])$
$\operatorname{End}_G(V)$	intertwining operators, $\cong\operatorname{End}_{F[G]}(V)$
$\Delta$, $\varepsilon$, $S$	comultiplication, counit, antipode of $F[G]$
$\chi_V(g)=\operatorname{Tr}\rho(g)$	character of $V$
$\langle\chi,\psi\rangle_G=\frac{1}{\|G\|}\sum_g\chi(g)\overline{\psi(g)}$	inner product of class functions
$\chi_i$, $d_i$	irreducible characters and their degrees
$r$	number of irreducible complex characters, $=$ number of conjugacy classes
$C_G(g)$	centralizer of $g$
$c_{ij}^k=\langle\chi_i\chi_j,\chi_k\rangle$	Clebsch–Gordan multiplicities
$V^$, $i^$	dual representation $V^*=\operatorname{Hom}_F(V,F)$ and the dual label
$W^G$	subspace of $G$-invariant vectors of a representation $W$
$R(G)$	representation ring
$\chi_{\mathrm{reg}}$	character of the regular representation
$\operatorname{Ind}_H^G$, $\operatorname{Res}_H^G$	induction and restriction
$J(F[G])$	Jacobson radical in the modular case
$\widehat{G}$	character group of an abelian group
$\mathbb{H}$	quaternions, as an endomorphism division ring over $\mathbb{R}$