The Group Algebra
Point of post: In this post we discuss the notion of the group algebra on a group.
We shall see that it will often be fruitful to consider functions from a finite group into 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 (the representation space of ). Technically the group algebra is considered to be an adaptation of the free vector space of over . We’ll take a slightly different version here and describe the group algebra as functions as was indicated in the first sentence.
Let be a finite group and define the group algebra over , denoted , to be the set
with addition and scalar multiplication defined point-wise in the sense that and where and . The multiplication on is often called convolution and is denoted and given by
and a conjugation given by . We define the mappings to be, unimaginatively,
With these concepts in mind we claim that is a -dimensional associative unital algebra over . More precisely:
Theorem: Let be a finite group. Then the group algebra with operations of point-wise addition, point-wise scalar multiplication, convolution, and conjugation is an associative unital algebra with identity .
Proof: It’s evident that the definitions of point-wise scalar multiplication and addition define a complex vector space structure on . 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 be arbitrary and . Then:
but, with equal validity
where we’ve used the trick that for each fixed as is a bijection and thus the last equality holds. Since was arbitrary it follows that . To prove distributivity we note that for any and we have that:
and since was arbitrary it follows that . The scalar distributivity follows easily since for any , , and we have
To see that is in fact an identity for convolution we merely note that for any and we have that
and since was arbitrary it follows that . It similarly follows that . From this we may conclude that is an associative unital algebra with identity .
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.