# Analogies supporting heuristic: Weyl groups = algebraic groups over field with one element?

There is well-known heuristic that Weyl groups are reductive algebraic groups over "field with one element".

Probably the best known analogy supporting that heuristic is the limit $$q\to1$$ for number of elements in $$G(F_q)$$ - for appropriate "m" it holds:

$$\lim_{q\to1} \frac { |G(F_q) | } { (q-1)^m} = |Weyl~Group~of~G|$$

For example: $$|GL(n,F_q)|= [n]_q! (q-1)^{n}q^{n(n-1)/2}$$ so divided by $$(q-1)^n$$ one gets $$[n]_q! q^{n(n-1)/2}$$ and at the limit $$q\to1$$, one gets $$n!$$ which is the size of $$S_n$$ (Weyl group for GL(n)). (For other groups see Lorscheid 2009 page 2 formula 1).

Question What are the other analogies supporting heuristics: Weyl groups = algebraic groups over field with one element ?

Subquestion once googling papers on F_1, I have seen quite an interesting analogy from representation theory point of view - it was some fact about induction from diagonal subgroups of symmetric groups $$S_{d_1}\times ... \times S_{d_k} \subset S_n$$ where $$\sum d_i = n$$ and similar fact for $$GL(n,F_q)$$ which was due to Steinberg or Springer or Carter (cannot remember). But I cannot google it again and cannot remember the details :( (Tried quite a lot - I was sure it was on the first or second page of Soule's paper on F_1 - but it is not there, neither many other papers).

Knowing that total element count is okay, we may ask about counting elements with certain properties - like: m-tuples of commuting elements (MO271752), involutions, elements of order $$m$$, whatever ... From answer MO272059 one knows that there are certain analogies for such counting, however it seems the limits $$q\to1$$ are not quite clear.

Question 2 Is there any analogy for counting elements with some reasonable conditions ? Hope to see that count for $$G(F_q)$$ (properly normalized) in the limit $$q\to1$$ gives answer for Weyl group.

• I'd be interested to know if one has some kind of limit-expression for the number of real elements in $G(\mathbb{F}_q)$... That seems a good test case for trying to extend this sort of heuristic. Jun 20, 2017 at 9:17
• @AlexanderChervov: A theorem of Carter says that in a Weyl group every element is "strongly real", i.e. a product $xy$ for some $x, y$ with $x^2 = y^2 = 1$. In particular every element of a Weyl group is real. Jun 20, 2017 at 10:51
• @AlexanderChervov: Conjugacy classes in the Weyl group, Seminar on Algebraic Groups and Related Finite Groups, Lecture Notes in Mathematics 131. Jun 20, 2017 at 11:11
• No, it's not true that all elements in $G(F_q)$ are strongly real -- the list of which quasisimple groups have all elements (strongly) real was answered by Tiep and Zalesski. This doesn't include all groups of Lie type of course, but it deals with some of them. Jun 20, 2017 at 11:23
• Sorry, my last comment was misleading -- you asked which groups have all real elements are strongly real, whereas Tiep and Zalesski showed which groups have all elements strongly real. In any case, it is an interesting and relevant paper! And, to address your question,it's not true that all real elements in $G(F_q)$ are strongly real for arbitrary $G$ -- see this paper on the unitary groups arxiv.org/pdf/1303.6085.pdf Jun 20, 2017 at 11:28

This is more an extended comment than a full-fledged answer, but OP encouraged me to write it, and maybe it can encourage someone to post a more precise account along those lines.

A parabolic subgroup of a Coxeter group $W$ with Coxeter generating set $S$ is simply the subgroup (also a Coxeter group) $W_J$ generated by a subset $J\subseteq S$. In case $W$ is a Weyl group, this means we select a subset $J$ of the set $S$ of nodes of the Dynkin diagram. The maximal parabolics are obtained by removing a single node from the Dynkin diagram. In the case of $\mathfrak{S}_n = W(A_{n-1})$ generated by transpositions of adjacent elements, a maximal parabolic is the stabilizer of $\{1,\ldots,k\}$ in $\{1,\ldots,n\}$ where $k$ is the removed node; so the corresponding quotient (I mean, set of cosets) can be viewed as the set of $k$-element subsets of $\{1,\ldots,n\}$, of which there are $\binom{n}{k}$. Now the corresponding parabolic subgroup of $\mathit{GL}_n(\mathbb{F}_q)$ is the stabilizer of a $k$-dimensional subspace, and the quotient is the set of such subspaces (set of points of the Grassmannian), of which there are $\binom{n}{k}_q$ (Gaussian binomial coefficient). The various analogies between ordinary and Gaussian binomial coefficients can then be construed as analogies between the Weyl group and the linear group. Similar things can be said for flag varieties and other Dynkin types, but I don't feel comfortable enough expanding this here.

Along different lines (or maybe not so different), (thick) Tits buildings of spherical type can be seen as a generalization of Coxeter complexes, i.e., "thin" buildings, (of spherical type), and the relation with the algebraic groups on the one hand, and finite Coxeter groups on the other clearly makes the Weyl groups appear similar to algebraic groups over the field with $1$ element. Again, I don't want to expand upon this for fear of saying something wrong, but this should at least suggest a way of looking at things.

