All Questions
Tagged with co.combinatorics ag.algebraic-geometry
292 questions
-1
votes
1
answer
304
views
Incidences of Lines / Circles in the Plane
During linear algebra class, I was explaining that given 2 equations + 2 unknowns where we expect there to be a unique solution but sometimes there can be 0 solutions or a line's worth.
At some ...
- 22.8k
3
votes
1
answer
1k
views
What is the Bruhat decomposition of the affine Grassmannian?
We define the affine Grassmannian to be the quotient $Gr = GL_n(\mathbb{C}((t)))/GL_n(\mathbb{C}[[t]])$ where $\mathbb{C}((t))$ is the field of formal Laurent series and $\mathbb{C}[[t]]$ is the ring ...
- 33
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
2
votes
0
answers
214
views
Non-regular (Non-coherent) subdivisions of a polygon.
There are many papers and books which study about the regular subdivision of a convex lattice polytope.
My question is about "Non"-regular subdivisions of a 2-dimensional convex lattice polygon.
I ...
- 21
6
votes
1
answer
599
views
Embedding $G(2,n)$ into $G(k,n)$
Let
$$M=\begin{pmatrix}
u_1 & u_2 & \ldots & u_n \\
v_1 & v_2 & \ldots & v_n \\
\end{pmatrix}$$
be a $2 \times n$ matrix. Define $\nu(M)$ to be the $k \times n$ matrix
$$\nu(M)...
- 156k
23
votes
4
answers
4k
views
What information is contained in the Kazhdan-Lusztig polynomials?
The Kazhdan-Lusztig polynomials contain all kinds of representation theoretic (and other kinds of) informations.
For example the character of a simple module over a Lie algebra with Weyl group $W$ ...
8
votes
2
answers
1k
views
What is known about zero-sets of Schur polynomials?
Consider a set S of partitions not containing the empty partition (I would be happy with, say, all the partitions of size less than k, except for the empty one).
Let $U_\lambda^{(r)}$ be the zero-...
- 81
4
votes
0
answers
168
views
A fast way to test whether a partial function can be extended to a chirotope of rank 3 ?
Hello,
What is a fast way to test whether a partial function can be extended to a chirotope of rank 3 ? That is: I have a domain
$E = \{1,...,n\}$
and a partial function
$f: E^3 \to \{-1, 0, 1\}$
...
- 191
12
votes
1
answer
949
views
Discrete version of Nullstellensatz?
Hi. I was reading the paper "On the foundations of combinatorial theory (VI): The idea of a generating function" by Doubilet, Rota and Stanley, and there is a relation treated which is very ...
- 902
2
votes
1
answer
199
views
Minimal number of nodes in a complex line arrangement.
Let $\mathcal{A}$ be a collection of $n$ lines. Assume that $\mathcal{A}$ is not a pencil. It is known (see http://www.springerlink.com/content/320p742475v6q746/) that if all lines are in $\mathbb{RP}...
- 2,444
12
votes
3
answers
2k
views
Mnev's universality corollaries, quantitative versions?
Mnev's universality theorem claims that any semialgebraic set is the realization space of some oriented matroid. Moreover, the rank of the or matroid can be prescribed in advance.
1.-Are there ...
- 1,303
1
vote
0
answers
513
views
Maximal disjoint hyperplanes
Assume a set of $n^{r}$ points $X_{r} = \{ x_{1}, \cdots, x_{n^{r}} \}$ is given occupying a codimension $t^{r}$ subspace in $\mathbb{R}^{n^{r}}$. Let $M_{r}$ be the set of $t^{r}$-tuples of these ...
- 800
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
9
votes
1
answer
1k
views
Finding the vertices of a polyhedral complex coming from a GIT wall and chamber decomposition
I am interested in a polyhedral/combinatorics problem that arises in algebraic geometry in the context of geometric invariant theory (GIT).
Algebro-geometric background: Consider the natural ...
- 1,871
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
13
votes
3
answers
1k
views
Reference for combinatorics of cell decomposition of the Hilbert scheme of points in the plane
It is known from either Morse theory or Bialynicki-Birula decomposition that the fixed points of a ${\mathbb{C}}^*$ action on a smooth algebraic variety over $\mathbb{C}$ determine a cell ...
- 2,964
3
votes
1
answer
542
views
A good vector field to calculate the Euler's number of a compact differentiable manifold
Given a triangulation of a compact, orientable and differentiable manifold $M^m$, it is possible to give a formula (in terms of real coordinates of the ambient space) for a vector field with only non-...
- 31
11
votes
2
answers
1k
views
Geometric motivation for the Stanley-Reisner correspondence
The Stanley-Reisner ring of an abstract simplicial complex $\Delta$ on the vertex set $\{1,...,n\}$ is the $k$-algebra
$$
k[X_1,...,X_n]/I_\Delta
$$
where $I_\Delta$ is the ideal generated by the $X_{...
- 2,782
1
vote
1
answer
1k
views
How to Tropicalize a Polynomial in Two Variables?
