The Group Algebra and Representations of Finite Groups

Let $R$ be a commutative ring (Commutative ring) and let $G$ be a group (Group and abelian group). The group ring $R[G]$ is the free left $R$-module on the set $G$ (The free module on a set and its standard basis).

For each $g\in G$, write $[g]\in R[G]$ for the standard basis vector indexed by $g$. Thus every element of $R[G]$ has a unique expression ${\sum }_{g\in F}{r}_g[g]$ with $F\subseteq G$ finite and $r_g\in R$.

The notation is deliberately that of formal finite sums: at this stage $R[G]$ is only the underlying free $R$-module with its distinguished basis. The multiplication satisfying $[g][h]=[gh]$ and the resulting $R$-algebra structure are constructed in The group ring $R[G]$ is a unital $R$-algebra with basis $G$, and each $g\in G$ is a unit of $R[G]$.

• The support is finite by definition because $R[G]$ is the direct sum, not the full product, of copies of $R$ indexed by $G$.

• The basis vectors are written $[g]$ to keep the group element $g$ distinct from its basis image in the free module.

Let $R$ be a commutative ring and let $G$ be a group. Write $[g]\in R[G]$ for the basis vector of The group ring $R[G]$ of finitely supported formal $R$-linear combinations of group elements indexed by $g\in G$.

There is a unique $R$-bilinear multiplication on $R[G]$ satisfying $[g][h]=[gh](g,h\in G)\mathrm{.}$ With this product, $R[G]$ is a unital $R$-algebra whose underlying $R$-module has basis $\{[g]:g\in G\}$. Its identity is $[e]$, where $e$ is the identity of $G$, and every basis element $[g]$ is a unit with inverse $[g^{-1}]$.

Let $R$ be a commutative ring and let $G$ be a group. For an element of $R[G]$ written uniquely as a finite sum $x={\sum }_{g\in F}{r}_g[g]$ (The group ring $R[G]$ of finitely supported formal $R$-linear combinations of group elements), define the augmentation map $\epsilon :R[G]⟶R,\epsilon (x):={\sum }_{g\in F}{r}_g\mathrm{.}$

Because the expansion in the basis $\{[g]\}$ is unique, this is a well-defined $R$-linear map. It is also a ring homomorphism in the sense of Ring homomorphism: additive, multiplicative, and required to send $1$ to $1$: additivity is immediate, it sends $[e]$ to $1_R$, and if $x={\sum }_g{r}_g[g]$ and $y={\sum }_h{s}_h[h]$, then $\epsilon (xy)={\sum }_{g,h}{r}_g{s}_h=\left({\sum }_g{r}_g\right)\left({\sum }_h{s}_h\right)=\epsilon (x)\epsilon (y)$ by the multiplication formula supplied by The group ring $R[G]$ is a unital $R$-algebra with basis $G$, and each $g\in G$ is a unit of $R[G]$.

The augmentation ideal of $R[G]$ is the kernel $I_G:=\ker \epsilon ,$ which is a two-sided ideal by The kernel of a ring homomorphism is a two-sided ideal.

• The basis element $[g]$ always satisfies $\epsilon ([g])=1_R$.

• When $G$ is finite, the element ${\sum }_{g\in G}[g]$ is the image under the basis sum of the constant function $1$ on $G$.

Let $k$ be a field and let $G$ be a finite group. Then the group algebra $k[G]$ is finite-dimensional over $k$, with ${\dim }_{k}k[G]=\mathrm{\mid }G\mathrm{\mid }\mathrm{.}$

Let $k$ be a field and let $G$ be a group. Then the group algebra $k[G]$ is commutative if and only if $G$ is abelian.

Let $R$ be a commutative ring (Commutative ring), let $G$ be a group (Group and abelian group), and let $M$ be a left $R$-module (Unital left and right modules over a ring; unqualified module means left module).

A left $G$-action on $M$ by $R$-linear maps is a left group action $G\times M\to M$, $(g,m)\mapsto g\cdot m$, in the sense of Left group actions, transitive actions, and faithful actions, such that for every $g\in G$ the map $m⟼g\cdot m$ is an $R$-module homomorphism $M\to M$ (Module homomorphism and isomorphism, kernel, image and cokernel).

Equivalently, the action satisfies $g\cdot (m+m^{\mathrm{\prime }})=g\cdot m+g\cdot m^{\mathrm{\prime }},g\cdot (rm)=r(g\cdot m)$ for all $g\in G$, $m,m^{\mathrm{\prime }}\in M$, and $r\in R$.