One last thing which comes to my mind is about generalized matroids: not only does the set of $k$-element subsets of $\{1,\ldots,n\}$ have a cardinal which has formal similarities with the set of $k$-dimensional subspaces of $\mathbb{F}_q^n$, but their sets also have a structure as a matroid, and again, matroids can be generalized to flag matroids and other Dynkin types.

• Thank you for your answer ! So for S_n maximal parabolics seems to be S_{k}\times S_{n-k} and general parabolic S_{k_1} \times ... \times S_{k_l}. So Borel becomes trivial (it is S_1\times ... \times S_1) , but parabolics still exists. There should be analogies with parabolic inductions I guess... Jun 20, 2017 at 19:55
• So for maximal parabolic the size of the factor space S_n / (S_k \times S_{n-k}) is obiously a binomial coefficient n! / k! (n-k)!. While the size of same space for GL(n,F_q) is q-binomial [n]_q! / k_q! (n-k)_q! That is nice.! Would be interesting to see what are properties can be deduced further.. Jun 20, 2017 at 20:01
• By the way definition of parabolic is consistent with naive embedding S_n to GL as permutation matrices and intersecting with standard parabolics in GL. Jun 20, 2017 at 20:04
• It seems Vipul Naik text COMBINATORICS OF THE GENERAL LINEAR GROUP cmi.ac.in/~vipul/writeupsandpresentations/… is something about that Jun 20, 2017 at 20:08
• It also becomes clearer for me about vector space over F_1 . Vector space over F_q we may think F_q \oplus ... F_q , so F_1 \oplus ... F_1 is over F_1 - this means just a set because: over F_q we can multiply by q each coordinate , but over F_1 we have only one element so multiplcation gives it itself. So what we can say is that preserving the vector space property still works over F_1 . In that way we get analogies with Grassman , binomial, q-binomial . Jun 20, 2017 at 20:13

[The following comment is too long for the comment box.]

On the subquestion: Zelevinsky's Representations of finite classical groups - a Hopf algebra approach (LNM) may be relevant to what you're thinking of. Zelevinsky builds two Hopf algebras: the first coming from induction and restriction of (complex) representations of the symmetric groups along the inclusion $S_n\times S_m\to S_{n+m}$, the second using parabolic induction and restriction (again, of complex representations) for finite general linear groups along the inclusion $GL_n(\mathbb F_q)\times GL_m(\mathbb F_q)\to GL_{n+m}(\mathbb F_q)$. Zelevinsky shows that the second algebra is a tensor product of copies of the first, with one copy for each pair $(n,\pi)$ where $\pi$ is a cuspidal representation of $GL_n(\mathbb F_q)$.

Here's an attempt, possibly completely misguided, to extract from this an analogy that would be relevant to your question. (I don't know how this relates to the existing literature on $\mathbb{F}_1$. Sorry if I am repeating something that is well known.) If we identify $S_n=GL_n(\mathbb F_1)$, Zelevinsky's result might be interpreted metaphorically as saying that ''the only cuspidal representation of $GL_n(\mathbb F_1)$ is the trivial representation of the trivial group''. Since cuspidal representations of $GL_n(\mathbb F_q)$ are associated to characters of anisotropic tori, and since (I suppose?) the ''group of $\mathbb F_1$-points of a torus over $\mathbb F_1$'' is always the trivial group, this makes some kind of sense.

• Thank you for your answer ! I was just thinking what is "cuspidal" for S_n ? So you suggest it is trivial one. Can we check your suggestion in the following way: inducation from S_a\times S_b \times ... S_l can be seen as parabolic induction. Cuspidal are those NOT obtained by parabolic induction. So your proposal means that all irreps of S_n are somehow obtained by "parabolic induction" - is that true ? Jun 21, 2017 at 9:39
• @AlexanderChervov Yes, it is true for the somewhat silly reason that every irreducible representation of $S_n$ is a subrepresentation of $Ind_{S_1\times\cdots\times S_1}^{S_n} 1$ (the representation induced from the trivial representation of the trivial subgroup---which is a ''Levi subgroup'' according to this analogy).
– t.c.
Jun 21, 2017 at 9:48
• Remark: it seems to me Levi = Parabolic for S_n , that is natural from Gro-Tsen answer and action on vector spaces over F_q and F_1 Jun 21, 2017 at 9:52
• Okay, but is true for GL that all NOT cuspidals can be parabolic induced from MAXIMAL parabolics ? Is the same true for S_n ? Maximal parabolics are GL(k)\times GL(n-k), and S_k \times S_{n-k} Jun 21, 2017 at 9:54
• @AlexanderChervov : I doubt that it will work in exactly the same way for SL_n, since SL_n x SL_m is not a Levi subgroup of SL_{n+m}, so in the formula for the composition of restriction and induction there will be a sum over S_n-double-cosets (like in GL_n and S_n), but also a sum over characters of the multiplicative group, which will complicate things. On the other hand, Zelevinsky's construction does have analogues for other groups: see e.g. Van Leeuwen (J. Algebra 1991). On the question of GL_n(F_1) vs SL_n(F_1), I'm afraid I don't know enough about F_1 to have an opinion.
– t.c.
Jun 23, 2017 at 8:10