# $G/[G,G]$, irreps and conjugacy classes

This question is motivated by this one. Basically, the remark of Bugs Bunny in comments made me think - OK, there is an ambiguity in choosing between a Young diagram and its transpose when constructing irreps of $S_n$, but it is somewhat remarkable: it corresponds in terms of irreps to tensoring representations with the sign representation. There is also of course the remark of Gjergji Zaimi suggesting that for abelian groups the ambiguity is maximal possible, since we are literally dealing with a group vs its dual. This of course makes one wonder if the group of characters $G/[G,G]$ is somehow responsible for the key ambiguities, hence this (possibly very naive) question.

Of course the dual group of $G/[G,G]$ acts on irreps of $G$: that dual group is just the group of one-dimensional irreps of $G$, and we can tensor irreps of $G$ with those. I thought I had a way to make $G/[G,G]$ act in a very easy way on conjugacy classes of $G$, but David Speyer drew my attention to the fact that it must be much less obvious. Hence for the moment my question is - is there a way to repair the original question along these lines? For instance, is there an action of $G/[G,G]$ on conjugacy classes of $G$ for which there is a natural (in some sense) bijection between the orbits of $G/[G,G]\simeq (G/[G,G])^\vee$ on the conjugacy classes of $G$ and the irreps of $G$?

• I'm confused. How does $G/[G,G]$ act on conjugacy classes? For example , if $G$ is $S_n$, how does the nontrivial element of $S_n/[S_n, S_n]$ act on the identity? – David E Speyer Jul 23 '12 at 14:48
• I know you want the answer to be "it takes it to the class of the $n$-cycle", but I don't understand how you get that from abstract group theory. – David E Speyer Jul 23 '12 at 14:49
• @David: yes you are right. There seems to be a problem. I shall for the moment formulate a less specific version of my question, and think of how to repair it. – Vladimir Dotsenko Jul 23 '12 at 15:22
• I also disagree that $G/[G, G]$ is responsible for the ambiguities. Can you write down a natural bijection from the conjugacy classes of the binary icosahedral group $\tilde{A_5}$ (which is perfect) to its irreps? – Qiaochu Yuan Jul 23 '12 at 15:58
• @Will: up to multiplication by scalars, not up to multiplication by a $1$-dimensional representation. For example ${\rm SL}(2,5)$ has a faithful $2$-dimensional complex representation, but no non-trivial one-dimensional representation. However ${\rm SL}(2,5)$ is not isomorphic to a subgroup of ${\rm PGL}(2,\mathbb{C}).$ – Geoff Robinson Jul 23 '12 at 18:53