All Questions
Tagged with co.combinatorics ag.algebraic-geometry
292 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
2
votes
0
answers
357
views
$G$-equivariant coherent sheaves on Bott$-$Samelson resolutions
Let $G$ be a Lie group and $B$ a Borel subgroup. $G/B$ is the corresponding flag variety.
Let $w$ be an element of the Weyl group $W$ with a reduced expression
$w = s_1 \cdots s_n$. Let $X_w$ be ...
- 1,719
1
vote
2
answers
374
views
How to understand a rooting of a dessin d'enfant?
As I understand it, rooted maps on surfaces were first introduced in enumerative combinatorics because they are easier to count than unrooted maps, which can have non-trivial symmetries. A map is a ...
- 3,393
10
votes
2
answers
495
views
Is there a hyperplane avoiding two independent sets?
Let $V$ be a vector space over a field with $5$ elements, $A,B \subseteq V$ independent subsets. Must there be a subspace of $V$ of codimension 1 disjoint from $A \cup B$?
- 11.3k
4
votes
1
answer
255
views
Is there a geometric meaning of the Major index?
The actual question I want to ask is whether there is a geometric proof of this famous identity
$$\sum_{\sigma \in S_n} q^{\operatorname{inv} \sigma}=\sum_{\sigma\in S_n}q^{\operatorname{maj}\sigma},$$...
- 85.6k
5
votes
1
answer
312
views
Non-vanishing of elements in cohomology of full Flag varieties
Consider the full flag variety $F_n$ consisting of full flags in $\mathbb C^n$. There is a collection of tautological bundles on $F_n$:
$0=U_0\subset U_1\subset ...\subset U_{n-1}\subset U_n=\mathbb ...
- 14.3k
8
votes
1
answer
487
views
Continued fraction expansion of an algebraic number and its conjugates
Let $w$ be an element of a Galois extension $L:\mathbb{Q}$ such that $\text{Gal}(L/\mathbb{Q})=\langle g\rangle$ is cyclic of order $n$ (here $\mathbb{Q}$ is rationals). Suppose we know the continued ...
- 641
4
votes
1
answer
178
views
Transalate of a Richardson Variety
For $v \leq w$ are Weyl group elements, the intersection of a Schubert variety $X^w$ and the opposite Schubert variety $X_v$ is called a Richardson variety. It is denoted by $X_v^w$. It is well know ...
- 41
3
votes
2
answers
437
views
convex polytope integer points
is there a simple proof for the following lemma:
An unbounded convex polytope (defined by linear constraints) has either zero integer points or infinite many integer points.
- 39
0
votes
0
answers
230
views
Toric morphism fiber and kernel dimensions
Given a morphism between two smooth toric varieties $f: X \rightarrow Y$, is the dimension of the kernel of $\mathrm{d}f$ at any point $p \in X$ equal to the dimension of the fiber at $f(p) \in Y$?
...
- 1,719
2
votes
3
answers
259
views
branching schubert calculus
Let $X=Gr(r,V), Y=Gr(r+1,W)$ where $V,W$ are complex vector spaces with $\dim V > r$ and $\dim W > \dim V$. Let $\phi:X\rightarrow Y$ be some embedding of varieties. This induces a morphism on ...
- 501
5
votes
0
answers
604
views
The twisted kiss of the curvaceous cubic and the staid tetrahedron (references)
(Migrated from MSE)
While investigating some operators, I came across some relations between the twisted cubic curve and the tetrahedron that link together some notions in differential geometry, ...
- 10.5k
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
4
votes
1
answer
383
views
Point-Hyperplane incidence in finite projective spaces
Let $P$ be a finite projective space of order $q$ and dimension $d$. I am interested in finding the least $k$ such that for any set $S$ of $k$ points of $P$, and for any set $S'$ of $k$ hyperplanes of ...
- 935
6
votes
2
answers
781
views
What is the combinatorial data classifying non-normal affine toric varieties?
Recall that a toric variety is a variety $V$ containing an open dense algebraic torus. Here an algebraic torus means a finite product of copies of the multiplicative group of the ground field (which I ...
