Characters and the Orthogonality Relations

The ordinary-character-theory items on this page work inside the following setting, fixed once here: $G$ is a finite group, the base field is $k=\mathbb{C}$, and every representation is finite-dimensional (A finite-dimensional representation $\rho :G\to \mathrm{GL}(V)$ over a field, and its degree). This is ordinary character theory in the sense of Webb, Chapter 3 and Etingof et al., Section 3.3: no infinite group, unitary-representation, or modular-character material is load-bearing in the character-theoretic arguments on the page. The quotient-factorisation result A representation with kernel containing a normal subgroup factors through the quotient, and irreducibility is unchanged by inflation is deliberately stated in the greater generality of an arbitrary group, field, and representation because its proof needs none of these standing restrictions.

The choice of $k=\mathbb{C}$ is what makes the hypotheses of the published representation-theory spine available. Since $\mathrm{char}\mathbb{C}=0$ and $G$ is finite, the characteristic does not divide $\mathrm{\mid }G\mathrm{\mid }$, so every finite-dimensional representation is completely reducible (If $\mathrm{char}k\nmid \mathrm{\mid }G\mathrm{\mid }$, every finite-dimensional representation of $G$ is completely reducible), and the field is algebraically closed, which feeds the count of irreducibles against conjugacy classes (If $k$ is algebraically closed and $\mathrm{char}k\nmid \mathrm{\mid }G\mathrm{\mid }$, the number of irreducible representations of $G$ equals the number of conjugacy classes). These hypotheses are restated where they are consumed; the present remark fixes the default scope without narrowing an item that explicitly states broader hypotheses.

Let $\rho :G\to \mathrm{GL}(V)$ be a finite-dimensional complex representation (A finite-dimensional representation $\rho :G\to \mathrm{GL}(V)$ over a field, and its degree). Its character is the function

${\chi }_V:G⟶\mathbb{C},{\chi }_V(g):=\mathrm{tr}\text{\hspace{0.17em}⁣}({\rho }_V(g)),$

where the trace is the basis-independent trace of an endomorphism (The basis-independent trace of an endomorphism of a finite-dimensional vector space); one writes ${\chi }_{\rho }$ or $\chi$ when the representation or the group is fixed.

The definition is well posed in two senses, recorded here because both are used throughout the page. First, the value does not depend on a choice of basis of $V$: matrices of one endomorphism in two ordered bases are similar, and similar matrices have equal trace (The basis-independent trace of an endomorphism of a finite-dimensional vector space). Second, equivalent representations have equal characters: if $T:V\to W$ is an invertible intertwiner, then ${\rho }_W(g)=T{\rho }_V(g)T^{-1}$, so $\mathrm{tr}({\rho }_W(g))=\mathrm{tr}({\rho }_V(g)T^{-1}T)=\mathrm{tr}({\rho }_V(g))$ by the identity $\mathrm{tr}(AB)=\mathrm{tr}(BA)$ (For $A\in M_{m\times n}(F)$ and $B\in M_{n\times m}(F)$, $\mathrm{tr}(AB)=\mathrm{tr}(BA)$). Thus the character depends only on the equivalence class of the representation.

Let $G$ be a finite group. A function $f:G\to \mathbb{C}$ is a class function when it is constant on every conjugacy class (The conjugacy class ${\mathrm{Cl}}_G(x)$ and centralizer $C_G(x)$ of an element):

$f(gx{g}^{-1})=f(x)\text{for all }g,x\in G\mathrm{.}$

The set of all class functions is written

$\mathrm{c}\mathrm{f}(G):=\{f:G\to \mathbb{C}:f(gx{g}^{-1})=f(x)\text{ for all }g,x\in G\}\mathrm{.}$

With pointwise addition $(f+h)(g):=f(g)+h(g)$ and pointwise scalar multiplication $(\lambda f)(g):=\lambda f(g)$, the set $\mathrm{c}\mathrm{f}(G)$ is a complex vector space (Vector space over a field): the axioms (V1)–(V5) are inherited from the field $\mathbb{C}$ at every argument $g\in G$. Because $G$ is finite, the evaluation functions $f\mapsto f(x)$ are complex-linear, and a class function is determined by its values on one representative of each conjugacy class.

