Linked Questions
10 questions linked to/from $q$-(and other)-analogs for counting index-$n$ subgroups in terms of Homs to $S_n$?
18
votes
5
answers
1k
views
Number of $3\times 3$ anticommuting matrices over finite fields $\mathbb{F}_p$ is (or is not?) polynomial in $p$?
There are rare algebraic varieties such that the number of points over finite fields $\mathbb{F}_p$ is given by a polynomial in $p$. One notable series of examples is the commuting variety: $[A,B]=0$ ...
- 24.9k
13
votes
2
answers
1k
views
Number of commuting pairs (triples, n-tuples) in GL_n(F_q) (and other groups)?
Question 1 What is the number of pairs of commuting elements in GL_n(F_q) ?
I am aware of many results concerning commuting elements in Mat_n(F_q), but I am interested in GL i.e. non-degenerate ...
- 24.9k
33
votes
2
answers
1k
views
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 $q\to1$ ...
- 24.9k
5
votes
2
answers
292
views
Field with one element look at counting index-$n$ subgroups in terms of Homs to $S_n$, generalization to $F_{1^k}$?
Main idea shortly: As we discussed recently MO272045, there is beautiful fomula which
counts index-n subgroups in terms of homomorphisms to $S_n$.
Let me give "field with one element" interpretation ...
- 24.9k
17
votes
0
answers
401
views
Number of $F_p$-matrices ac=ca, bd = db , ad - da = cb - bc is polynomial in p ? ("Manin matrix variety" - normal ? Cohen–Macaulay ? )
Consider four $n\times n$ matrices $a,b,c,d$ over finite field $F_q$ (or $F_p$ for simplicity), such that they satisfy three equations: $ac=ca,bd=db, ad-da=cb-bc $. Thus an affine algebraic manifold ...
- 24.9k
10
votes
0
answers
436
views
Commuting matrix variety $[A,B]=0$ - can one geometrically explain divisibility of $F_ q$ point count by high powers of $q$?
$\DeclareMathOperator\Comm{Comm}\DeclareMathOperator\Id{Id}$Consider the variety $\Comm$ of commuting matrices $[A,B]=0$ over some field $K$. It is much studied, and interesting for various reasons.
...
- 24.9k
15
votes
0
answers
271
views
Lie theoretic meaning to $e^{\text{cycle}} = \text{permutation}$?
It is well known that exponentiating the EGF(exponential generating function) for cycles gives the EGF for permutations: link here. Usually summarized under the catchy slogan ...
- 1,221
2
votes
1
answer
184
views
Count N-tuples of commuting matrices over $F_q$ is given by polynomials with pattern $\sum q^{A_i(N)} P_{i}(q) $, where $P_i$ - do not depend on $N$?
Count pairs of $k \times k$ commuting matrices over finite field $F_q$ is given by certain polynomials in $q$ (which is quite rare phenomena for algebraic varieties) and have interesting generating ...
- 24.9k
2
votes
0
answers
148
views
Nilpotent polynomial matrices over $F_q$ - polynomial count variety ? ( Nilpotent cone for Hitchin-Gaudin like integrable system)
Context: Number of nilpotent $n\times n $ matrices over $F_q$ is $q^{n(n-1)}$ classical result due to Ph.Hall, M.Gerstenhaber (see very nice exposition by T.Leinster at n-cat-cafe/arxiv) which have ...
- 24.9k
3
votes
0
answers
97
views
What algebras generate polynomial count varieties as their representations spaces ? Is it preserved by the Koszul duality, Manin's endomorphisms?
Consider for example commutative polynomial algebra $C[x,y]: xy=yx$, look on $F_p$-matrices satisfying that relation - the number over $F_p$ will be given by polynomial in $p$ (classical result due to ...
- 24.9k