- 38.6k
4
votes
2
answers
270
views
Is there a theory of oriented subspace arrangements?
The theory of hyperplane arrangements is a rich and intensely studied subject, especially from the perspective of combinatorics; see e.g. this wonderful monograph of Stanley. Oriented hyperplane ...
- 24.2k
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
4
votes
1
answer
485
views
Intersection theory on moduli spaces of curves without marked points
1. There are a lot of works concerning the intersection theory on the moduli spaces of curves $\mathcal M_{g,n}$ (and their Deligne-Mumford compactifications $\overline{\mathcal M}_g$), for $n>0$.
...
- 838
3
votes
2
answers
362
views
Kahler differentials on cluster varieties
On affine toric varieties there is a classical theorem of Danilov which gives some combinatorial ways to describe the global sections of an appropriate sheaf of Kahler differentials as a vector space. ...
- 1,331
1
vote
0
answers
96
views
Finding stable ideals of $\mathbb{F}_3[[X,S]]$ by group action
Let $k > 1$ be a positive integer and define the action $\sigma_k$ on $\mathbb{F}_3[[X,S]]$ by:
$\sigma_k: X \mapsto X + S + X^k$
$\sigma_k: S \mapsto S + S^3$.
Conjecture: There exists a ...
- 87
3
votes
2
answers
795
views
computing monodromy of branched cover of sphere specified by a polynomial
We know by the Riemann Existence Theorem that any Riemann surface can arise holmorphically as the branched cover of a sphere:
Which Riemann surfaces arise from the Riemann existence theorem?
...
- 22.8k
22
votes
2
answers
5k
views
What is Tropicalization, and how is it applied
My question is:
What is Tropicalization, how is it done, and what are some basic applications of it?
motivation
I am interested especially in how questions about enumerative algebraic geometry ...
- 24.7k
9
votes
0
answers
247
views
Degree of a cone over the set of rank $r$ $n\times n$ matrices
Let $X_r\subset Mat_{n\times n}(C)$ denote the variety of rank at most $r$ matrices, set $k=n-r$ and assume $n\geq k^2-1$. Consider the cone over $X_r$ with vertex spanned by the first $k^2-1$ entries ...
- 835
13
votes
1
answer
698
views
Counting representations of $k[x,y]$ when $k$ is finite
$\newcommand{\GFq}{\mathbf F_q}$
Let $r_n(q)$ denote the number of isomorphism classes of $n$-dimensional modules of the $\GFq$-algebra $\GFq[x,y]$. Is it known whether there exists a polynomial $p_n(...
- 5,654
2
votes
0
answers
203
views
Schemes defined by a collection of Plücker coordinates
If $C \subset {[n]\choose k}$ is any collection of $k$-element sets, we can define a scheme $$ W(C) = \bigcap_{S\notin C} \{V \in Gr(k,n) : p_S(V)=0\} \qquad \subseteq Gr(k,n), $$ where $p_S$ is the ...
- 27.8k
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
0
answers
137
views
What are the Voronoi cones in 4 variables?
Question: What are the top dimensional cones of the 2nd Voronoi decomposition of the space of positive definite forms in $4$ variables?
The 2nd Voronoi decomposition of the cone of positive definite ...
- 31
10
votes
4
answers
1k
views
zeros of a homogeneous polynomial
Hi All,
Let $F$ be a finite field, $\lambda\in F$, and $$p_\lambda (x,y,z)=\left|\begin{array}{ccc}x & y & z \\ y & z & x +\lambda z \\ z& x+\lambda z & y+\lambda x+\lambda ^...
- 103
5
votes
1
answer
527
views
global sections of structure sheaf on the toric Calabi-Yau
Let P be a lattice polytope and lying in $ N \times {1} \subset N \times \mathbb{R}$. Let $\sigma$ be the cone over this polytope and $X_\sigma$ be the corresponding toric variety, which is an affine,...
0
votes
1
answer
79
views
Connection between the number of vertices and the number of lattice points of the integer hull of a polytope?