For two class functions $\phi ,\psi \in \mathrm{c}\mathrm{f}(G)$ (Class functions and the complex vector space $\mathrm{c}\mathrm{f}(G)$), define

$⟨\phi ,\psi ⟩:=\frac{1}{\mathrm{\mid }G\mathrm{\mid }}{\sum }_{g\in G}\phi (g)\overline{\psi (g)},$

the finite sum being that of (A finite sum in a commutative monoid indexed by an arbitrary finite set) for the additive commutative monoid of $\mathbb{C}$. This is the standard Hermitian form on the complex vector space of class functions.

This assignment is an inner product in the exact sense of the published definition (Real and complex inner product spaces, with the inner product linear in the first argument), with the inner product linear in the first argument. Linearity in the first slot is immediate from the pointwise vector-space structure of $\mathrm{c}\mathrm{f}(G)$ and linearity of the finite sum. The conjugate-symmetry clause holds because conjugation (Real and imaginary parts, complex conjugation, and modulus) is an involution: $⟨\psi ,\phi ⟩=\frac{1}{\mathrm{\mid }G\mathrm{\mid }}{\sum }_g\psi (g)\overline{\phi (g)}=\overline{\frac{1}{\mathrm{\mid }G\mathrm{\mid }}{\sum }_g\phi (g)\stackrel{‾}{\psi (g)}}=\overline{⟨\phi ,\psi ⟩}$. For definiteness, $⟨\phi ,\phi ⟩=\frac{1}{\mathrm{\mid }G\mathrm{\mid }}{\sum }_g\mathrm{\mid }\phi (g){\mathrm{\mid }}^2$ is a sum of nonnegative real numbers, hence nonnegative real; it is $0$ exactly when every term $\mathrm{\mid }\phi (g){\mathrm{\mid }}^2$ is $0$, and since $\mathrm{\mid }z\mathrm{\mid }=0$ holds only for $z=0$ in $\mathbb{C}$, that says exactly $\phi =0$ as a function.

A complex character $\chi$ (The character ${\chi }_V(g)=\mathrm{tr}({\rho }_V(g))$ of a finite-dimensional complex representation) is irreducible when $\chi ={\chi }_V$ for an irreducible representation $V$ of $G$ over $\mathbb{C}$ (Subrepresentations, direct sums of representations, and irreducibility).

The phrase is well defined for characters rather than for individual representations: if $V$ and $W$ are equivalent, then ${\chi }_V={\chi }_W$ by the invariance clause of (The character ${\chi }_V(g)=\mathrm{tr}({\rho }_V(g))$ of a finite-dimensional complex representation), and equivalence preserves irreducibility. Throughout the page, "the irreducible characters" means one representative ${\chi }_1,\dots ,{\chi }_r$ from each equivalence class of irreducible complex representations of the finite group $G$; the number $r$ is finite and equals the number of conjugacy classes of $G$ by the published count (If $k$ is algebraically closed and $\mathrm{char}k\nmid \mathrm{\mid }G\mathrm{\mid }$, the number of irreducible representations of $G$ equals the number of conjugacy classes).

Let $G$ be a finite group, let ${\chi }_1,\dots ,{\chi }_r$ be the irreducible complex characters of $G$, one per equivalence class (An irreducible complex character), and let $g_1,\dots ,g_s$ be representatives of the distinct conjugacy classes of $G$ (The conjugacy class ${\mathrm{Cl}}_G(x)$ and centralizer $C_G(x)$ of an element). The character table of $G$ is the array whose rows are indexed by ${\chi }_1,\dots ,{\chi }_r$, whose columns are indexed by $g_1,\dots ,g_s$, and whose $(i,j)$ entry is ${\chi }_i(g_j)$.

The array is written with a top row recording the conjugacy-class sizes $\mathrm{\mid }{\mathrm{Cl}}_G(g_j)\mathrm{\mid }$ above the column labels, and with the first column recording the degrees ${\chi }_i(1)$ beside the row labels. The ordering of the rows and of the columns is arbitrary: permuting either does not change the table as data. Because each ${\chi }_i$ is constant on each conjugacy class, the entry ${\chi }_i(g_j)$ is independent of the choice of representative $g_j$. That the number of rows equals the number of columns is a theorem about the table, proved later on this page as The character table is square and invertible, not part of the present definition.

