All Questions
Tagged with co.combinatorics ag.algebraic-geometry
45 questions
19
votes
3
answers
2k
views
Guises of the Stasheff polytopes, associahedra for the Coxeter $A_n$ root system?
Richard Stanley keeps a famous running compilation of different guises of the celebrated Catalan numbers. The number of vertices of the associahedron is one instantiation among the multitude, and the ...
- 10.5k
17
votes
0
answers
402
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
19
votes
2
answers
1k
views
Explicit invariant of tensors nonvanishing on the diagonal
The group $SL_n \times SL_n \times SL_n$ acts naturally on the vector space $\mathbb C^n \otimes \mathbb C^n \otimes \mathbb C^n$ and has a rather large ring of polynomial invariants. The element $$\...
- 148k
14
votes
1
answer
801
views
Is there a lift of the q-Vandermonde identity to some geometric (motivic) identity for Grassmannians over $F_q$?
The q-Vandermonde identity reads:
$$ \binom{m + n}{k}_{\!\!q} =\sum_{j} \binom{m}{k - j}_{\!\!q} \binom{n}{j}_{\!\!q} q^{j(m-k+j)} $$
The q-binomial coefficients:
$$ \binom{ a }{ b}_{\!\!q} $$
...
- 24.9k
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
11
votes
0
answers
629
views
Inversion, Koszul duality, combinatorics and geometry
According to this MO answer Koszul duality is related to operations on generating series;
1) multiplicative inversion for quadratic algebras,
2) compositional inversion for quadratic operads,
3) ...
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
103
votes
3
answers
6k
views
Why do combinatorial abstractions of geometric objects behave so well?
This question is inspired by a talk of June Huh from the recent "Current Developments in Mathematics" conference.
Here are two examples of the kind of combinatorial abstractions of geometric ...
- 24.2k
18
votes
1
answer
1k
views
A natural refinement of the $A_n$ arrangement is to consider all $2^n-1$ hyperplanes given by the sums of the coordinate functions. Have you seen this arrangement? Is it completely intractable?
The short version
Here is an extremely natural hyperplane arrangement in $\mathbb{R}^n$, which I will call $R_n$ for resonance arrangement.
Let $x_i$ be the standard coordinates on $\mathbb{R}^n$. For ...
- 2,372
18
votes
1
answer
1k
views
Smith Normal Form of powers of a matrix
What invariants of a matrix determine the Smith Normal Form (SNF) of all the powers of a matrix?
The question makes sense over any PID $R$. If we let $M = M_n(R)$ and $G=Gl_n(R)$, then SNF is a ...
- 3,526
16
votes
4
answers
3k
views
How many minors I need to check to conclude all minors will vanish ?
Given a $m \times n$ matrix $n>m$, I was trying to check if all its $m \times m$ minor vanish.
I remember hearing that one really does not need to check all possible minors in order to conclude ...
- 1,795
8
votes
1
answer
1k
views
Grassmann–Plücker relations for permanents
Let $K$ be a field, $1 \leq d \leq n$ integers and $V$ an $n$-dimensional vector space. The Grassmann–Plücker relations are quadratic forms on $\bigwedge^d V$ whose zero set is exactly the set of ...
- 24.7k
3
votes
1
answer
607
views
Automorphism groups of indefinite non-unimodular integer lattices
Does anyone know of any papers in which structural aspects of the orthogonal group of some indefinite non-unimodular integral lattice are calculated? The exact lattice isn't so important and they don'...
- 31
3
votes
2
answers
452
views
Guess the next polynoms in the sequence (MO vs. AI :), count anticommuting $F_p$-matrices, P. Hrubeš conjecture
Here is a sequence of polynoms - (presumably) counting N-tuples of ANTI-commuting 2x2 matrices over $F_p, p>2$. (That is just the case of 2x2 matrices, and (surprisingly) it is not so easy to see a ...
- 24.9k
3
votes
0
answers
203
views
What d.o. $\sum_i f_i(z)\partial_z^i$ correspond to subalgebras $M$ in polynoms $C[x_i]$ being Langlands dual to motive of $Spec(M) \to X$?
Briefly: The question is about presenting explicit examples of the construction discussed in the recent MO question "Relation between motives and geometric Langlands" and Will Sawin's asnwer ...
- 24.9k
0
votes
0
answers
61
views
Combinatorial counting question related to count (anti)commuting N-tuples of matrices (more generally $(X_1,...X_n): F(X_i,X_j)=0$ - only one F)
Consider some finite set $S$ (can be matrices over $F_p$), consider some symmetric relation $F(s1,s2)$ which values are True or False (for example - matrices (anti)commutate or not).
Question 1: can ...
- 24.9k
62
votes
7
answers
7k
views
Euler-Maclaurin formula and Riemann-Roch
Let $Df$ denote the derivative of a function $f(x)$ and $\bigtriangledown f=f(x)-f(x-1)$ be the discrete derivative. Using the Taylor series expansion for $f(x-1)$, we easily get $\bigtriangledown = ...
