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For a finite group, the class sums form a basis of $Z (k [G])$

Statement

Let $G$ be a finite group and let $k$ be a field. Then the class sums $\hat{C}$ , as $C$ ranges over the conjugacy classes of $G$ , form a $k$ -basis of the center $Z (k [G])$ .

Facts & Assumptions

Given: A finite group $G$ and a field $k$ .

[L1]

The center $Z (k [G])$ consists of the elements of $k [G]$ that commute with every element of $k [G]$ (The center $Z (k [G])$ of the group algebra).

[L2]

For a conjugacy class $C$ , its class sum is $\hat{C} = \sum_{g \in C} [g]$ (The class sum $\hat{C}$ of a conjugacy class $C$ ).

[L3]

The group algebra $k [G]$ has basis ${[g] : g \in G}$ and multiplication $[g] [h] = [g h]$ (The group ring $R [G]$ is a unital $R$ -algebra with basis $G$ , and each $g \in G$ is a unit of $R [G]$ ).

Proof

technique · direct

1.1

Let $C$ be a conjugacy class and $h \in G$ . Using [L3], $[h] \hat{C} [h]^{- 1} = \sum_{g \in C} [h g h^{- 1}] .$ Because conjugation by $h$ permutes the elements of $C$ , this sum is again $\hat{C}$ . Hence $[h] \hat{C} = \hat{C} [h]$ for every basis element $[h]$ , so $\hat{C} \in Z (k [G])$ by [L1] and [L3].

L1L2L3givenalgebra

2.1

Now let $x = \sum_{g \in G} a_{g} [g]$ be any central element. For every $h \in G$ , centrality gives $[h] x = x [h]$ , so multiplying on the right by $[h]^{- 1}$ and using [L3] yields $[h] x [h]^{- 1} = x$ . Comparing coefficients in the basis ${[g]}$ shows $a_{h g h^{- 1}} = a_{g}$ for all $g, h \in G$ . Thus the coefficient function $g \mapsto a_{g}$ is constant on conjugacy classes, and $x$ is a $k$ -linear combination of the class sums.

step 1.1L1L2L3givenalgebra

3.1

Distinct conjugacy classes are disjoint subsets of $G$ , so their class sums have disjoint supports in the basis ${[g]}$ . Therefore a linear relation among class sums forces every coefficient to vanish. Combined with step 2.1, this shows that the class sums form a basis of $Z (k [G])$ .

step 2.1L2L3givenalgebra∎

Depends on

Used by

The dimension of Z(k[G]) is the number of conjugacy classes of G Corollary

Dependency tree · two levels

7 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Peter Webb, A Course in Finite Group Representation Theory, Lemma 3.4.2 (standard reference, not scraped)

For a finite group, the class sums form a basis of Z(k[G])Z(k[G])