All Questions
Tagged with co.combinatorics ag.algebraic-geometry
115 questions with no upvoted or accepted answers
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
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
12
votes
0
answers
731
views
Product of a Schubert polynomial and a double Schubert polynomial
Let $S_u(x)$ be a Schubert polynomial and let $S_v(x;y)$ be a double Schubert polynomial. Then their product can be expressed in terms of the double Schubert polynomials as
$$S_u(x)S_v(x;y)=\sum_w{c_{...
- 2,168
12
votes
0
answers
529
views
A commutative monoid associated with a finite abelian group
Let $M$ be a finite abelian group, and denote by $e_m$, for $m \in M$, the canonical basis of $\mathbb{Z}^M$. For $m, n \in M$ define elements $v_{m,n} \in \mathbb{Z}^M/\langle e_0\rangle$ as
$$
v_{m,...
- 181
11
votes
0
answers
242
views
Coefficients of universal Schubert polynomials
Let $e_i^j$ be the elementary symmetric polynomial in $x_1,x_2,\ldots,x_j$. Then the ordinary Schubert polynomial has an expansion of the form
$$S_u(x)=\sum_{i_1,i_2,\ldots,i_n}{a^{i_1,i_2,\ldots,i_n}...
- 2,168
11
votes
0
answers
629
views
Inversion, Koszul duality, combinatorics and geometry
According to this MO answer Koszul duality is related to operations on generating series;
1) multiplicative inversion for quadratic algebras,
2) compositional inversion for quadratic operads,
3) ...
10
votes
0
answers
436
views
Commuting matrix variety $[A,B]=0$ - can one geometrically explain divisibility of $F_ q$ point count by high powers of $q$?
$\DeclareMathOperator\Comm{Comm}\DeclareMathOperator\Id{Id}$Consider the variety $\Comm$ of commuting matrices $[A,B]=0$ over some field $K$. It is much studied, and interesting for various reasons.
...
- 24.9k
9
votes
0
answers
361
views
Bernoulli-like polynomials
Let $\psi_0 (x,t)=\frac{te^{xt}}{1-e^{-t}}$. Then
$$\psi_0(0,t)=\frac{t}{1-e^{-t}};$$
$$\psi_0(x,t)=1+\sum_{n=1}^\infty \frac{t^n}{n!} B_n(x)$$
where $B_n$ is a monic polynomial of degree $n.$
Now ...
- 401
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
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
8
votes
0
answers
313
views
Cohomology of the complement of the resonance hyperplane arrangement
Here was a question about resonance arrangement. It is defined as follows.
Let $x_i$ be the standard coordinates on $\mathbb{C}^n$. For each nonempty $I\subseteq\{1,\dots,n\}$, define the hyperplane $...
- 743
8
votes
0
answers
202
views
Does anyone know of this manifestation of the Littlewood-Richardson coefficients for the complete flag variety?
This is the culmination of about 11 years of research but after I discovered it I found a proof that was extremely trivial, so I'm wondering if it's already known.
Let $(a,b)$ with $a < b$ ...
- 2,168
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
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
7
votes
0
answers
179
views
Combinatorics of enumerating theta characteristics - a meaningless coincidence?
A Riemann surface $X$ of genus $g$ has $2^{g-1}(2^g +1)$ theta characteristics (henceforth conflated with spin structures) of even parity, and $2^{g-1}(2^g-1)$ odd theta characteristics. These numbers ...
- 2,830
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
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
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
4
votes
0
answers
259
views
Road map for learning cluster algebras
I'm a PhD student and I would like learn about cluster algebras. I'm wondering what is a good reference (i.e., has detailed explanations, examples, etc) to learn from the basic and what are some of ...
- 839
4
votes
0
answers
258
views
A technical question about a paper by Gross-Hacking-Keel
I have a technical question on the commutativity of diagrams (2.11) and (2.12) in the paper "Birational geometry of cluster algebras" by Gross-Hacking-Keel:
For the leftmost square in (2.11),...
- 61
4
votes
0
answers
152
views
How to show the set of stable polynomials equals to the set of Lorentzian polynomials in degree 2
Give a homogenous polynomial $f\in \mathbb{R}[x_1,\dots,x_n]$ of degree $2$ in $n$ variables, we can consider $f$ as a quadratic form.
