The Unapologetic Mathematician

Mathematics for the interested outsider

Characters of Permutation Representations

Let’s take (\mathbb{C}S,\rho) to be a permutation representation coming from a group action on a finite set S that we’ll also call \rho. It’s straightforward to calculate the character of this representation.

Indeed, the standard basis that comes from the elements of S gives us a nice matrix representation:

\displaystyle\rho(g)_s^t=\delta_{\rho(g)s,t}

On the left \rho(g) is the matrix of the action on \mathbb{C}S, while on the right it’s the group action on the set S. Hopefully this won’t be too confusing. The matrix entry in row s and column t is 1 if \rho(g) sends s to t, and it’s 0 otherwise.

So what’s the character \chi_\rho(g)? It’s the trace of the matrix \rho(g), which is the sum of all the diagonal elements:

\displaystyle\mathrm{Tr}\left(\rho(g)\right)=\sum\limits_{s\in S}\rho(g)_s^s=\sum\limits_{s\in S}\rho(g)_s^s=\sum\limits_{s\in S}\delta_{\rho(g)s,s}

This sum counts up 1 for each point s that \rho(g) sends back to itself, and 0 otherwise. That is, it counts the number of fixed points of the permutation \rho(g).

As a special case, we can consider the defining representation V^\mathrm{def} of the symmetric group S_n. The character \chi^\mathrm{def} counts the number of fixed points of any given permutation. For instance, in the case n=3 we calculate:

\displaystyle\begin{aligned}\chi^\mathrm{def}\left((1)(2)(3)\right)&=3\\\chi^\mathrm{def}\left((1\,2)(3)\right)&=1\\\chi^\mathrm{def}\left((1\,3)(2)\right)&=1\\\chi^\mathrm{def}\left((2\,3)(1)\right)&=1\\\chi^\mathrm{def}\left((1\,2\,3)\right)&=0\\\chi^\mathrm{def}\left((1\,3\,2)\right)&=0\end{aligned}

In particular, the character takes the value 3 on the identity element e\in G, and the degree of the representation is 3 as well. This is no coincidence; \chi(e) will always be the degree of the representation in question, since any matrix representation of degree n must send e to the n\times n identity matrix, whose trace is n. This holds both for permutation representations and for any other representation.

October 19, 2010 Posted by | Algebra, Group theory, Representation Theory, Representations of Symmetric Groups | 3 Comments

   

Follow

Get every new post delivered to your Inbox.

Join 224 other followers