{"id":2739,"date":"2011-08-30T16:04:23","date_gmt":"2011-08-30T14:04:23","guid":{"rendered":"https:\/\/wpethzprd.ethz.ch\/kowalski\/?p=2739"},"modified":"2024-02-20T13:40:36","modified_gmt":"2024-02-20T11:40:36","slug":"whats-special-with-commutators-in-the-weyl-group-of-c5","status":"publish","type":"post","link":"https:\/\/blogs.ethz.ch\/kowalski\/2011\/08\/30\/whats-special-with-commutators-in-the-weyl-group-of-c5\/","title":{"rendered":"What’s special with commutators in the Weyl group of C5?"},"content":{"rendered":"

I have just added to my notes on representation theory the very cute formula of Frobenius that gives, in terms of irreducible characters, the number $latex N(g)$ of representations of a given element $latex g$ as a commutator $latex g=[x,y]=xyx^{-1}y^{-1}$ in a finite group $latex G$:
\n$latex N(g)=|G|\\sum_{\\chi}\\frac{\\chi(g)}{\\chi(1)},$
\nwhere $latex \\chi$ runs over the irreducible (complex) characters of $latex G$ (this is Proposition 4.4.3 on page 118 of the last version of the notes<\/a>).<\/p>\n

I wanted to mention some applications, and had a vague memory that this was used to show that most or all elements in various simple groups are actual commutators. By searching around a bit, I found out easily that, indeed, there was a conjecture of Ore from 1951 to the effect that the set of commutators is equal to $latex G$ for any non-abelian finite simple group $latex G$, and that (after various earlier works) this has recently been proved by Liebeck, O’Brien, Shalev and Tiep<\/a>.<\/p>\n

I mentioned this of course, but then I also wanted to give some example of non-commutator, and decided to look for this using Magma<\/a> (the fact that I am recovering from a dental operation played a role in inciting me to find something distracting to do). Here’s what I found out.<\/p>\n

First, a natural place to look for interesting examples is the class of perfect groups, of course not simple. This is also easy enough to implement since Magma has a database of perfect groups of “small” order. Either by brute force enumeration of all commutators or by implementing the Frobenius formula, I got the first case of a perfect group $latex G$, of order $latex 960$, which contains only $latex 840$ distinct commutators.<\/p>\n

Then I wanted to know “what” this group really was. Magma gave it to me as a permutation group acting on $latex 16$ letters, with an explicit set of $latex 6$ generators, and with a list of $latex 21$ relations, which was not very enlightening. However, looking at a composition series, it emerged that $latex G$ fits in an exact sequence
\n$latex 1\\rightarrow (\\mathbf{Z}\/2\\mathbf{Z})^4\\rightarrow G\\rightarrow A_5\\rightarrow 1.$
\nThis was much better, since after a while it reminded me of one of my favorite types of groups: the Weyl groups $latex W_{g}$ of the symplectic groups $latex \\mathrm{Sp}_{2g}$ (equivalently, the “generic” Galois group for the splitting field of a palindromic rational polynomial of degree $latex 2g$), which fit in an relatively similar exact sequence
\n$latex 1\\rightarrow (\\mathbf{Z}\/2\\mathbf{Z})^g\\rightarrow W_g\\rightarrow S_g\\rightarrow 1.$
\nFrom there, one gets a strong suspicion that $latex G$ must be the commutator subgroup of $latex W_5$, and this was easy to check (again with Magma, though this is certainly well-known; the drop of the rank of the kernel comes from looking at the determinant in the signed-permutation $latex 5$-dimensional representation, and the drop from $latex S_5$ to $latex A_5$ is of course from the signature.)<\/p>\n

This identification is quite nice, obviously. In particular, it’s now possible to identify concretely which elements of $latex G$ are not commutators. It turns out that a single conjugacy class, of order $latex 120$, is the full set of missing elements. As a signed permutation matrix, it is the conjugacy class of
\n$latex g=\\begin{pmatrix} 0& -1 & 0 & 0 & 0\\\\ 1& 0 & 0 & 0 & 0\\\\ 0& 0 & 0 & 1 & 0\\\\ 0& 0 & 1 & 0 & 0\\\\ 0& 0 & 0 & 0 & -1\\end{pmatrix},$
\nand the reason it is not a commutator is that Magma tells us that all commutators in $latex G$ have trace in $latex \\{-3,-2,0,1,2,5\\}$ (always in the signed-permutation representation). Thus the trace $latex -1$ doesn’t fit…<\/p>\n

