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Let $B_n$ be the group of upper matrices and $U_n$ the subgroup of unipotent upper triangular matrices. I would like some references which discusses complex character theory of $B_n(\mathbb{F}_q)$ for small $n$, say, $n=2, 3, 4, 5$ and $U_n(\mathbb{F}_q)$ for $n=4, 5$. I'm mostly interested in the character degrees and how many characters of each degree there are. But if the full character table is available, that would be great.

Dr. Evil
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  • This is probably such a do-able exercise that there's no big-time reference for it, apart from textbooks. That is, you can just do it yourself, and thereby also be confident... But in terms of cite-able sources (so that your readers don't have to necessarily trust you for this) probably some people on this site will know something. :) – paul garrett Apr 12 '22 at 02:02
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    Have you checked the following MO questions:

    https://mathoverflow.net/questions/106521/representation-theory-of-p-groups-in-particular-upper-tringular-matrices-over-f-p https://mathoverflow.net/questions/68207/irreducible-representations-of-the-unitriangular-group https://mathoverflow.net/questions/126932/finite-unipotent-groups-references ? I think that for your purposes various references mentioned there are almost sufficient. It would be good to not create duplicates unnecessarily.

     – Vladimir Dotsenko Apr 12 '22 at 02:19
  • Thanks. That is the same question for $U_n$. But doesn't discuss $B_n$. – Dr. Evil Apr 12 '22 at 02:28
  • Passing from $U_{n}$ to $B_{n}$ is a matter of doing Clifford theory, using the action of the torus $T_{n}$ on the irreducible characters of $U_{n}.$ Since $T_{n}$ and $U_{n}$ have coprime orders, this is fairly easy to analyze. – Geoff Robinson Apr 13 '22 at 13:51
  • Thanks Geoff. It would still be nice to have a reference if it exists. – Dr. Evil Apr 15 '22 at 06:21

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