The (Left) Regular Representation
Now it comes time to introduce what’s probably the most important representation of any group, the “left regular representation”. This arises because any group acts on itself by left-multiplication. That is, we have a function — given by . Indeed, this is an action because first ; and second , and as well.
So, as with any group action on a finite set, we get a finite-dimensional permutation representation. The representing space has a standard basis corresponding to the elements of . That is, to every element we have a basis vector . But we can recognize this as the standard basis of the group algebra . That is, the group algebra itself carries a representation.
Of course, this shouldn’t really surprise us. After all, representations of are equivalent to modules for the group algebra; and the very fact that is an algebra means that it comes with a bilinear function , which makes it into a module over itself.
We should note that since this is the left regular representation, there is also such a thing as the right regular representation, which arises from the action of on itself by multiplication on the right. But by itself right-multiplication doesn’t really give an action, because it reverses the order of multiplication. Indeed, for a group action as we’ve defined it first acting by and then acting by is the same as acting by the product . But if we first multiply on the right by and then by we get , which is the same as acting by . The order has been reversed.
To compensate for this, we define the right regular representation by the function . Then , and as well.
As an exercise, let’s work out the matrices of the left regular representation for the cyclic group with respect to its standard basis. We have four elements in this group: and . Thus the regular representation will be four-dimensional, and we will index the rows and columns of our matrices by the exponents , , , and . Then in the matrix the entry in the th row and th column is . The multiplication rule tells us that , where the exponent is defined up to a multiple of four, and so the matrix entry is if , and otherwise. That is:
You can check for yourself that these matrices indeed give a representation of the cyclic group.