By the Burnside theorem, we know that we can decompose the order of a group in to a sum of some integers' square, and these integers are the dimensions of the group's irreducible representations . But for a fixed number n, there are a lot of decomposition of this form. So which decomposition are corresponding to some group's all irreducible representations? thank you
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$\begingroup$ The numbers have to be divisors of $n$. $\endgroup$– S. Carnahan ♦Nov 20, 2014 at 13:30
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$\begingroup$ What you mentioned is a mere consequence of the representation theory. The decomposition you are asking about depends on the group: say, for $S_3$ and $\Bbb Z/6$ they would be different. $\endgroup$– Alex DegtyarevNov 20, 2014 at 13:39
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2$\begingroup$ @AlexDegtyarev : I interpret the question as asking whether, given $k$ (not necessarily distinct), integers $m_{1},m_{2}, \ldots, m_{k}$ with $\sum_{i=1}^{k}m_{k}^{2} = n,$ can we finite group $G$ such that the $m_{i}$ are the degrees of the complex irreducible characters of $G$ (including multiplicities). I think that question is actually quite difficult, if not impossible, to answer fully, though there are many necessary conditions. $\endgroup$– Geoff RobinsonNov 20, 2014 at 14:23
1 Answer
I think this question is rather difficult to answer in full generality. Here are some partial comments. Nilpotent groups give a rich supply of finite groups which might realize certain potential integer $k$ tuples $m_{1},m_{2},\ldots m_{k}$ of irreducible character degrees (including multiplicities). For these to have a chance of being genuine character degrees for a (general) finite group of order $n$, there are obvious restrictions. As already noted in comments, each $m_{i}$ must divide $n.$ Also, the number of occurrences of $1$ must be a divisor of $n.$ To be (irreducible) character degrees of a nilpotent group is even more restrictive. For example, each $m_{i}$ must divided the largest $m_{i}.$ on the other hand, solving the problem for nilpotent groups reduces to solving it for $p$-groups. If my memory is correct, M. Slattery has proved (possibly with a co-author or two) that any collection of distinct positive powers of $p$ can occur as the set of distinct irreducible character degrees of some finite $p$-group, so for $p$-groups, the question comes down to restricting the possible multiplicities with which each character degree can occur. This seems to me to already be quite difficult in itself.