The Inner Product of Characters

When we’re dealing with characters, there’s something we can do to rework our expression for the inner product on the space of class functions.

Let’s take a $G$ -module $(W,\sigma)$ , with character $\psi$ . Before, we’ve used Maschke’s theorem to tell us that all $G$ -modules are completely reducible, but remember what it really tells us that there is some $G$ -invariant inner product on $W$ (we’ll have to keep straight the two inner products by which vector space they apply to). With respect to the inner product on $W$ , every transformation $\sigma(g)$ with $g\in G$ is unitary, and if we pick an orthonormal basis to get a matrix representation $Y$ each of the matrices $Y(g)$ will be unitary. That is:

$\displaystyle Y\left(g^{-1}\right)=Y(g)^{-1}=\overline{Y(g)}^\top$

So what does this mean for the character $\psi$ ? We can calculate

$\displaystyle\begin{aligned}\overline{\psi(g)}&=\overline{\mathrm{Tr}\left(Y(g)\right)}\\&=\mathrm{Tr}\left(\overline{Y(g)}\right)\\&=\mathrm{Tr}\left(Y\left(g^{-1}\right)^\top\right)\\&=\mathrm{Tr}\left(Y\left(g^{-1}\right)\right)\\&=\psi\left(g^{-1}\right)\end{aligned}$

And so we can rewrite our inner product

$\displaystyle\begin{aligned}\langle\chi,\psi\rangle&=\frac{1}{\lvert G\rvert}\sum\limits_{g\in G}\overline{\chi(g)}\psi(g)\\&=\frac{1}{\lvert G\rvert}\sum\limits_{g\in G}\chi\left(g^{-1}\right)\psi(g)\end{aligned}$

The nice thing about this formula is that it doesn’t depend on complex conjugation, and so it’s useful for any base field (if we were using other base fields).

The catch is that for class functions in general we have no reason to believe that this is an inner product. Indeed, if $g\in G$ is some element that isn’t conjugate to its inverse then we can define a class function $f$ that takes the value $1$ on the class $K_g$ of $g$ , $-1$ on the class $K_{g^{-1}}$ of $g^{-1}$ , and $0$ elsewhere. Our new formula gives

$\displaystyle\langle f,f\rangle=\frac{1}{\lvert G\rvert}\sum\limits_{g\in G}f\left(g^{-1}\right)f(g)=-\lvert K_{g^{-1}}\rvert\lvert K_g\rvert$

so this bilinear form isn’t positive-definite.

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October 18, 2010 - Posted by John Armstrong | Algebra, Group theory, Representation Theory

4 Comments »

[…] now we can recognize the left hand side as our alternate expression for the inner product of characters of . If the functions and were characters, this would be an […]

Pingback by Irreducible Characters are Orthogonal « The Unapologetic Mathematician | October 22, 2010 | Reply
[…] Where the left inner product is that of class functions on , while the right is that of class functions on . We calculate the inner products using our formula […]

Pingback by (Fake) Frobenius Reciprocity « The Unapologetic Mathematician | November 30, 2010 | Reply
I’m a graduate student from brazil, so sorry about my english, I’m from UFVJM
this post help me so much on my work.
I used to work whith this book:
“Representations and characters group” by gordon james and Martin Liebeck, but me and my teacher’ve found so many mistakes on this book and the language it’s strange, do you have a hint about other book that I could use to study group theory?
thank you

Comment by Pedro Borges | March 28, 2011 | Reply
Pedro, group theory on its own is commonly taught as part of an overall course on abstract algebra. I think a commonly-recommended book for that is Michael Artin’s.

On the other hand, if you mean to study representation theory, Fulton and Harris’ “Representation Theory: a First Course” is a good place to start for that.

Comment by John Armstrong | March 28, 2011 | Reply

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