Let $\rho :G\to \mathrm{GL}(V)$ and $\sigma :G\to \mathrm{GL}(W)$ be finite-dimensional complex representations of a group $G$ (A finite-dimensional representation $\rho :G\to \mathrm{GL}(V)$ over a field, and its degree). Their tensor product representation $\rho \otimes \sigma$ is the representation of $G$ on the complex tensor product $V{\otimes }_{\mathbb{C}}W$ given on elementary tensors by

$g\cdot (v\otimes w):=(\rho (g)v)\otimes (\sigma (g)w)(g\in G,v\in V,w\in W)\mathrm{.}$

The action is well defined. For fixed $g\in G$, the assignment $(v,w)\mapsto \rho (g)v\otimes \sigma (g)w$ is additive in each variable and $\mathbb{C}$-balanced, because $\rho (g)$ and $\sigma (g)$ are $\mathbb{C}$-linear: $\rho (g)(v\lambda )\otimes \sigma (g)w=(\rho (g)v)\lambda \otimes \sigma (g)w$. By Universal property of the tensor product for balanced maps into abelian groups there is a unique $\mathbb{C}$-linear map $\rho (g)\otimes \sigma (g):V{\otimes }_{\mathbb{C}}W\to V{\otimes }_{\mathbb{C}}W$ with the displayed value on every elementary tensor. Because the two sides of $g\cdot (h\cdot (v\otimes w))=(gh)\cdot (v\otimes w)$ agree on every elementary tensor and the elementary tensors span $V{\otimes }_{\mathbb{C}}W$, the assignment $g\mapsto \rho (g)\otimes \sigma (g)$ is a group homomorphism into $\mathrm{GL}(V{\otimes }_{\mathbb{C}}W)$, with $\rho (1)\otimes \sigma (1)=\mathrm{id}$. The underlying space is finite-dimensional: bases of $V$ and $W$ give the basis $(v_i\otimes w_j{)}_{i,j}$, so $\dim (V{\otimes }_{\mathbb{C}}W)=(\dim V)(\dim W)$.

Let $\rho :G\to \mathrm{GL}(V)$ be a finite-dimensional complex representation of a group $G$ (A finite-dimensional representation $\rho :G\to \mathrm{GL}(V)$ over a field, and its degree). Its dual or contragredient representation is the representation ${\rho }^{\ast }$ of $G$ on the algebraic dual space $V^{\ast }$ (Linear functionals and the algebraic dual $V^{\ast }=\mathcal{L}(V,F)$) defined by

$(g\cdot f)(v):=f(g^{-1}\cdot v)(g\in G,f\in V^{\ast },v\in V),$

equivalently $g\cdot f=f\circ \rho (g^{-1})$.

Each operator ${\rho }^{\ast }(g)$ is $\mathbb{C}$-linear because $f$ and $\rho (g^{-1})$ are. The inverse appears so that composition is order-preserving: writing $g\cdot f=f\circ \rho (g^{-1})$,

$(gh)\cdot f=f\circ \rho ((gh{)}^{-1})=f\circ \rho (h^{-1})\circ \rho (g^{-1})=g\cdot (h\cdot f),$

and $1\cdot f=f\circ {\mathrm{id}}_V=f$. Hence ${\rho }^{\ast }$ is a group homomorphism $G\to \mathrm{GL}(V^{\ast })$, a representation of the same degree as $\rho$.

Let $\rho :G\to \mathrm{GL}(V)$ be a finite-dimensional complex representation of a finite group $G$, with character $\chi ={\chi }_V$ (The character ${\chi }_V(g)=\mathrm{tr}({\rho }_V(g))$ of a finite-dimensional complex representation). The kernel of the character is

$\ker \chi :=\{g\in G:\chi (g)=\chi (1)\}\mathrm{.}$

The value $\chi (1)$ is the reference point of the definition, not a random number: The kernel of a complex character agrees with the kernel of any representation affording it proves that this set equals the kernel $\ker \rho$ of the representation, so the two readings of "kernel" never conflict on this page.

