# Finite Unipotent Groups: References

It will be a great pleasure for me if one can suggest "Survey Articles" on following topics related to the finite unipotent group $U(n,\mathbb{F}_q)$. (Thanks in advance!!!)

1. The number of conjugacy classes in $U(n,\mathbb{F}_q)$.

(I saw some papers of Vera-Lopez, which give the number for $n\leq 5$; but it is published around 1992, so possibly, more work may have been done, I couldn't find it.)

2. The number of conjugacy classes of cyclic subgroups in $U(n,\mathbb{F}_q)$.

3. Complex Irreducible representations (or characters) of $U(n,\mathbb{F}_q)$.

(From the comment of Nick Gill, and others) Here $U(n,\mathbb{F}_q)$ is the group of upper uni-triangular matrices over finite field $\mathbb{F}_q$ .

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Could you define $U(n, \mathbb{F}_q)$ please? Are you talking about upper-triangular matrices? –  Nick Gill Apr 9 '13 at 10:33
Related question mathoverflow.net/questions/106521/… in comments under it I have collected some references, which might be of interest –  Alexander Chervov Apr 9 '13 at 12:37
And also mathoverflow.net/questions/68207/… irreducible-representations-of-the-unitriangular-group –  Alexander Chervov Apr 9 '13 at 13:27
I would guess that $U(n,\mathbf{F}_q)$ means upper triangular $n\times n$ matrices with $1$'s on the diagonal. –  Keenan Kidwell Apr 9 '13 at 18:42
And also mathoverflow.net/questions/127010/… classification for coadjoint orbits of lower or upper triangular matrices –  Alexander Chervov Apr 10 '13 at 9:16

I guess U(n,F_q) - upper triangular matrices over F_q. I am not an expert in the field, but was recently interested in similar question, so I'll put these remarks.

There is certain amount of quite recent works related to representations of U(n,F_q) and some "big names" involved. There are certain conjectures which are easy to state, but hard to prove, and some more conceptual problems and recent breakthroughs.

One of the origins of the modern interest are papers by A.A. Kirillov (1995-2005), e.g. Variation on a triangular theme, Two more variations on a triangular theme, where he considered a question whether the "orbit method" can be extended to finite Lie groups such as U(n,F_q). (Originally (60-ies) A.A. Kirillov proposed "orbit method" for Lie groups over R, and U(n,R) where the first groups where he demonstrated its work).

Kirillov's papers are always pleasure to read, he puts the accent not on technical details, but on new ideas, problems and insighting observations. The moral that in certain cases "orbit method" can be applied, but there some problems, which are deserved to be further studied to achieve further progress. See e.g. Vipul Naik's site, where one can find lots of interesting and understandable information, worked out examples and further references: orbit method for U(3,p), orbit method for finite Lazard groups.

One of the major breakthroughs in the subject is an idea of "supercharacters". Let me quote from E. Arias-Castro P. Diaconis R. Stanley

The character theory of U(n,F_q) is a well known nightmare. In recent work, Carlos Andre, Roger Carter and Ning Yan have developed a theory based on certain unions of conjugacy classes (here called super-classes) and sums of irreducible characters (here called super-characters).

The main point is that "classification of characters" is to certain extent "wild" problem, while "supercharacters" are quite manageable to classify.

See more comments in MO question- "Irreducible representations of the unitriangular group" .

Let me also quote from F. Ladisch answer on MO question - "Representation theory of p-groups in particular upper triangular matrices over F_p"

There are, by now, many papers about the character theory of the upper triangular group and related topics, which is in part motivated by Higman's conjecture that for every n, the number of conjugacy classes of Un(Fq) is a polynomial in q with integer coefficients.

Let me also mention "23 page article with 28 authors :)" which is devoted to the subject: Supercharacters, symmetric functions in noncommuting variables, and related Hopf algebras

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This paper is well-written and has some intereseting and relevant results: Marberg, Eric, Heisenberg characters, unitriangular groups, and Fibonacci numbers. J. Combin. Theory Ser. A 119 (2012), no. 4, 882–903. I reviewed it for mathscinet here: ams.org/mathscinet-getitem?mr=2881233 –  Nick Gill Apr 9 '13 at 14:17