At least, this is the numerical reason. I feel I should be able to give a theoretical explanation of this, but I haven’t succeeded for the moment. Part of the puzzlement is that this behavior seems to be special to $latex W_5$, the Weyl group of the root system $latex C_5$. Indeed, for $latex g\\in\\{2,3,4\\}$, the corresponding derived subgroup is not perfect, so the question does not arise (at least in the same way). And when $latex g\\geq 6$, the derived subgroup $latex G_g$ of $latex W_g$ is indeed perfect, but — experimentally! — it seems that all elements of $latex G_g$ are then commutators. <\/p>\n

I haven’t found references to a study of this Ore-type question for those groups, so I don’t know if these “experimental” facts are in fact known to be true. Another question seems natural: does this special fact have any observable consequence, for instance in Galois theory? I don’t see how, but readers might have better insights…<\/p>\n

(P.S.<\/b> I presume that GAP<\/a> or Sage<\/a> would be equally capable of making the computations described here; I used Magma mostly because I know its language better.<\/p>\n

P.P.S<\/b> And the computer also tells us that even for the group $latex G$ above, all elements are the product of at most two commutators, which a commenter points out is also a simple consequence of the fact that there are more than $latex 480$ commutators…<\/b>.<\/p>\n

P.P.P.S<\/b> To expand one of my own comments: the element $latex g$ above is<\/i> a commutator in the group $latex W_5$ itself. For instance $latex g=[x,y]$ with
\n$latex x=\\begin{pmatrix} 0& 0 & 0 & 0 & -1\\\\ 0& 1 & 0 & 0 & 0\\\\ 1& 0 & 0 & 0 & 0\\\\ 0& 0 & 1 & 0 & 0\\\\ 0& 0 & 0 & 1 & 0\\end{pmatrix},$
\nand
\n$latex y=\\begin{pmatrix} 1& 0 & 0 & 0 & 0\\\\ 0& 0 & 0 & 0 & -1\\\\ 0& 1 & 0 & 0 & 0\\\\ 0& 0 & 1 & 0 & 0\\\\ 0& 0 & 0 & -1 & 0\\end{pmatrix},$
\nwhere $latex y\\notin G$.)<\/p>\n","protected":false},"excerpt":{"rendered":"

I have just added to my notes on representation theory the very cute formula of Frobenius that gives, in terms of irreducible characters, the number $latex N(g)$ of representations of a given element $latex g$ as a commutator $latex g=[x,y]=xyx^{-1}y^{-1}$ in a finite group $latex G$: $latex N(g)=|G|\\sum_{\\chi}\\frac{\\chi(g)}{\\chi(1)},$ where $latex \\chi$ runs over the irreducible … Continue reading What’s special with commutators in the Weyl group of C5?<\/span><\/a><\/p>\n","protected":false},"author":625,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1086,2],"tags":[],"class_list":["post-2739","post","type-post","status-publish","format-standard","hentry","category-exercise","category-blogroll"],"_links":{"self":[{"href":"https:\/\/blogs.ethz.ch\/kowalski\/wp-json\/wp\/v2\/posts\/2739","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/blogs.ethz.ch\/kowalski\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/blogs.ethz.ch\/kowalski\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/blogs.ethz.ch\/kowalski\/wp-json\/wp\/v2\/users\/625"}],"replies":[{"embeddable":true,"href":"https:\/\/blogs.ethz.ch\/kowalski\/wp-json\/wp\/v2\/comments?post=2739"}],"version-history":[{"count":0,"href":"https:\/\/blogs.ethz.ch\/kowalski\/wp-json\/wp\/v2\/posts\/2739\/revisions"}],"wp:attachment":[{"href":"https:\/\/blogs.ethz.ch\/kowalski\/wp-json\/wp\/v2\/media?parent=2739"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blogs.ethz.ch\/kowalski\/wp-json\/wp\/v2\/categories?post=2739"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blogs.ethz.ch\/kowalski\/wp-json\/wp\/v2\/tags?post=2739"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}