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I was thinking about how to compute the number of coniugacy classes of a group $G$, in particular I'm interested in the number of coniugacy classes of $GL_n(\mathbb{F}_p)$. This is clearly related to the number of irreducible representations of this group, but searching on the internet, I found out that this is actually an "open" problem.

If the field in which we are considering our matrices is algebrically closed, we can use an argument like the Jordan form to attack this problem, but this doesn't work in $\mathbb{F}_p$.

Can you suggest me some papers about this topic? I found that there are only upper and lower bounds for this number, but not an asymptotic behaviour or something like this.

Given a general group $GL_n(\mathbb{F}_p)$ how many coniugacy classes are there?

Do you think this is a question for StackExchange or should I ask this on Overflow? Any hint/paper/actual research on this topic?

Thanks in advance.

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You might have a look at Macdonald's classic book "Symmetric functions and Hall polynomials," in which this material is covered in detail and used to give a beautiful description of the irreducible complex characters of the general linear groups parallel to the symmetric function description of those of the symmetric group.

If you do, you'll see that conjugacy classes of elements of $\mathrm{GL}_n(\mathbf{F}_q)$ are indexed by partition-valued functions $$\lambda: \{ \text{irreducible monic polynomials} \ f \in \mathbf{F}_q[x], \ f \neq x \} \rightarrow \{\text{integer partitions} \} $$ such that $$\sum_{f,i} \mathrm{deg}(f)\lambda(f)_i=n. $$ (Note in particular that only polynomials $f$ of degree at most $n$ actually contribute to this sum).

This bijection is obtained as follows: given a matrix $g \in \mathrm{GL}_n(\mathbf{F}_q)$, regard $V=\mathbf{F}_q^n$ as a module over $\mathbf{F}_q[x]$ by letting $x$ act as $g$. The structure theorem for modules over a Euclidean domain shows that as a $\mathbf{F}_q[x]$-module, $$V \cong \bigoplus_{f,i} \mathbf{F}_q[x]/(f^{\lambda(f)_i})$$ for some partitions $\lambda(f)$. This defines the bijection (one checks that it is indeed a bijection).

In particular, computing the number of conjugacy classes for $n$ and $q$ fixed requires knowing the set of monic irreducible polynomials, and the set of partitions. I regard this as only slightly more problematic than computing the number of integer partitions of $n$ (and presumably there are known asymptotic formulas similar to Hardy-Ramanujan's formula for integer partitions).

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  • $\begingroup$ This is a good result, thanks for answering. Searching deeply, this may be related with the values of the Dirichlet's $L$ function, but there isn't any known formula for computing this number. So I think it might be interesting to find an explicit formula for this. Thanks again, I'll accept your answer. Any other paper/research/result is really appreciated. $\endgroup$ Oct 25, 2017 at 21:59

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