# Abstract Nonsense

## The Group Algebra

Point of post: In this post we discuss the notion of the group algebra on a group.

Motivation

We shall see that it will often be fruitful to consider functions from a finite group $G$ into $\mathbb{C}$ as a perfect setting for our discussion of representation theory. We shall see that there are canonical ways to define representations with this space of functions both as a group (precisely how the group structure works shall be seen below) and as the target space for the group $G$ (the representation space of $G$). Technically the group algebra is considered to be an adaptation of the free vector space of $G$ over $\mathbb{C}$. We’ll take a slightly different version here and describe the group algebra as functions as was indicated in the first sentence.

Group Algebra

Let $G$ be a finite group and define the group algebra over $G$, denoted $\mathcal{A}(G)$, to be the set

$\mathbb{C}^{G}=\left\{a:G\to \mathbb{C}\right\}$

with addition and scalar multiplication defined point-wise in the sense that $(a+b)(g)=a(g)+b(g)$ and $(\alpha a)(g)=\alpha a(g)$ where $a,b\in\mathcal{A}\left(G\right)$ and $\alpha\in\mathbb{C}$. The multiplication on $\mathcal{A}\left(G\right)$ is often called convolution and is denoted $a\ast b$ and given by

$\displaystyle \left(a\ast b\right)(h)=\sum_{g\in G}a\left(hg^{-1}\right)b(g)$

and a conjugation given by $a^\ast (g)=\overline{a\left(g^{-1}\right)}$. We define the mappings $\delta_g$ to be, unimaginatively,

$\displaystyle \delta_g:G\to\mathbb{C}:h\mapsto \begin{cases}1 & \mbox{if}\quad g=h\\ 0 & \mbox{if}\quad g\ne h\end{cases}$

With these concepts in mind we claim that $\mathcal{A}(G)$ is a $|G|$-dimensional associative unital algebra over $\mathbb{C}$. More precisely:

Theorem: Let $G$ be a finite group. Then the group algebra $\mathcal{A}\left(G\right)$ with operations of point-wise addition, point-wise scalar multiplication, convolution, and conjugation is an associative unital algebra with identity $\delta_e$.

Proof: It’s evident that the definitions of point-wise scalar multiplication and addition define a complex vector space structure on $\mathcal{A}\left(G\right)$. Less trivial is to see that convolution interacts with addition and scalar multiplication in such a way to make it into an associative algebra. But, these are just calculations. Namely to prove associativity let $a,b,c\in\mathcal{A}\left(G\right)$ be arbitrary and $h\in G$. Then:

\begin{aligned}\left(a\ast(b\ast c)\right)(h) &= \sum_{g\in G}a\left(hg^{-1}\right)\left(b\ast c\right)(g)\\ &= \sum_{g\in G}a\left(hg^{-1}\right)\left(\sum_{k\in G}b\left(g k^{-1}\right) c(k)\right)\\ &= \sum_{g\in G}\sum_{k\in G}a\left(h g^{-1}\right)b\left(gk^{-1}\right)c\left(k\right)\\ &= \sum_{k\in G}\left(\sum_{g\in G}a\left(hg^{-1}\right)b\left(gk^{-1}\right)\right)c(k)\end{aligned}

but, with equal validity

\begin{aligned}\left(\left(a\ast b\right)\ast c\right)(h) &= \sum_{k\in G}\left(a\ast b\right)\left(hk^{-1}\right)c(k)\\ &= \sum_{k\in G}\left(\sum_{g\in G}a\left(hk^{-1}g^{-1}\right)b(g)\right)c(k)\\ &= \sum_{k\in G}\left(\sum_{g\in G}a\left(h\left(gk\right)^{-1}\right)b\left(gk k^{-1}\right)\right)c(k)\\ &= \sum_{k\in G}\left(\sum_{g\in G}a\left(hg^{-1}\right)b\left(gk^{-1}\right)\right)c(k)\end{aligned}

where we’ve used the trick that for each fixed $k\in G$ as $g\mapsto gk$ is a bijection and thus the last equality holds. Since $h$ was arbitrary it follows that $\left(\left(a\ast b\right)\ast c\right)=\left(a\ast\left(b\ast c\right)\right)$. To prove distributivity we note that for any $h\in G$ and $a,b,c\in\mathcal{A}\left(G\right)$ we have that:

\begin{aligned}a\ast\left(b+c\right)(h) &= \sum_{g\in G}a\left(h g^{-1}\right)\left(b+c\right)(g)\\ &= \sum_{g\in G}a\left(h g^{-1}\right)\left(b(g)+c(g)\right)\\ &= \sum_{g\in G}\left(a\left(hg^{-1}\right)b(g)+a\left(hg^{-1}\right)c(g)\right)\\ &= \sum_{g\in G}a\left(hg^{-1}\right)b(g)+\sum_{g\in G}a\left(hg^{-1}\right)c(g)\\ &= \left(a\ast b\right)(h)+\left(a\ast c\right)(h)\end{aligned}

and since $h\in G$ was arbitrary it follows that $a\ast \left(b+c\right)=a\ast b+a\ast c$. The scalar distributivity follows easily since for any $\alpha\in\mathbb{C}$, $a,b\in\mathcal{A}\left(G\right)$, and $h\in G$ we have

