# Regular semisimple elements in $SL(n,q)$

Consider $$G=GL(n,q)$$. A regular semisimple element of this group, is a matrix, whose characterestic polynomial is square-free and same as minimal polynomial. Results show that the number of such matrices in $$G$$ is a polynomial in $$q$$, with degree $$n^2$$, and coefficient of the leading term being 1.

Does similar hold, when we take $$G=SL(n,q)$$, that is, the number of such matrices in $$G$$ is given by a polynomial in $$q$$ of degree $$n^2-1$$, with the leading term being 1?

I appreciate any kind of help or references that can be of use. Thank you.

• If n=2 then the number of non-regular-semisimple elements is q^2-q+1 for odd q and q^2 for even q. So what sort of answer are you looking for? Do you care if there is a different formula for a few small characteristics? I expect the general formula is of similar qualitative behaviour to this case. – Peter McNamara Apr 16 at 1:01
• @PeterMcNamara, I'm not familiar with these counting problems. Does that mean that the statement in the question about the number of rss elements in $\operatorname{GL}(n, q)$ is also true only for large $q$, or does passing to $\operatorname{GL}(n, q)$ magically sort out $\operatorname{SL}(n, q)$ problems? – LSpice Apr 16 at 1:07
• I can't find a reference on the number of rss elements, but the number of rss classes is well studied. I'm familiar with Steinberg (see, for example, §14 of Steinberg - Endomorphisms of linear algebraic groups); but some Googling also turned up Fleischmann, Janisczczak, and Knörr - The number of regular semisimple classes of special linear and unitary groups. – LSpice Apr 16 at 1:27
• @LSpice I suspect that for SL_N one will get a different formula for q a power of a prime dividing N, and this phenomenon does not appear for GL_N, where the OP indicates there is a general formula that works for all q. – Peter McNamara Apr 16 at 1:53
• number of regular semisimple elements is always asymptotic to q^{n^2-1} by Deligne's proof of Weil conjectures. – Peter McNamara Apr 17 at 0:58

## 1 Answer

This is just a comment but in community wiki format. Most studies of semisimple (or reductive) algebraic groups and finite groups of Lie type emphasize counting the number of classes of various elements with a view toward representation theory. So it's tmportant to consider motivation when counting elememts.

Concerning references, much of this goes back to Steinberg. A short summary of further work is given in my 1995 AMS book on conjugacy classes, e.g., section 8.9. In spite of their misleading title, classes are the subject of a paper bu Peter Fleischmann and Ingo Janisczak here. Note too Steinberg's theorem stating that the set of regular semisimple elements in any semisimple (or reductive) group is open and dense, consistent with your observation: see 2.5 in my book.