A question on the representation theory of finite group 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
 A: 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.
