Editing Group ring (section)

==Group algebra over a finite group==
Group algebras occur naturally in the theory of [[group representation]]s of [[finite group]]s. The group algebra ''K''[''G''] over a field ''K'' is essentially the group ring, with the field ''K'' taking the place of the ring. As a set and vector space, it is the [[free vector space]] on ''G'' over the field ''K''. That is, for ''x'' in ''K''[''G''],
:<math>x=\sum_{g\in G} a_g g.</math>

The [[algebra over a field|algebra]] structure on the vector space is defined using the multiplication in the group:
:<math>g \cdot h = gh,</math>
where on the left, ''g'' and ''h'' indicate elements of the group algebra, while the multiplication on the right is the group operation (denoted by juxtaposition).

Because the above multiplication can be confusing, one can also write the [[basis vector]]s of ''K''[''G''] as ''e''<sub>''g''</sub> (instead of ''g''), in which case the multiplication is written as:
:<math>e_g \cdot e_h = e_{gh}.</math>

===Interpretation as functions===
Thinking of the [[free vector space]] as ''K''-valued functions on ''G'', the algebra multiplication is [[convolution]] of functions.

While the group algebra of a ''finite'' group can be identified with the space of functions on the group, for an infinite group these are different. The group algebra, consisting of ''finite'' sums, corresponds to functions on the group that vanish for [[cofinitely]] many points; topologically (using the [[discrete topology]]), these correspond to functions with [[compact support]].

However, the group algebra ''K''[''G''] and the space of functions {{nowrap|1=''K''<sup>''G''</sup> := Hom(''G'', ''K'')}} are dual: given an element of the group algebra

:<math>x = \sum_{g\in G} a_g g</math>

and a function on the group {{nowrap|''f'' : ''G'' → ''K''}} these pair to give an element of ''K'' via

:<math>(x,f) = \sum_{g\in G} a_g f(g),</math>

which is a well-defined sum because it is finite.

=== Representations of a group algebra ===
Taking ''K''[''G''] to be an abstract algebra, one may ask for [[group representation|representations]] of the algebra acting on a ''K-''vector space ''V'' of dimension ''d''. Such a representation

:<math>\tilde{\rho}:K[G]\rightarrow \mbox{End} (V)</math>

is an algebra homomorphism from the group algebra to the algebra of [[endomorphism]]s of ''V'', which is isomorphic to the ring of ''d × d'' matrices: <math>\mathrm{End}(V)\cong M_{d}(K) </math>. Equivalently, this is a [[module (mathematics)|left ''K''[''G'']-module]] over the abelian group ''V''.

Correspondingly, a group representation

:<math>\rho:G\rightarrow \mbox{Aut}(V),</math>

is a group homomorphism from ''G'' to the group of linear automorphisms of ''V'', which is isomorphic to the [[general linear group]] of invertible matrices: <math>\mathrm{Aut}(V)\cong \mathrm{GL}_d(K) </math>. Any such representation induces an algebra representation

:<math>\tilde{\rho}:K[G]\rightarrow \mbox{End}(V),</math>

simply by letting <math>\tilde{\rho}(e_g) = \rho(g)</math> and extending linearly. Thus, representations of the group correspond exactly to representations of the algebra, and the two theories are essentially equivalent.

=== Regular representation ===
{{Main|Regular representation}}
The group algebra is an algebra over itself; under the correspondence of representations over ''R'' and ''R''[''G''] modules, it is the [[regular representation]] of the group.

Written as a representation, it is the representation ''g'' {{mapsto}} ''ρ''<sub>''g''</sub> with the action given by <math>\rho(g)\cdot e_h = e_{gh}</math>, or

:<math>\rho(g)\cdot r =  \sum_{h\in G} k_h \rho(g)\cdot e_h = \sum_{h\in G} k_h e_{gh}. </math>

===Semisimple decomposition===
The dimension of the vector space ''K''[''G''] is just equal to the number of elements in the group. The field ''K'' is commonly taken to be the complex numbers '''C''' or the reals '''R''', so that one discusses the group algebras '''C'''[''G''] or '''R'''[''G''].

The group algebra '''C'''[''G''] of a finite group over the complex numbers is a [[semisimple ring]]. This result, [[Maschke's theorem]], allows us to understand '''C'''[''G''] as a finite [[Product of rings|product]] of [[matrix ring]]s with entries in '''C'''. Indeed, if we list the complex [[irreducible representation]]s of ''G'' as ''V<sub>k</sub>'' for ''k'' = 1, . . . , ''m'', these correspond to [[group homomorphism]]s <math>\rho_k: G\to \mathrm{Aut}(V_k)</math> and hence to algebra homomorphisms <math>\tilde\rho_k: \mathbb{C}[G]\to \mathrm{End}(V_k)</math>. Assembling these mappings gives an algebra isomorphism
:<math>\tilde\rho : \mathbb{C}[G] \to \bigoplus_{k=1}^m \mathrm{End}(V_k)
\cong \bigoplus_{k=1}^m M_{d_k}(\mathbb{C}),  </math>
where ''d<sub>k</sub>'' is the dimension of ''V<sub>k</sub>''. The subalgebra of '''C'''[''G''] corresponding to End(''V<sub>k</sub>'') is the [[Ideal (ring theory)|two-sided ideal]] generated by the [[Idempotent (ring theory)|idempotent]]
:<math>\epsilon_k = \frac{d_k}{|G|}\sum_{g\in G}\chi_k(g^{-1})\,g,   </math>
where <math>\chi_k(g)=\mathrm{tr}\,\rho_k(g)  </math> is the [[Character theory|character]] of ''V<sub>k</sub>''. These form a complete system of orthogonal idempotents, so that <math>\epsilon_k^2 =\epsilon_k  </math>, <math>\epsilon_j \epsilon_k = 0  </math> for ''j ≠ k'', and <math>1 = \epsilon_1+\cdots+\epsilon_m   </math>. The isomorphism <math>\tilde\rho</math> is closely related to [[Fourier transform on finite groups]].

For a more general field ''K,'' whenever the [[characteristic (algebra)|characteristic]] of ''K'' does not divide the order of the group ''G'', then ''K''[''G''] is semisimple. When ''G'' is a finite [[abelian group]], the group ring ''K''[G] is commutative, and its structure is easy to express in terms of [[root of unity|roots of unity]].

When ''K'' is a field of characteristic ''p'' which divides the order of ''G'', the group ring is ''not'' semisimple: it has a non-zero [[Jacobson radical]], and this gives the corresponding subject of [[modular representation theory]] its own, deeper character.

===Center of a group algebra===
The [[center of a group|center]] of the group algebra is the set of elements that commute with all elements of the group algebra:
:<math>\mathrm{Z}(K[G]) := \left\{ z \in K[G] : \forall r \in K[G], zr = rz \right\}.</math>

The center is equal to the set of [[class function]]s, that is the set of elements that are constant on each conjugacy class
:<math>\mathrm{Z}(K[G]) = \left\{ \sum_{g \in G} a_g g :  \forall g,h \in G, a_g = a_{h^{-1}gh}\right\}.</math>

If {{nowrap|1=''K'' = '''C'''}}, the set of irreducible [[character theory|characters]] of ''G'' forms an orthonormal basis of Z(''K''[''G'']) with respect to the inner product
:<math>\left \langle \sum_{g \in G} a_g g, \sum_{g \in G} b_g g \right \rangle = \frac{1}{|G|} \sum_{g \in G} \bar{a}_g b_g.</math>