We call $L_n^2:=$ the set of quadratic forms with nonnegative ...
- 41
4
votes
0
answers
87
views
Toric Bézier patches
Toric Bézier patches (as described in https://arxiv.org/abs/0706.2116) are maps from a lattice polytope $P$ to the positive part of its associated toric variety $X_P$. While they are not the inverse ...
- 1,435
4
votes
0
answers
144
views
Generalized Catalan generating series
Let
$$
\mathscr{B}_k(z) = \sum_{n\geq 0}{kn+1\choose n}\frac{1}{kn+1}z^n\,,
$$
then it is well known that
$$
\tag{1}\label{1}
\text{log}\mathscr{B}_k(z)= \sum_{n\geq 1}{kn\choose n}\frac{1}{kn}z^n\,.
$...
- 988
4
votes
0
answers
206
views
A conjectural inequality of the constant terms of functions
Could someone help me with the following question? This is equivalent to my previous question
A conjecture about the barycenter of a polytope
Let $D$ be a differential operator defined as follows,
\...
- 1,121
4
votes
0
answers
130
views
program to compute hurwitz numbers
Is there a computer program available to compute Hurwitz numbers easily? In fact I only care about counting covers $C\to\mathbb{P}^1$ branched over $0,1,\infty$, and am even willing to restrict to the ...
- 735
4
votes
0
answers
127
views
Measuring the failure of basepoint independence of the rotor-routing model for non-planar ribbon graphs
In this question from 2012, Jordan Ellenberg asks if the set of spanning trees of a graph $G$ is naturally a torsor for the critical group (also called the sandpile group or the picard group $Pic^0(G)$...
4
votes
0
answers
102
views
Bounds on k-tuple points for intersections of hyperplanes
Suppose that $H_1$,...,$H_d$ are hyperplanes in $\mathbb P^n$ (over some field -- you can pick). For $k \geq n$, let $t_k$ denote the number of points through which there pass exactly $k$ hyperplanes....
- 41
4
votes
0
answers
95
views
Topological hyperfields
I am trying to generalize the notion of reorientation class of an oriented matroid to the context of matroids over hyperfields (compare Baker and Bowler, 2016). I have already got some results in this ...
- 741
4
votes
0
answers
384
views
Parity in degrees of determinantal varieties
Let $M_{m,n}(\Bbb{C})$ be the space of $m\times n$ matrices with entries in $\Bbb{C}$, and let $U_{k,m.n}(\Bbb{C})\subset M_{m,n}(\Bbb{C})$ be the variety of matrices of rank $\leq k\leq\min(m,n)$. ...
- 43.2k
4
votes
0
answers
76
views
Comparing parametrizations of unipotent radical
Let ${G}$ be a simple algebraic group over $\mathbb{C}$ with maximal torus $T$ and set of simple roots $\{\alpha_i\}_{i\in \Delta}$. We then have a Borel supgroup $B=TU$ with unipotent radical $U$. ...
- 2,516
4
votes
0
answers
341
views
Confusion with proof of Pieri's Formula
In Manivel's Symmetric Functions, Schubert Polynomials, and Degeneracy Loci, I am confused with some parts of his proof of Pieri's Formula. It is given as Pieri's Formula $3.2.8$ (p. $109$):
If $\...
- 349
4
votes
0
answers
189
views
Fibers of torus equivariant moment maps
Given a closed (possibly singular) projective variety $V$ with a symplectic structure and a torus action, there is a moment map
$\mu: V \rightarrow Lie(T)^*$. Note that the dimension of $T$ could be ...
- 1,719
4
votes
0
answers
371
views
How is $ \sum_{x \in X(\mathbb{F}_q)} \dots $ a generalization of cardinality?
This quarter Maxim Kontsevich is offering a course on exponential integral. There is not much in the way of notes, but is one page with mysterious comments.
Let $X$ be an algebraic variety over $\...
- 22.8k
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
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
3
votes
0
answers
202
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
3
votes
0
answers
93
views
Minimal set of geometric moves in various equivalence classes of triangulated geometries
I would like to get to know what is the minimal set of geometric changes "aka. moves" (topology preserving modifications / Pachner moves / bistellar moves) that can transform any 3-...
- 183