- 13.1k
51
votes
3
answers
4k
views
What is the sandpile torsor?
Let G be a finite undirected connected graph. A divisor on G is an element of the free abelian group Div(G) on the vertices of G (or an integer-valued function on the vertices.) Summing over all ...
- 19.2k
41
votes
6
answers
4k
views
Why do Littlewood-Richardson coefficients describe the cohomology of the Grassmannian?
I'm looking for a "conceptual" explanation to the question in the title. The standard proofs that I've seen go as follows: use the Schubert cell decomposition to get a basis for cohomology and show ...
- 10.7k
34
votes
2
answers
3k
views
Shimura-Taniyama-Weil VS Grothendieck's dessins
When listening to the beautiful lectures by Gilles Schaeffer at
the SLC68, the following (perhaps crazy) question occurred to me:
did anyone attempt (succeed?) to combinatorially prove modularity of ...
19
votes
2
answers
1k
views
About a Delzant polytope. (In particular dodecahedron)
Hi. I have a question.
Definition. Delzant polytope $P$ is a rational convex simple polytope with the smooth condition. Here, "smooth" means that for each vertex $v$, the $n$ edges containing $v$ ...
- 1,037
18
votes
0
answers
579
views
What is the geometric intuition behind Wilf-Zeilberger theory?
This problem is somehow inspired by a bunch of impressive posts of combinatorial identities by T. Amdeberhan. Earlier this month I learnt from computer scientists that they have a generic algorithmic ...
- 8,071
18
votes
2
answers
1k
views
Deligne-Simpson problem in the symmetric group
Question.
Let $C_1,\dots,C_k$ be conjugacy classes in the symmetric group $S_n$. (More explicitly,
each $C_i$ is given by a partition of $n$; $C_i$ consists of permutations whose cycles
have the ...
- 4,540
18
votes
2
answers
983
views
A direct proof of the Harer-Zagier recursion enumerating the ways to paste a 2n-gon to get a genus g surface?
In a 1986 paper, Harer and Zagier proved the recursion:
$$(n+1)e(g,n)=(4n-2)e(g,n-1)+(2n-1)(n-1)(2n-3)e(g-1,n-2)$$
where e(g,n) is the number of ways of grouping sides $S_1...S_{2n}$ of a 2n-gon ...
- 1,303
17
votes
2
answers
3k
views
What are some open problems in toric varieties?
In light of the nice responses to this question, I wonder what are some open problems in
the area of toric geometry? In particular,
What are some open problems relating to the algebraic ...
15
votes
3
answers
892
views
Log-concavity of matroids: characterization of equality?
Let $M$ be a (loopless) matroid of rank $r$.
The characteristic polynomial $\chi_M(x)$ is defined by $\chi_M(x)=\sum_{F \in \mathcal{L}(M)}\mu(\hat{0},F) \cdot x^{\mathrm{rk}(F)}$, where $ \mathcal{L}(...
- 24.2k
11
votes
4
answers
1k
views
What is the correspondence between combinatorial problems and the location of the zeroes of polynomials called?
(From MSE)
In the wikipedia article on the Italian-born American mathematician and philosopher Gian-Carlo Rota, it is stated that the one combinatorial idea he would like to be remembered for
".....
- 4,829
11
votes
1
answer
980
views
Fomin-Kirillov algebras and Schubert calculus
In
Fomin, Sergey; Kirillov, Anatol N. Quadratic algebras, Dunkl elements, and
Schubert calculus. Advances in geometry, 147--182, Progr. Math., 172,
Birkhäuser Boston, Boston, MA, 1999. MR1667680 (...
- 3,078
10
votes
2
answers
2k
views
Expressions involving Eulerian numbers of the second kind: trying to show $\sum_{m=0}^{n} (-1)^m(m)m!(2n-m-2)!\left\langle\left\langle n\atop m\right\rangle\right\rangle\neq0$ for even $n$.
Considering the success of a previous question involving Eulerian numbers, I thought I might throw this question into the mix. It comes from some localization computations in GW theory, but in this ...
- 103
10
votes
1
answer
461
views
Reduction formula for Schubert polynomials
In my endless fiddling with formulas I discovered one that fills in the blanks in a generic formula I saw in a paper, but I'm wondering if maybe it's already known and the paper was just mentioning ...
- 2,168
9
votes
0
answers
2k
views
Exactly Counting the Number of Lattice Points in an $n$-Dimensional Sphere
Let $S_n(R)$ denote the number of lattice points in an $n$-dimensional "sphere" with radius $R$. For clarification, I am interested in lattice points found both strictly inside the sphere, and on its ...
- 311
7
votes
1
answer
489
views
Combinatorial meaning of Kazhdan-Lusztig-Stanley polynomial
This question is motivated by
Why do combinatorial abstractions of geometric objects behave so well?
