Group ring

The given group and crowd ring (ぐんかん British: group ring) for the ring uses a group and the structure of the ring given naturally and, in algebra, is comprised. That, as for the group ring, pro-generation with the ring that itself was given in a coefficient ring and the group where do it, and was given it; is a module freely, and "is linear" as operation between the generators by the operation of the group given it besides, and form the product and a ring to do with the thing which extended. The group ring is the whole of the form sum to assume the cause of the ring given the given group "heaviness" commonly if I say. When a given ring is commutation, the group ring has the structure of the multi dimensions ring (algebra) in the ring given and is called group multi dimensions ring (ぐんたげんかん British: group algebra; group algebra) (or it is short group ring ^{[note 1]}).

The group ring is algebraic structure to play a role that is important in an expression theory of the particularly limited group. I hand over the group of infinity group ring to the clause of the group ring of the topological group to need the discussion that often added phase, and this clause treats a group ring of the limitedness group mainly. In addition, the general discussion look at the group hop algebra (English version).

Table of contents

Definition

I refer to "the monoid ring # definition"

I assume ring, G a group in R.

1. Write the whole (vector space free R-freedom module on G particularly R at the time of health) of the linear combination that is formal ("is limited") of the R-coefficient pro-generation in G with R[G] (is written as RG); [¹]. In other words, former x ∈ R[G] arbitrary

   ${\displaystyle x=\sum _{g\in G}a_{g}\,g\quad (a_{g}\in R)}$ 

   I can write it in the の form. But it must be ag = 0 in the friendship of the right side for all g except the exception of the limited unit. When you make the distinction with the cause of G and the cause of R[G] clear, write a generator corresponding to each former g ∈ G as e_g

   ${\displaystyle x=\sum _{g\in G}a_{g}\,e_{g}}$ 

   Write のようにも [²]; [^{note 2}]. It is the sum every clause on this meeting R[G]

   ${\displaystyle (\sum _{g\in G}a_{g}\,g)+(\sum _{g\in G}b_{g}\,g):=\sum _{g\in G}(a_{g}+b_{g})\cdot g}$ 

   I assumed it を addition and it was linear and expanded the product of G

   ${\displaystyle (\sum _{g\in G}a_{g}\,g)(\sum _{g\in G}b_{g}\,g):=\sum _{g,h\in G}(a_{g}b_{h})\cdot gh=\sum _{g\in G}(\sum _{h\in G}a_{h}b_{h^{-1}g})\cdot g}$ 

   I make を multiplication and a ring to do and double a scalar more

   ${\displaystyle r\cdot (\sum _{g\in G}a_{g}\,g):=\sum _{g\in G}(ra_{g})\cdot g}$ 

   I make a multidimensional ring (linear ring) on により R. I call this multidimensional ring R[G] with the group ring of the R-coefficient on G, a group ring on R generated in G.
2. Former f of space C_c(G; R) which the whole continuation function with the R-value compact stand on group G forms (about disintegration phase) is representation f: from group G to commutation ring R ようなものである which it is G → R, and has a limited stand (in other words, it becomes f(g) = 0 (g ∈ G) except the exception of the limited unit). The sum every point

   ${\displaystyle (f+h)(g):=f(g)+h(g)\quad (g\in G)}$ 

   I enfold と

   ${\displaystyle (f\ast h)(g):=\sum _{\gamma \in G}f(\gamma) h(\gamma ^{-1}g)}$ 

   And it is doubled a scalar

   ${\displaystyle (rf)(g):=r(f(g))\quad (r\in R)}$ 

   のもと C_c(G; R) becomes the multidimensional ring on R.

R-value instructions function (delta function of the D rack) of set {g} of G a piece of for each former g

${\displaystyle \delta _{g}(h):={\begin{cases}1=1_{R}&(h=g)\\0=0_{R}&(h\neq g)\end{cases}}}$ 

When think about を; C_c(G; R) as a standard base on R {δ_g | Having g ∈ G},

${\displaystyle R[G]\to C_{c}(G;R);\;\sum _{g\in G}a_{g}\,g\mapsto \sum _{g\in G}a_{g}\delta _{g}}$ 

It is the same model of は 多元環. I often write C_c(G; R) here as R[G] (1.as well as の case) and call it a group of G ring on R [²].

Space RG (= R(G) = Hom(G, R) where the whole representation from G to R makes if G is a limited group this C_c(G; R)) There are not に et al. In the case of infinity group, this is not managed generally, but still group ring R[G] and representation space R^G are in a relation of 双対 in the meaning that they show below each other:

The cause of the group ring

${\displaystyle x=\sum _{g\in G}a_{g}\,g}$ 

と R- level representation f: For a pair of G → K, it is dot product

${\displaystyle (x,f)=\sum _{g\in G}a_{g}f(g)\quad \in R}$ 

But, it is decided without contradiction (be careful about the right side being the real limited sum).

Example

Cyclic group G = 〈 g of order of magnitude 3 | I take g3 = 1>, and ω = exp(2πi/3) is far. Then

${\displaystyle {\begin{aligned}e_{1}&={\tfrac {1}{3}} (1+g+g^{2})\\e_{2}&={\tfrac {1}{3}} (1+\omega g+\omega ^{2}g^{2})\\e_{3}&={\tfrac {1}{3}} (1+\omega ^{2}g+\omega g)\end{aligned}}}$ 

These give you idempotent element resolution 1 = e1 + e2 + e₃ for the central orthogonality beginning when it determines the cause of the と group ring CG, and the next direct existing reduction to its lowest terms solution and same model are provided.

${\displaystyle \mathbb {C} G=e_{1}\mathbb {C} G\oplus e_{2}\mathbb {C} G\oplus e_{3}\mathbb {C} G\cong {\begin{pmatrix}\mathbb {C} &0&0\\0&\mathbb {C} &0\\0&0&\mathbb {C} \end{pmatrix}}}$ 

Module in the group ring

The details refer to "the module on the group"

When I consider group ring K[G] on ring K to be a ring, the module on ring K[G] is called a module on group G. I can read it for the expression of group G by the words of the G-module. In particular

• The simple G-module is G-existing about expression.
• When expression space of G is K-module V₁, V₂, the associate same model between the expression is the K-alignment associate same model between G-module V₁, V₂; the whole is HomG
  I am expressed in
  K

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