An $R$-linear $G$-module, or simply a $G$-module over $R$, is a left $R$-module together with such an action.

• Because $g^{-1}$ acts as an inverse in the group-action sense, each map $m\mapsto g\cdot m$ is automatically an $R$-module automorphism.

• When $R$ is a field, this is the module-language form of a representation.

Let $k$ be a field (Field) and let $G$ be a group. A finite-dimensional representation of $G$ over $k$ is a finite-dimensional $k$-vector space $V$ (Vector space over a field, Finite-dimensional vector space, and its dimension ${\dim }_{F}V$; infinite-dimensional means having no finite basis) together with a group homomorphism $\rho :G⟶\mathrm{GL}(V),$ where $\mathrm{GL}(V)$ denotes the group of invertible $k$-linear maps $V\to V$ (Invertible linear maps, linear isomorphisms, and inverse linear maps, Monoid homomorphism and group homomorphism).

The associated action is $g\cdot v:=\rho (g)(v)\mathrm{.}$ This makes $V$ into a $G$-module over $k$ in the sense of An $R$-linear action of $G$ on a left $R$-module, and a $G$-module over $R$.

The degree of the representation is the dimension of its underlying vector space: $\mathrm{deg}(\rho ):={\dim }_{k}V\mathrm{.}$

• The finite-dimensionality convention is part of the definition on this page, not a later standing assumption.

• The representation is determined equally well by the homomorphism $\rho$ or by the associated $k$-linear action of $G$ on $V$.

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

A subrepresentation is a linear subspace $W\le V$ (Linear subspace of a vector space) such that $\rho (g)(W)\subseteq W$ for every $g\in G$. Equivalently, $g\cdot w\in W$ for all $g\in G$ and $w\in W$.

If $\sigma :G\to \mathrm{GL}(W)$ is another representation, the direct sum representation $\rho \oplus \sigma$ on $V\oplus W$ is defined by $g\cdot (v,w):=(\rho (g)v,\sigma (g)w)\mathrm{.}$

The representation $\rho$ is irreducible when $V\mathrm{\ne }0$ and its only subrepresentations are $0$ and $V$ itself.

• The zero representation on the zero vector space is not irreducible, by the explicit nonzero requirement.

• A decomposition $V=U\oplus W$ into nonzero subrepresentations is exactly a nontrivial decomposition of the representation as a direct sum.

Let $\rho :G\to \mathrm{GL}(V)$ and $\sigma :G\to \mathrm{GL}(W)$ be representations of $G$ over the same field $k$.

A linear map $f:V\to W$ (Linear map between vector spaces over the same field) is an intertwiner, or $G$-equivariant map, when $f(\rho (g)v)=\sigma (g)f(v)(g\in G,v\in V)\mathrm{.}$ Equivalently, $f\rho (g)=\sigma (g)f(g\in G)\mathrm{.}$

Write ${\mathrm{Hom}}_G(V,W)$ for the set of all intertwiners. It is a subset of the vector space $\mathcal{L}(V,W)$ of all linear maps (The space $\mathcal{L}(V,W)$ of linear maps with pointwise addition and scalar multiplication). When $V=W$, write ${\mathrm{End}}_G(V):={\mathrm{Hom}}_G(V,V)\mathrm{.}$

The two representations are equivalent if there is an invertible intertwiner $V\to W$.

A representation $\rho$ is faithful when $\rho (g)={\mathrm{id}}_V$ implies $g=e$.

• Faithfulness says that the action remembers every group element.

• The equality $f\rho (g)=\sigma (g)f$ is the representation-language form of a module homomorphism condition.

Let $R$ be a commutative ring, let $G$ be a group, and let $M$ be a left $R$-module. Then:

1. every $R$-linear action of $G$ on $M$ extends uniquely to a left $R[G]$-module structure on $M$ satisfying $(r[e])\cdot m=rm(r\in R,m\in M);$
2. every left $R[G]$-module structure on $M$ satisfying $(r[e])\cdot m=rm(r\in R,m\in M)$ restricts to an $R$-linear action of $G$ on $M$;
3. these two constructions are inverse to each other.

Moreover, if $M$ and $N$ carry the corresponding structures, then an $R$-linear map $f:M\to N$ is $G$-equivariant if and only if it is an $R[G]$-module homomorphism.