Is there a connections between the number of vertices and the number of lattice points of $P_I$, the integer hull of a polytope $P$? Which is usually more difficult to determine?
Or if I have a bound ...
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
4
votes
3
answers
223
views
Generate a higher degree symmetric polynomial from an existing one
Suppose $p(x_1, x_2, \cdots, x_n)$ is a symmetric polynomial. Given any univariate polynomial $u$, we can define a new polynomial $q(x_1, x_2, \cdots, x_{n+1})$ as
$q(x_1, x_2, \cdots, x_{n+1}) = u(...
- 61
0
votes
3
answers
622
views
When does the rigidity matrix of a graph have full row rank?
Intuitive description: In the 2D plane, there are $m$ bars connected by $n$ joints. The length of each bar is fixed. These joints and bars can be viewed as a graph (see the figures below). Denote $s_i$...
- 61
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
3
votes
0
answers
424
views
Tropicalization of the Grassmannian
Let $Trop(Gr(m,n))$ denote the tropicalization of the grassmannian $Gr(n,m)$. Let $\phi^m : \mathbb R^{n \choose 2} \rightarrow \mathbb R^{n \choose m}$ such that $X_{i,j} \rightarrow X_{i_1,...,i_m}$...
- 1,277
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
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
1
vote
0
answers
56
views
Combinatorical surface is restricted to a closed face an injection
Hello :)
I'm third year student of mathematics. In my own intrest i'm studying topology in combinatorical sense. Herefore i found also an lecture note in knot theory from Roberts. I want to understand ...
- 11
11
votes
2
answers
910
views
Access to a preprint by D. N. Verma
Some work I am doing is connected with a sequence 1, 3, 40, 1225, 67956, $\dots$ which agrees with http://oeis.org/A012250 for all eight terms. The only useful information in OEIS on this sequence is ...
- 50.8k
4
votes
1
answer
272
views
Family of hypersurfaces in (C^*)^2 corresponding to tropical family
Edit: I realize the mathematics below is lacking a precise phrasing. I hope that the intuitiion behind the question is clear enough that a reader will understand the question and provide guidance. The ...
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/...
11
votes
2
answers
1k
views
Tensor rank of anti-symmetric tensor
Let $V$ be a vector space of dimension $n$. Let us consider $V^{\otimes n}=V\otimes V \ldots \otimes V$. This vector space contains one dimentional vector space $\wedge^n V$. My question is does it ...
- 2,179
2
votes
2
answers
533
views
elements in the weyl group
Let W be the Weyl a group of a semisimple simply connected group over C.
Let I={1,...,r} the set of simple roots.
For $w\in W$, I denote by supp(w) the subset of I corresponding to the simple ...
- 3,472
8
votes
0
answers
636
views
Chern Classes of Exterior Products of a vector bundle.
This is mostly a question in combinatorics. Is there a clean way in terms of determinantal identities to write down $c(\wedge^k V)$ i.e. the individual summands in terms of the individual summands of $...
- 327
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
15
votes
1
answer
690
views
Smooth bases of matroids
Motivated by algebraic geometry, I've come up with a purely
combinatorial definition within the theory of matroids.
The question is: is this concept known?
If you like matroids but not algebraic ...
- 27.8k
20
votes
5
answers
1k
views
From convex polytopes to toric varieties: the constructions of Davis and Januszkiewicz
One of the most useful tools in the study of convex polytopes is to move from polytopes (through their fans) to toric varieties and see how properties of the associated toric variety reflects back on ...
- 24.7k
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 ...
0
votes
1
answer
367
views
On the number of lines of given points
Hi all, I have a question Concerning Beck's theorem. I have read it from http://en.wikipedia.org/wiki/Beck%27s_theorem and I have two questions :
I suppose Beck's theorem doesn't hold when instead ...
- 1
12
votes
2
answers
664
views
Detecting tilings by toric geometry
This is probably a silly question, but I figured that if there is a good answer, this would be a good place to ask.
Ever since I got my hands on the book "Toric Varieties" by Cox, Little and Schenck, ...
- 85.6k