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

Question

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).

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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.

Answer 1

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.

Answer 2

[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.