Trying to draw the Amoeba
With Mathematica, it's possible to graph $e^{-k x} + e^{-k y} = 1,e^{-k x} - e^{-k y} = 1$ and $e^{-k x} + e^{-k y} = -1$ to get the amoeba of 1 + x + y when k = 1. Then by ...
- 22.8k
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
5
votes
0
answers
268
views
Unicity of branched covering of sphere, and Hurwitz numbers
Hurwitz's encoding counts the number of branched self-coverings of a sphere, with prescribed ramification degrees at the critical points, as numbers of factorizations of the identity in a symmetric ...
- 2,519
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
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 ...
1
vote
2
answers
307
views
Subset higher power sum question (related to quadratic forms)
Let $\mathbb N_{n} = \{1,2,\cdots,n\}$.
Let $S$ be of cardinality $n$ where elements of $S$ are integers from $\mathbb N_{n}$ and at least one element of $S$ is repeated (That is at least one integer ...
- 13.9k
4
votes
2
answers
442
views
A mapping from a lattice to itself
Consider $\mathbb{Z}^{n}$ for $n = 2^r$ where $r \geq 1$ . Look at the iterates of the following function $T$ from $\mathbb{Z}^n$ to itself.
$T((a_1, a_2, \ldots, a_n)) = (|a_1 - a_n|, |a_2 - a_1|, |...
- 73
0
votes
1
answer
225
views
Is the Euler characteristic of a certain nonlinear variety related to that of a certain linear variety?
(This is a generalization of a question I posted a week ago.)
I'm looking at a variety sitting inside the algebraic torus $(\mathbb{C}\setminus 0)^n$ generated by the ideal $I = (*x_1^{\alpha_1} + \...
8
votes
1
answer
841
views
Virasoro constraints for the generating function of Hurwitz numbers.
Generating function of the simple Hurwitz numbers is known to be connected with Gromov-Witten potential of the point (Kontsevich $\tau$-function) (see e.g. Ian Goulden, David Jackson and Ravi Vakil). ...
- 1,343
12
votes
1
answer
566
views
Counting branched covers of the projective line and Spec Z
I've asked a question like this before, but now I'm more interested in counting the number of covers.
We suppose given the following data.
A positive integer $d$
A finite set of closed points $B= (...
- 9,497
7
votes
2
answers
3k
views
On the cohomology ring of the Grassmannian
The basis of Schubert classes for the cohomology ring $H^*(\text{Gr}(m,N))$ of the Grassmannian of $m$-dimensional subspaces of $\mathbb{C}^N$ is indexed by $L(m,N-m)$, the poset of all partitions ...
- 952
12
votes
0
answers
529
views
A commutative monoid associated with a finite abelian group
Let $M$ be a finite abelian group, and denote by $e_m$, for $m \in M$, the canonical basis of $\mathbb{Z}^M$. For $m, n \in M$ define elements $v_{m,n} \in \mathbb{Z}^M/\langle e_0\rangle$ as
$$
v_{m,...
- 181
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
12
votes
1
answer
939
views
Which Steiner systems come from algebraic geometry?
This question is motivated by the ongoing discussion under my answer to this question. I wrote the following there:
A $(p, q, r)$ Steiner system is a collection of $q$-element subsets $A$ (called ...
- 23k
7
votes
1
answer
384
views
Why do non-equioriented A<sub>n</sub> quivers have singularities identical to the singularities of Schubert varieties?
In the general case, quiver cycles are of the form of orbit closures of $GL\cdot V_{\vec{r}}$, where $GL= \prod_{i=0}^n GL_{r_i}$ is the possible changes of basis on all of the vector spaces on each ...
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
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
8
votes
2
answers
692
views
Enumeration of graphs arising in invariant theory
I've been working on a talk based on some stuff in Olver's "Classical Invariant Theory" book and have been wondering about a related graph enumeration problem.
Start with a triple $(n,v,e)$ of ...
- 16k
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
16
votes
5
answers
2k
views
Elliptic Curves over F_1?
Is there an notion of elliptic curve over the field with one element? As I learned from a previous question, there are several different versions of what the field with one element and what schemes ...
- 27.5k
16
votes
7
answers
2k
views
Learning About Schubert Varieties
I am a combinatorist by training and I am interested in learning about the connections between combinatorics and Schubert varieties. The theory of Schubert varieties seems to be a difficult area to ...
- 163
16
votes
3
answers
2k
views
The Sylvester Gallai Theorem and Sections of Varieties with "Simple Topology".
The Sylvester-Gallai theorem asserts that for every collection of points in the plane, not all on a line, there is a line containing exactly two of the points.
One high dimensional extension ...
6
votes
3
answers
1k
views
Is there a software package that does Schubert Calculus computations?
Is there a good software package for doing computations in the cohomology ring of Grassmannians? Things like, I can write down a polynomial in, in fact, special Schubert classes, but it's one where ...
- 16k
0
votes
0
answers
2k
views
Ignore this question [closed]
This question is a hacky way to create some tags for you to use. Move along.
