All Questions
Tagged with co.combinatorics ag.algebraic-geometry
292 questions
6
votes
2
answers
519
views
Seeking for a meaning: a curious symmetry
Suppose $\Phi(m,n)=(2m)!^n\prod_{k=1}^n\binom{2m+2k+x}{2k+x}$.
Then, algebraically, it is trivial to see that
$$(2m)!^n\prod_{k=1}^n\binom{2m+2k+x}{2k+x}=(2n)!^m\prod_{k=1}^m\binom{2n+2k+x}{2k+x}.$$
...
- 43.2k
6
votes
1
answer
678
views
Combinatorics of the Cohomology Ring of the Lagrangian Grassmannians
The Lagrangian Grassmannian is an important example in symplectic geometry, see here or here for details. It shares many similarities with the ordinary Grassmannians (as one would expect from the name)...
- 449
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
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
0
answers
314
views
Number of square-free polynomials over a finite field - a combinatorial interpretation?
Cross-posted from MSE. The question has remained unanswered for six years but I still like it!
One can show using zeta functions that the number of (monic)square-free polynomials of degree $n$ over a ...
- 7,746
6
votes
0
answers
192
views
Polynomial count varieties and affine paving (e.g., determinantal varieties)
Let $X$ be a variety defined over $\mathbb{Z}$, $X_{\mathbb{F}}$ be the base change $X\times_{\mathrm{Spec}(\mathbb{Z})} \mathrm{Spec}(\mathbb{F})$. We say $X$ is of polynomial count if there is a ...
- 653
6
votes
0
answers
217
views
Nonclassical polynomials, circles, and groups
Tao and Ziegler have introduced a generalization of polynomials over a prime field called nonclassical polynomials, useful for studying the Gowers norm.
A nonclassical polynomial of degree $d$ is a ...
- 539
6
votes
0
answers
161
views
LS paths construction
Let $W$ be the Weyl group of a simple Lie algebra $\mathfrak L$, and for a dominant weight $\lambda$ denote by $W_{\lambda}$ the stabilizer of $\lambda$ in $W$. Let $\leq$ be the Bruhat order on $W/W_{...
- 61
5
votes
1
answer
357
views
Asymptotics of degree of $\textrm{SO}_n$?
(This is an offshoot of Degree of parametrization of $\textrm{SO}_n$?)
Consider $G=\textrm{SO}_n$ as an affine subvariety of the affine space of $N$-by-$N$ matrices. There is an explicit expression ...
- 20.2k
5
votes
1
answer
733
views
To derive or not to derive, that is the question
What are concrete and abstract examples of problems (even whole programmes of inquiry) when one has a choice to use a "derived" theory (e.g., $\infty$-categories, DAG, HAG, $DRep_k(G)$, "higher" ...
- 639
5
votes
2
answers
639
views
Matroids relaxations of a given matroid
Let $\mathcal{M}$ be a rank-$d$ matroid on $[n]$. Say a matroid $\mathcal{N}$ is a relaxation of $\mathcal{M}$ if $\mathrm{rank}(\mathcal{N})=d$, $\mathrm{groundset}(\mathcal{N})=[n]$, and every ...
- 902
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
5
votes
1
answer
163
views
Polynomials vanishing on prescribed layers
Given a prime $p$ and an integer $n\ge p$, what is the smallest possible degree of a polynomial $Q\in\mathbb F_p[x_1,\dotsc, x_n]$ such that $Q$ vanishes on every vector $x\in\{0,1\}^n$ of weight $w(x)...
- 23k
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
1
answer
145
views
Reference for restriction formula in terms of double Schubert polynomials
Everyone (that is, everyone who cares) knows that double Schubert polynomials represent Schubert classes in equivariant cohomology in type $A$. We also know that we can restrict Schubert classes to ...
- 2,168
5
votes
1
answer
247
views
Generating function for lattice paths making aribitrary (i,j)-up-right move in one step and fitting rectangular (m,n)?
There is the following beautiful formula (see Qiaochu Yuan excellent blog):
$$ \sum_{\lambda \in Young~diagrams~fitting~rectangle~m~n} q^{Box~count(="area~under~the~curve")~of~\lambda} = \binom{n+m}{...
- 24.9k
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,...
5
votes
1
answer
368
views
Six people standing on earth
Consider 6 people $p_i$, $i=1,\dots 6$, standing on a sphere $S^2$. We label the positions of these people by $p_i$ again. Suppose no pair of these points $p_i$ are antipodal. At each point $p_i$ ...
- 434
5
votes
1
answer
253
views
Permutations of points in the projective plane
Let $p_1,...,p_7\in\mathbb{P}^{2}$ be seven general points in the projective plane $\mathbb{P}^{2}$ over the complex numbers.
Let $f$ be an automorphism of $\mathbb{P}^{2}$ inducing a permutation of $...
- 51
5
votes
1
answer
208
views
Zariski openness of Newton non-degenerate polynomials
Suppose you are given a convex polyhedron $\Delta$ in $\mathbb{R}^n$ (i.e. a convex hull of finitely many points in $\mathbb{Z}^n$) and consider a finite dimensional vector space $V$ over $\mathbb{C}$ ...
- 53
5
votes
0
answers
83
views
Nonnegativity of the coefficients of the commuting difference operators of Fomin, Gelfand, and Postnikov evaluated on quantum Schubert polynomials
This post is about quantum Schubert polynomials. Fomin, Gelfand, and Postnikov defined operators in the nil-Hecke ring with coefficients in $\mathbb{Z}[x,q]$ denoted by $\chi_k$ for $1\leq k\leq n$ ...