\displaystyle \begin{aligned}\left(\left(\alpha a\right)\ast b\right)(h) &=\sum_{g\in G}\left(\alpha a\right)\left(hg^{-1}\right)b(g)\\ &=\sum_{g\in G}\alpha a\left(h g^{-1}\right)b(g)\\ &=\alpha\sum_{g\in G}a\left(h g^{-1}\right)b(g)\\ &= \alpha \left(a\ast b\right)(h)\end{aligned}

To see that $\delta_e$ is in fact an identity for convolution we merely note that for any $a\in\mathcal{A}\left(G\right)$ and $h\in G$ we have that

$\displaystyle \left(\delta_e\ast a\right)(h)=\sum_{g\in G}\delta_e\left(h g^{-1}\right) a(g)= \delta_e\left(h h^{-1}\right)a(h) =a(h)$

and since $h\in G$ was arbitrary it follows that $\delta_e\ast a=a$. It similarly follows that $a\ast \delta_e=a$. From this we may conclude that $\mathcal{A}\left(G\right)$ is an associative unital algebra with identity $\delta_e$. $\blacksquare$

References:

1.Simon, Barry. Representations of Finite and Compact Groups. Providence, RI: American Mathematical Society, 1996. Print.

2. Serre, Jean Pierre. Linear Representations of Finite Groups. New York: Springer-Verlag, 1977. Print.

January 20, 2011 -

1. […] Point of post: This is a continuation of this post. […]

Pingback by Representation Theory: The Group Algebra (Pt. II) « Abstract Nonsense | January 20, 2011 | Reply

2. […] be a finite group and the group algebra on . If is a pre-Hilbert space then a mapping  is called a -representation of if the following […]

Pingback by Representation Theory: *-representations of the Group Algebra « Abstract Nonsense | January 21, 2011 | Reply

3. […] Point of post: In this post we shall discuss the notion of the left and right regular representations on the group algebra […]

Pingback by Representation Theory: Left Regular Representation « Abstract Nonsense | January 23, 2011 | Reply

4. […] be a finite group and the group algebra. Then, if is a pre-Hilbert space and   a  -representation we call  -invariant (or invariant […]

Pingback by Representation Theory: Schur’s Lemma (*-representation Form) « Abstract Nonsense | January 27, 2011 | Reply

5. […] and we’re discussing we simply write . From this we then have defined elements of the group algebra . Namely, if we denote to be the entry of then the mapping is a mapping and thus an element of […]

Pingback by Representation Theory: Matrix Entry Functions « Abstract Nonsense | February 22, 2011 | Reply

6. […] be a finite group. Then, if is a member of the group algebra which satisfies for every then we call a class function. We denote the set of all class […]

Pingback by Representation Theory: Class Functions « Abstract Nonsense | February 24, 2011 | Reply

7. […] As a step toward this we prove in this post that the dimension of thought of as a subspace of the group algebra is the number of conjugacy classes of […]

Pingback by Representation Theory: Dimension of the Space of Class Functions « Abstract Nonsense | February 24, 2011 | Reply

8. […] be a finite group and . Consider then the quotient group and its associated group algebra . We define then […]

Pingback by Representation Theory: Relation Between the Kernels of Characters and Normal Subgroups « Abstract Nonsense | March 7, 2011 | Reply

9. […] is meant to recall that we’re taking the inner product considering an element of the group algebra  and the inner product on the group algebra . Moreover, equality holds if and only if vanishes […]

Pingback by Representation Theory: The Index of the Center of a Character « Abstract Nonsense | March 9, 2011 | Reply

10. […] Recall that is real the center of the group algebra . But, recall that the center of an algebra is a subalgebra. In particular since is a subalgebra […]

Pingback by Representation Theory: A ‘Lemma’ (pt. II) « Abstract Nonsense | March 10, 2011 | Reply

11. […] past posts we obainted certain relations between the pairwise inner product (as elements of the group algebra) of matrix entry functions and irreudcible characters. We shall use these relations to compute the […]

Pingback by Representation Theory: Using Orthogonality Relations to Compute Convolutions of Characters and Matrix Entry Functions « Abstract Nonsense | March 29, 2011 | Reply

12. […] have seen that the matrix entry functions form an orthonormal basis for the group algebra.  From this we got the important result that . It thus follows from basic linear algebra that as […]

Pingback by Representation Theory: Decomposing the Group Algebra Into the Direct Sum of Matrix Algebras « Abstract Nonsense | April 6, 2011 | Reply

13. […] saw in our last post that the group algebra is isomorphic to a direct sum of matrix algebras. We shall use this fact to derive an interesting […]

Pingback by Representation Theory: Consequence of the Decomposition of the Group Algebra Into Matrix Algebras « Abstract Nonsense | April 9, 2011 | Reply

14. […] example one could (and is often–almost always) define the group algebra  to be . Then, the left regular representation  can be thought of as where […]

Pingback by Free Vector Spaces « Abstract Nonsense | April 19, 2011 | Reply

15. […] basis for . Moreover, we know that since we have proven that is a subalgebra of the group algebra . So, explicitly we have a map given […]

Pingback by Induced Class Functions and the Space of Integral Class Functions (Pt. I) « Abstract Nonsense | April 27, 2011 | Reply

16. […] them in the sense of taking the free vector space  (which of course we can identify with the group algebra by ) and is the sign function.  We lastly define a third function given by (order matters, so […]

Pingback by Row and Column Stabilizer « Abstract Nonsense | May 20, 2011 | Reply

17. […] to a -representation, which can be thought of as extending an -module (where is the group algebra) to an […]

Pingback by Extension of Scalars and Change of Ring (Pt. I) « Abstract Nonsense | January 24, 2012 | Reply