Coset Representations

Next up is a family of interesting representations that are also applicable to any group $G$ . The main ingredient is a subgroup $H$ — a subset of the elements of $G$ so that the inverse of any element in $H$ is also in $H$ , and the product of any two elements of $H$ is also in $H$ .

Our next step is to use $H$ to break $G$ up into cosets. We consider $g_1$ and $g_2$ to be equivalent if $g_2^{-1}g_1\in H$ . It’s easy to check that this is actually and equivalence relation (reflexive, symmetric, and transitive), and so it breaks $G$ up into equivalence classes. We write the coset of all $g'\in G$ that are equivalent to $g$ as $gH$ , and we write the collection of all cosets of $H$ as $G/H$ .

We should note that we don’t need to worry about $H$ being a normal subgroup of $G$ , since we only care about the set of cosets. We aren’t trying to make this set into a group — the quotient group — here.

Okay, now multiplication on the left by $G$ shuffles around the cosets. That is, we have a group action of $G$ on the quotient set $G/H$ , and this gives us a permutation representation of $G$ !

Let’s work out an example to see this a bit more explicitly. For our group, take the symmetric group $S_3$ , and for our subgroup let $H=\{e,(2\,3)\}$ . Indeed, $H$ is closed under both inversion and multiplication. And we can break $G$ up into cosets:

$\displaystyle\begin{aligned}\{e,(1\,2),(1\,3),(2\,3),(1\,2\,3),(1\,3\,2)\}&=\{e,(2\,3)\}\uplus\{(1\,2),(1\,2\,3)\}\uplus\{(1\,3),(1\,3\,2)\}\\&=H\uplus(1\,2)H\uplus(1\,3)H\end{aligned}$

where we have picked a “transversal” — one representative of each coset so that we can write them down more simply. It doesn’t matter whether we write $(1\,2)H$ or $(1\,2\,3)H$ , since both are really the same set. Now we can write down the multiplication table for the group action. It takes $g_1\in G$ and $g_2H\in G/H$ , and tells us which coset $g_1g_2$ falls in:

$\displaystyle\begin{tabular}{c|rrr}&H&(1\,2)H&(1\,3)H\\\hline e&H&(1\,2)H&(1\,3)H\\(1\,2)&(1\,2)H&H&(1\,3)H\\(1\,3)&(1\,3)H&(1\,2)H&H\\(2\,3)&H&(1\,3)H&(1\,2)H\\(1\,2\,3)&(1\,2)H&(1\,3)H&H\\(1\,3\,2)&(1\,3)H&H&(1\,2)H\end{tabular}$

This is our group action. Since there are three elements in the set $G/H$ , the permutation representation we get will be three-dimensional. We can write down all the matrices just by working them out from this multiplication table:

$\displaystyle\begin{aligned}\rho_H(e)&=\begin{pmatrix}1&0&0\\{0}&1&0\\{0}&0&1\end{pmatrix}\\\rho_H((1\,2))&=\begin{pmatrix}{0}&1&0\\1&0&0\\{0}&0&1\end{pmatrix}\\\rho_H((1\,3))&=\begin{pmatrix}{0}&0&1\\{0}&1&0\\1&0&0\end{pmatrix}\\\rho_H((2\,3))&=\begin{pmatrix}1&0&0\\{0}&0&1\\{0}&1&0\end{pmatrix}\\\rho_H((1\,2\,3))&=\begin{pmatrix}{0}&0&1\\1&0&0\\{0}&1&0\end{pmatrix}\\\rho_H((1\,3\,2))&=\begin{pmatrix}{0}&1&0\\{0}&0&1\\1&0&0\end{pmatrix}\\\end{aligned}$

It turns out that these matrices are the same as we saw when writing down the defining representation of $S_3$ . There’s a reason for this, which we will examine later.

As special cases, if $H=\{e\}$ , then there is one coset for each element of $G$ , and the coset representation is the same as the left regular representation. At the other extreme, if $H=G$ , then there is only one coset and we get the trivial representation.

5 Comments »

[…] by shuffling around the subspaces that correspond to the cosets of . In fact, this is exactly the coset representation of corresponding to ! If we write for some , then this uses up the transversal element . The is […]

Pingback by Induced Matrix Representations « The Unapologetic Mathematician | November 25, 2010 | Reply
[…] we know that left-multiplication by permutes the cosets of . That is, for some . Thus we […]

Pingback by (Real) Frobenius Reciprocity « The Unapologetic Mathematician | December 3, 2010 | Reply
[…] on whether is in or not. But if , then latex g(t_jH)=(t_iH)$. That is, this is exactly the coset representation of corresponding to . And so all of these coset representations arise as induced […]

Pingback by Inducing the Trivial Representation « The Unapologetic Mathematician | December 6, 2010 | Reply
Hello. Thanks for the great series on representation theory! Just to let you know, I think there is a slight typo in this post. In decomposing your set into cosets, in the indented equation a few lines before the multiplication table, I believe (12) should be paired with (132) and (13) with (123).

Comment by Jay Cummings | February 8, 2012 | Reply
It’s a matter of convention which order the multiplication in a symmetric group is written. When I introduced the symmetric group, I chose the composition to reflect functional notation. That is, just like we write $f(g(x))$ to say “do $g$ first, then $f$ “, I write $(1\,2)(2\, 3)$ to say “do $(2\,3)$ first, then $(1\,2)$ “.

Comment by John Armstrong | February 8, 2012 | Reply

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