- 2,168
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
5
votes
0
answers
270
views
Connected relative Gromov Witten invariants
I am currently interested to compute relative Gromov Witten invariants(GW) over $\mathbb{P}^1$.
In the paper
https://arxiv.org/pdf/math/0204305.pdf
eq 3.1 gives the count of relative disconnected GW ...
- 685
5
votes
1
answer
277
views
Set-theoretic generation by circuit polynomials
Let $P$ be a prime ideal in $S=\mathbb{C}[x_1,\ldots , x_n],$ and write $[n] = \{ 1, \ldots , n \}.$ The algebraic matroid of $P$ can be defined according to circuit axioms as follows: $C\subset [n]$ ...
- 396
5
votes
0
answers
99
views
Kac-Moody groups for non-crystallographic root systems
Given a finite-dimensional crystallographic root system, we can construct an associated Kac-Moody group, with a corresponding flag variety and Littlewood-Richardson coefficients. Between a pair of ...
- 2,168
5
votes
0
answers
230
views
Minimal algebraic degree of symmetric unit distance embedding of Heawood graph
I'm looking at embeddings of the Heawood graph in the plane as unit distance graph. Apparently the first such embedding was given by Gerbracht, 2009 and has algebraic (over the rationals) coordinates ...
- 10.7k
5
votes
0
answers
222
views
Littlewood-Richardson rule for the complete flag variety: GapP complete?
The cohomology ring of a complete flag variety $X$ has a basis of Schubert classes $S_u$ for permutations $u$. Define the Littlewood-Richardson coefficient $c_{uv}^w$ for permutations $u,v,w$ to be ...
- 2,168
5
votes
0
answers
171
views
Intersections of the B-orbits and the orbits of some other Borel subgroups in the flag variety G/B
This is a follow-up of this previous question below:
Intersections of $B$ and $B^-$ orbits in the flag variety $G/B$
Let $G = SL_n(\mathbb{C})$, $B$ be the standard Borel subgroup, and consider some ...
- 1,719
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
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
4
votes
1
answer
783
views
Three dimensional representations of Alternating group
The alternating group $A_5$ has $2$ irreducible representation of degree $3$. The characters for these representations have irrational values. I guess the ring of invariants of these representations ...
- 125
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
2
answers
144
views
Covering all except one of the purple intersection points of $n$ red and $m$ blue lines efficiently
Consider a set of $n$ red lines and $m$ blue lines, suppose there are $nm$ distinct red-blue intersections.
What is the minimum number of lines $L_1,L_2,\dots, L_n$ such that the union contains all $...
- 1,835
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
4
votes
1
answer
189
views
2-faces of reflexive Delzant polytopes
Question 1. Can a reflexive Delzant polytope of some dimension contain a $2$-face with more than $11$ edges?
Motivation. I would like more generally to get an answer to the following question:
...
- 14.3k
4
votes
1
answer
210
views
A map on Grassmannian
Let $G=SL_{2n}$ and let $\sigma:G \to G$ be defined by $\sigma (A)= E(A^t)^{-1}E^{-1}$, where $E=antidiag(1,1, ... ,1,-1,-1,...,-1)$. Then the maximal parabolic associated to the simple root $\...
- 121
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
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
4
votes
1
answer
233
views
What's known about the matroid induced by the Plücker coordinates of the representation of a matroid?
Let $M$ be a linear matroid with ground set $E$ and independent subsets $\mathcal I$, represented by $\rho: E \rightarrow V$.
This induces a map
$$
\hat\rho: \mathcal I \rightarrow \mathbf P(\Lambda V)...
- 213
4
votes
1
answer
297
views
Big cells in a Grassmann and permutations
In the lecture notes, it is said that (Theorem 3.1.3) the set of positroid cells in $Gr(k,n)$ are in one to one correspondence with the set of bounded affine permutations of type $(k,n)$. In Example 4....
- 6,201
4
votes
1
answer
491
views
What is the permutation group generated by those three given morphisms of the affine space $\mathbb{F}_q^3$?
Let $\mathbb{A}^3 = \mathbb{F}_q^3$. Consider the following three functions $\mathbb{A}^3\to\mathbb{A}^3$:
\begin{eqnarray*}
h: (x, y, z) &\mapsto& (x, y, xy - z) \\
u: (x, y, z) &\mapsto&...
user4324
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
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
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
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
4
votes
1
answer
196
views
Polynomial time decodable binary linear codes achieving $GV$ bound?
Are there explicit or random construction of linear codes that achieve the $GV$ bound with polynomial time decodable property with alphabet size $q=2$?
Tsfasman, Manin, Vladut beat the bound at ...
- 13.9k
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
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 ...
4
votes
0
answers
81
views
Classification of nilpotent orbits over local fields (for type ABCD via partitions )
Let $\mathfrak g$ be a simple Lie algebra over a char $0$ local field $F$ (e.g. $F=\mathbb R$ or $F=\mathbb Q_p$) with its adjoint group $G$. Let $\mathcal N \subseteq \mathfrak g$ be its nilpotent ...
- 6,622
4
votes
0
answers
255
views
Economic equilibrium and tropical geometry
There is a famous saying in economics: When everyone pursues his or her own interests, there is an invisible hand that brings the market to equilibrium. However, this is not always the case. Here is ...
- 123