All Questions
31 questions
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
8
answers
3k
views
Are there any algebraic geometry theorems that were proved using combinatorics?
I'm collaborating with some algebraic geometers in a paper, and when writing the introduction I mentioned the interaction of combinatorics and algebraic geometry, and gave some examples like the ...
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
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
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
12
votes
2
answers
556
views
Sylvester–Gallai theorem with circle version, plane version and curve version?
The Sylvester–Gallai theorem in geometry states that, given a finite number of points in the Euclidean plane, either
All the points are collinear; or
There is a line which contains exactly two of the ...
- 477
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
11
votes
2
answers
977
views
Reference for combinatorics with view towards representation theory/algebraic geometry
I'm making this post to ask for a reference about combinatorics: I'm a PhD student in representation theory/algebraic geometry. My background is mostly in algebra and geometry (and also mostly ...
- 1,061
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
2
answers
353
views
Rigid line arrangements
What is already known about rigid line arrangements? By line arrangement, I mean a unions of lines in $\mathbb{P}^2_{\mathbb{C}}$ with fixed incidences. (Written in notation, I mean a collection of ...
- 1,537
10
votes
1
answer
299
views
Map from Bruhat stratification to Catalan stratification for the space of totally nonnegative upper-triangular matrices
$\DeclareMathOperator\SL{SL}$This question came up in a class "Total Positivity and Cluster Algebras" being taught by Chris Fraser.
Let $N^+$ denote the space of uni-upper-triangular ...
- 24.2k
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
7
votes
1
answer
440
views
Road map and references for combinatorial Hodge theory
I'm a PhD student. I'm familiar with graduate level algebraic geometry and toric varieties.
I wanted to know a road map for getting into combinatorial Hodge theory and other prerequisites that I'll ...
- 839
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
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
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
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
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
3
votes
1
answer
237
views
Singularity of torus fixed points from combinatorial data
May I ask what are the relations between the geometry and combinatorics near a torus fixed point? Any references?
In particular, let $S$ be a scheme that is torus invariant with finitely many zero and ...
- 1,719
3
votes
1
answer
221
views
Alternating multisymmetric functions
I am looking for a reference on certain modules of invariants. I think that the question is quite natural so that I believe there should be some results already, but I am not able to find anything.
...
- 577
3
votes
1
answer
451
views
Commutative algebra for the Conway games
I was reading the book On Number And Games and I have some question. In this book Conway constructed the set of "games" with a addition and a multiplication. I understand that the surreal numbers are ...
- 527
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
179
views
Polytope algebra and toric vareties
Let $\Pi$ denote the McMullen polytope algebra (over the field of rationals $\mathbb{Q}$) generated by convex polytopes with rational vertices in $\mathbb{R}^n$.
For a simple polytope $P$ let us ...
- 21.8k
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
239
views
T-equivariant homology of affine Grassmannian
Let $G=SL_n$, denote the affine Grassmannian $Gr:=Gr_{G}=\mathcal{G}/\mathcal{P}$, where $\mathcal{G}=G(\mathbb{C}((z)))$ and $\mathcal{P}=G(\mathbb{C}[[z]])$. We know that $R:=H_*^T(Gr)\cong H_*^T(\...
- 849
2
votes
1
answer
255
views
Polya-MacMahon-Burnside's generating function at "-1"
$\mathbb{Z}_n$, as a cyclic subgroup of symmetric group $\mathfrak{S}_n$, acts on $[n] :=\{1, 2,\dots,n\}$. Hence $\Bbb{Z}_n$ permutes the elements of the Boolean algebra $2^{[n]}$ of all subsets of $[...
- 43.2k
2
votes
1
answer
577
views
The concept "opposite" of Cohen-Macaulayness
Let $S = \mathbb k[x_1, \ldots, x_n]$ be a polynomial ring over a field $\mathbb k$ and let $I \subseteq S$ be a monomial ideal.
For a monomial ideal $J$, let $\#(J)$ be the smallest number of ...
- 459
2
votes
0
answers
150
views
Projection of conormal bundle of Schubert variety under Springer resolution
Let $G=\mathrm{GL}_n(\mathbb{C})$ and $X_{\omega}=\overline{B_-wB/B}\subset G/B$ be a Schubert variety. Denote by $C(X_\omega)$ the conormal variety inside $T^*(G/B)$ ,
$\mu:T^*(G/B)\to \mathcal{N}$ ...
- 849
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
0
votes
0
answers
57
views
Reference for packing property and König property
Can someone please suggest reference material to study about the packing property and König property of ideals and some examples?
- 21