Let $G$ be a finite group, let $V$ be a finite-dimensional complex representation of $G$, and let $\chi ={\chi }_V$ be its character. Then, for all $g,h\in G$:

1. $\chi (1)=\dim V$;
2. $\chi (gh{g}^{-1})=\chi (h)$, so $\chi$ is a class function;
3. $\chi (g)$ is a sum of $\dim V$ roots of unity, namely the eigenvalues of $\rho (g)$ counted with multiplicity;
4. $\mathrm{\mid }\chi (g)\mathrm{\mid }\le \chi (1)$, with equality if and only if $\rho (g)$ is a scalar operator;
5. $\chi (g^{-1})=\overline{\chi (g)}$.

Let $\rho :G\to \mathrm{GL}(V)$ be a finite-dimensional complex representation of a finite group $G$, with character $\chi ={\chi }_V$. Then

$\ker \chi =\ker \rho ,$

where $\ker \rho$ is the kernel of the group homomorphism $\rho$ and $\ker \chi =\{g\in G:\chi (g)=\chi (1)\}$ is the kernel of the character.

Let $G$ be a finite group and let $V$ and $W$ be finite-dimensional complex representations of $G$. Then, for every $g\in G$:

1. ${\chi }_{V\oplus W}(g)={\chi }_V(g)+{\chi }_W(g)$;
2. ${\chi }_{V\otimes W}(g)={\chi }_V(g){\chi }_W(g)$;
3. ${\chi }_{V^{\ast }}(g)=\overline{{\chi }_V(g)}$.

Let $X$ be a finite left $G$-set and let ${\mathbb{C}}^{(X)}$ be the permutation representation over $\mathbb{C}$. Then the character of ${\mathbb{C}}^{(X)}$ is

${\chi }_{{\mathbb{C}}^{(X)}}(g)=\mid \{x\in X:g\cdot x=x\}\mid ,$

the number of fixed points of $g$ acting on $X$.

Let $\rho :G\to \mathrm{GL}(V)$ be a representation of a group $G$ over a field $k$ (A finite-dimensional representation $\rho :G\to \mathrm{GL}(V)$ over a field, and its degree). The fixed subspace of $V$ is

$V^G:=\{v\in V:g\cdot v=v\text{ for every }g\in G\}\mathrm{.}$

The set $V^G$ is a linear subspace of $V$ (Linear subspace of a vector space): it contains $0_V$ because every operator $\rho (g)$ is linear, it is closed under addition and scalar multiplication for the same reason, and the defining equations are the required pointwise identities. When $G$ is finite and $k=\mathbb{C}$, The averaging operator projects onto the fixed subspace shows that an explicit projection of $V$ has image exactly $V^G$.

Let $\rho :G\to \mathrm{GL}(V)$ be a finite-dimensional representation of a finite group $G$ over $\mathbb{C}$. The averaging operator

$P:=\frac{1}{\mathrm{\mid }G\mathrm{\mid }}{\sum }_{g\in G}\rho (g)$

satisfies $P^2=P$ and has image exactly $V^G$; consequently $\mathrm{tr}P=\dim V^G$.

Let $G$ be a finite group, let $V$ and $W$ be finite-dimensional complex representations of $G$, and let

$\mathrm{\Phi }:V^{\ast }{\otimes }_{\mathbb{C}}W⟶{\mathrm{Hom}}_{\mathbb{C}}(V,W)$

be the natural isomorphism of For finite-dimensional $V$, the canonical map $V^{\ast }{\otimes }_{F}W\to {\mathrm{Hom}}_F(V,W)$ is an isomorphism, sending $f\otimes w$ to the map $v\mapsto f(v)w$. Then $\mathrm{\Phi }$ is an intertwiner between the diagonal representation on $V^{\ast }\otimes W$ and the conjugation representation on ${\mathrm{Hom}}_{\mathbb{C}}(V,W)$, and it carries the fixed subspace $(V^{\ast }\otimes W{)}^G$ bijectively onto ${\mathrm{Hom}}_G(V,W)$.

Let $G$ be a finite group and let $V$ and $W$ be finite-dimensional complex representations of $G$ with characters ${\chi }_V$ and ${\chi }_W$. Then

$⟨{\chi }_V,{\chi }_W⟩=\dim {\mathrm{Hom}}_G(W,V)\mathrm{.}$

Let $G$ be a finite group and let ${\chi }_1,\dots ,{\chi }_r$ be the irreducible complex characters of $G$, one from each equivalence class. Then

$⟨{\chi }_i,{\chi }_j⟩={\delta }_{ij}(1\le i,j\le r)\mathrm{.}$