The algebraic geometry of Kazhdan-Lusztig-Stanley polynomials
Kazhdan-Lusztig-Stanley polynomials ...
- 5,230
7
votes
1
answer
390
views
Combinatorial/probabilistic statements having $F_{\text{un}}$/$F_q$ geometric interpetation
$\newcommand{\Fun}{F_\text{un}}$There was lots of "Fun with $\Fun$" (field with one element) in recent years.
One of the points is that it provides bridge between geometrical and ...
- 24.9k
7
votes
0
answers
276
views
Cyclic shift acting on finite Grassmannian
The (twisted) cyclic shift $(v_1,v_2,\ldots,v_n) \mapsto (v_2,v_3,\ldots,v_n,(-1)^{k-1}v_1)$ acting on the Grassmannian $\mathrm{Gr}(\mathbb{C};k,n)$ of $k$-planes in $\mathbb{C}^n$ is an important ...
- 24.2k
6
votes
2
answers
531
views
Finding the matroids with a specified set of non-bases
I'm a grad student in algebraic geometry, and I've encountered a problem which requires me to produce an algorithm involving matroids. Since this isn't my area of expertise, I'm hoping someone knows ...
- 1,510
6
votes
1
answer
778
views
Dimension of the span of all partial derivatives of a given homogeneous symmetric polynomial $f$ and the polynomial $E(f)$
I need some help about the problem below.
Let $d\geq 4$ and $f$ a symmetric polynomial, homogeneous of degree $d$, in $n$ variables $x_1,\dots,x_n$, with real coefficients. We set
$$ E(f):=\sum_{j=1}^{...
- 307
6
votes
2
answers
1k
views
Systems of simultaneous real quadratic equations
Starting from a problem in spectral graph theory, I got dragged into a problem in combinatorial matrix theory about constructing $n\times n$ real orthogonal matrices with a specified pattern of zero/...
5
votes
1
answer
453
views
Polynomial defined recursively by a resultant
Cross posting from MSE.
Definition:
For any natural number $n\ge 3$, define the polynomial $P_{n}\left(x_1,x_2,...,x_{n-1},x_{n} \right)$, with indeterminates $x_{i}$, where $i\in\{1,2,...,n-1,n\}$, ...
- 149
5
votes
0
answers
107
views
Generalized Puiseux series for diagonal reflections of the curves $y = \frac{x}{(1-ax)(1-bx)^m}$
Reflection of the curve $y = f_m(x) = \frac{x}{(1-ax)(1-bx)^m}$ through the diagonal line $y=x$ in the $xy$-plane can be regarded as local compositional inversion of the curve $y=f_m(x)$. ($x,y,a,b$ ...
- 10.5k
4
votes
2
answers
2k
views
Reference request: Lascoux's formulas for Chern classes of tensor products and symmetric powers
Let $E$ and $F$ be vector bundles on a smooth projective variety, say.
A. Lascoux ("Classes de Chern d'un produit tensoriel", C. R. Acad. Sci. Paris Sér. A-B 286 (1978), no. 8, A385–A387) gave ...
user5117
4
votes
1
answer
1k
views
Solving a Diophantine equation related to Algebraic Geometry, Steiner systems and $q$-binomials?
The short version of my question is:
1)For which positive integers $k, n$ is there a solution to the equation $$k(6k+1)=1+q+q^2+\cdots+q^n$$ with $q$ a prime power?
2) For which positive ...
- 23k
3
votes
0
answers
102
views
The ring generated by a convex polytope and its faces
Let $V=\Bbb R^n$. Morelli defined the (commutative unital) ring $L(V)$ to be the additive group generated by the indicator functions of convex polytopes in $V$ with multiplication induced by Minkowski ...
- 3,079
3
votes
0
answers
233
views
A bridge between the algebraic / differential geometry of $\frak{sl}_2(\mathbb{C})$ and the Sheffer-Appell calculus and combinatorics
In "Four examples of Beilinson-Bernstein localization", Anna Romanov introduces the basis
$m_k = \frac{(-1)^k}{k!} \partial^k \delta $
on p. 9, where $\partial$ is a partial derivative and $\...
- 10.5k
3
votes
0
answers
683
views
Counting / characterizing the isolated points of a particular algebraic variety
I'm not a professional geometer / topologist, so please thanks for your patience :)
Setup
The following questions are the first in a series of steps I'm undertaking in an attempt to break down a ...
- 6,853
2
votes
0
answers
208
views
Sylvester-Gallai-type theorem for quadratic polynomials
Let $F_1, F_2$ and $F_3$ be finites sets of irreducible polynomials in $\mathbb{C}[x_0, \ldots, x_n]$ of degree at most $2$ such that $F_1 \cap F_2 \cap F_3 = \varnothing$ and for every $Q_1, Q_2$ ...
- 2,576