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 $G$ acts on itself by left-multiplication. That is, we have a function $G\times G\to G$ — given by $(g,h)\mapsto gh$ . Indeed, this is an action because first $(e,g_1)\mapsto g_1$ ; and second $(g_1,(g_2,h))\mapsto(g_1,g_2h)\mapsto g_1g_2h$ , and $(g_1g_2,h)\mapsto g_1g_2h$ as well.

So, as with any group action on a finite set, we get a finite-dimensional permutation representation. The representing space $\mathbb{C}G$ has a standard basis corresponding to the elements of $G$ . That is, to every element $g\in G$ we have a basis vector $\mathbf{g}\in\mathbb{C}G$ . But we can recognize this as the standard basis of the group algebra $\mathbb{C}[G]$ . That is, the group algebra itself carries a representation.

Of course, this shouldn’t really surprise us. After all, representations of $G$ are equivalent to modules for the group algebra; and the very fact that $\mathbb{C}[G]$ is an algebra means that it comes with a bilinear function $\mathbb{C}[G]\times\mathbb{C}[G]\to\mathbb{C}[G]$ , 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 $G$ 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 $g_2$ and then acting by $g_1$ is the same as acting by the product $g_1g_2$ . But if we first multiply on the right by $g_2$ and then by $g_1$ we get $hg_2h_1$ , which is the same as acting by $g_2g_1$ . The order has been reversed.

To compensate for this, we define the right regular representation by the function $(g,h)\mapsto hg^{-1}$ . Then $(g_1,(g_2,h))\mapsto(g_1,hg_2^{-1})\mapsto hg_2^{-1}g_1^{-1}=h(g_1g_2)^{-1}$ , and $(g_1g_2,h)\mapsto h(g_1g_2)^{-1}$ as well.

As an exercise, let’s work out the matrices of the left regular representation for the cyclic group $\mathbb{Z}_4$ with respect to its standard basis. We have four elements in this group: $\{g^0,g^1,g^2,g^3\}$ and $g^4=g^0$ . Thus the regular representation will be four-dimensional, and we will index the rows and columns of our matrices by the exponents $0$ , $1$ , $2$ , and $3$ . Then in the matrix $\rho(g^k)$ the entry in the $i$ th row and $j$ th column is $\delta_{g^kg^j,g^i}$ . The multiplication rule tells us that $(g^i,g^j)\mapsto g^{i+j}$ , where the exponent is defined up to a multiple of four, and so the matrix entry is $1$ if $i=j+k$ , and $0$ otherwise. That is:

$\displaystyle\begin{aligned}\rho(g^0)&=\begin{pmatrix}1&0&0&0\\{0}&1&0&0\\{0}&0&1&0\\{0}&0&0&1\end{pmatrix}\\\rho(g^1)&=\begin{pmatrix}{0}&0&0&1\\1&0&0&0\\{0}&1&0&0\\{0}&0&1&0\end{pmatrix}\\\rho(g^2)&=\begin{pmatrix}{0}&0&1&0\\{0}&0&0&1\\1&0&0&0\\{0}&1&0&0\end{pmatrix}\\\rho(g^3)&=\begin{pmatrix}{0}&1&0&0\\{0}&0&1&0\\{0}&0&0&1\\1&0&0&0\end{pmatrix}\end{aligned}$

You can check for yourself that these matrices indeed give a representation of the cyclic group.

September 17, 2010 - Posted by John Armstrong | Algebra, Group theory, Representation Theory

11 Comments »

I know people have suggested before that you write a book, which I realize may take a while to happen, but in the meantime, how about compiling a large PDF of your blog entries? It’d be a substantially smaller responsibility (well, OK, that could depend on what your LaTeX code is like) but would be very useful to readers.

Comment by Akhil Mathew | September 18, 2010 | Reply
[…] if , then there is one coset for each element of , and the coset representation is the same as the left regular representation. At the other extreme, if , then there is only one coset and we get the trivial […]

Pingback by Coset Representations « The Unapologetic Mathematician | September 20, 2010 | Reply
[…] an example, let be any finite group, and let be its group algebra, which carries the left regular representation . Now, consider the subspace spanned by the […]

Pingback by Submodules « The Unapologetic Mathematician | September 22, 2010 | Reply
[…] if it contains a nontrivial submodule. Thus our examples last time show that the left regular representation is always reducible, since it always contains a copy of the trivial representation as a nontrivial […]

Pingback by Reducibility « The Unapologetic Mathematician | September 23, 2010 | Reply
I think that you interchanged rows and columns: shouldn’t $\rho(g^k)$ have entry $\delta_{g^kg^j,g^i}$ in its i-th row and j-th column? I.e. entry 1 if i=j+k and 0 otherwise?

Comment by Landau | October 28, 2010 | Reply
You know, you’re right.. and I got this backwards when I defined permutation representations before. Fixing…

Comment by John Armstrong | October 28, 2010 | Reply
[…] right away is the right regular representation, which (predictably enough) corresponds to the left regular representation. In fact, when I introduced that representation I even mentioned the right action in passing. At […]

Pingback by Right Representations « The Unapologetic Mathematician | November 2, 2010 | Reply
[…] the Left Regular Representation Let’s take the left regular representation of a finite group on its group algebra and decompose it into irreducible […]

Pingback by Decomposing the Left Regular Representation « The Unapologetic Mathematician | November 17, 2010 | Reply
[…] Endomorphism Algebra of the Left Regular Representation Since the left regular representation is such an interesting one — in particular since it contains all the irreducible […]

Pingback by The Endomorphism Algebra of the Left Regular Representation « The Unapologetic Mathematician | November 18, 2010 | Reply
[…] show up in . In particular, since we know that every irreducible representation shows up in the left regular representation , the number of irreducible representations is . Thus to calculate this number , we must understand […]

Pingback by The Character Table is Square « The Unapologetic Mathematician | November 19, 2010 | Reply
[…] The action of on these tabloids is basically the same as left-multiplication on the underlying set . And so we find the left regular representation. […]

Pingback by Permutation Representations from Partitions « The Unapologetic Mathematician | December 14, 2010 | Reply

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