Let $k$ be a field, let $G$ be a group, and let $V$ be a finite-dimensional representation of $G$ over $k$. Under the correspondence of For a commutative ring $R$, $R$-linear $G$-actions are exactly the compatible left $R[G]$-module structures:

1. the subrepresentations of $V$ are exactly the $k[G]$-submodules of $V$;
2. $V$ is irreducible if and only if it is simple as a $k[G]$-module.

Let $V$ and $W$ be representations of a group $G$ over a field $k$.

1. The intertwiner space ${\mathrm{Hom}}_G(V,W)$ is a $k$-vector space.
2. The endomorphism space ${\mathrm{End}}_G(V)$ is a $k$-algebra.

Let $k$ be a field and let $G$ be a finite group.

The trivial representation of $G$ over $k$ is the one-dimensional representation on the vector space $k$ in which every $g\in G$ acts as the identity map.

The regular representation of $G$ over $k$ is the representation on the vector space $k[G]$ (The group ring $R[G]$ of finitely supported formal $R$-linear combinations of group elements) given by left multiplication by the basis units: $g\cdot x:=[g]x(g\in G,x\in k[G])\mathrm{.}$ Equivalently, on the basis vectors of $k[G]$, $g\cdot [h]=[gh]\mathrm{.}$ This uses the ring structure and unit property proved in The group ring $R[G]$ is a unital $R$-algebra with basis $G$, and each $g\in G$ is a unit of $R[G]$.

More generally, if $X$ is a finite left $G$-set (Left group actions, transitive actions, and faithful actions), the free $k$-module $k^{(X)}$ on $X$ (The free module on a set and its standard basis) becomes a representation by $g\cdot e_x:=e_{g\cdot x}(g\in G,x\in X)\mathrm{.}$ This is the permutation representation attached to the action of $G$ on $X$.

• The regular representation is the permutation representation of $G$ acting on itself by left translation.

• Because $G$ and $X$ are finite, these constructions are finite-dimensional.

Let $k$ be a field.

For the symmetric group $S_n$ (The symmetric group $\mathrm{Sym}(X)$: the bijections of a set $X$ under composition), the sign representation over $k$ is the one-dimensional representation on $k$ in which $\sigma \cdot a:=\mathrm{sgn}(\sigma )a\mathrm{.}$ This is a representation because $\mathrm{sgn}:S_n\to \{\pm 1\}$ is a group homomorphism (The sign is a homomorphism $S_n\to \{+1,-1\}$, surjective exactly when $n\ge 2$).

Now let $\rho :G\to \mathrm{GL}(V)$ be a representation of a group $G$ over $k$, and let $H\le G$ be a subgroup (Subgroup). The restriction of $\rho$ to $H$ is the representation ${\mathrm{Res}}_H^G(V):=\rho {\mathrm{\mid }}_H:H⟶\mathrm{GL}(V),h⟼\rho (h)\mathrm{.}$ It has the same underlying vector space $V$ and only forgets the action of the elements outside $H$.

• In characteristic $2$, the sign representation coincides with the trivial representation because $-1=1$ in the field.

• Restriction changes the acting group, not the underlying vector space.

Let $G$ be a finite group and let $k$ be a field. Then the regular representation of $G$ over $k$ is faithful.

Let $G$ be a finite group and let $V$ be an irreducible representation of $G$ over a field $k$. Then there is a surjective morphism from the regular representation of $G$ over $k$ onto $V$.

Let $G$ be a finite group and let $V$ be an irreducible representation of $G$ over a field $k$. Then $\mathrm{deg}(V)\le \mathrm{\mid }G\mathrm{\mid }\mathrm{.}$

Let $G$ be a finite group and let $k$ be a field. The field $k$ is a splitting field for $G$ when every irreducible representation $V$ of $G$ over $k$ satisfies ${\mathrm{End}}_G(V)=k,$ meaning that every $G$-endomorphism of $V$ is a scalar operator $\lambda {\mathrm{id}}_V$ with $\lambda \in k$ (A finite-dimensional representation $\rho :G\to \mathrm{GL}(V)$ over a field, and its degree, Intertwiners, the spaces ${\mathrm{Hom}}_G(V,W)$ and ${\mathrm{End}}_G(V)$, equivalent representations, and faithful representations).

• This is the representation-theoretic condition used on the next pages. It is weaker than algebraic closedness, and Brauer's theorem records one sufficient roots-of-unity criterion for it without making that criterion the definition.

• The definition quantifies only over irreducible representations of the fixed finite group $G$.