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2 votes
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211 views

Combinatorial and computational problem related to Weyl groups and the coroot lattice

Let $W$ be a Weyl group with root system $R$ and with set of positive roots $R^+$. Let $\tilde{R}^+$ be the set of $B$-cosmall roots, i.e. positive roots $\alpha$ which satisfy $\ell(s_\alpha)=2\...
Christoph Mark's user avatar
2 votes
0 answers
121 views

Hamming weights of special vectors

The motivation of this question comes from number theory (I add the tag number theory for this reason, in that it is possible that someone with a number-theoretic background has already thought about ...
Ted90's user avatar
  • 21
2 votes
0 answers
89 views

Group actions on polytopes in indefinite integer lattices

Is anything at all known about polytopes in indefinite integer lattices? I'm interested in lattice automorphisms which preserve certain polytopes of "high regularity" (e.g. cones). As a first step, I'...
MRD1729's user avatar
  • 393
1 vote
1 answer
144 views

On parametrization of a type of unimodular $2\times2$ integral matrix

A matrix $\begin{bmatrix}w&x\\y&z\end{bmatrix}\in\mathbb Z^{2\times 2}$ is unimodular if $$|wz-xy|=1$$ holds. Is there a parametrization of such matrices with $|w||z|-xy=1$ $$w,z<0<\max(...
Turbo's user avatar
  • 13.9k
1 vote
3 answers
470 views

Triangulations of lattice polygons

Let L be a 2-dimensional lattice and P- a lattice polygon. Suppose, it is triangulated into lattice tiangles. What are restrictions on their areas? For instance, can a lattice triangle of even area ...
Daniil Rudenko's user avatar
1 vote
1 answer
111 views

Existence of some lattice path connecting all given lattice paths

My daily work concerns analysis on metric spaces and some time ago it turned out that the problem I am dealing with boils down to a certain combinatorial problem. I've checked a lot of examples and it ...
elsnar's user avatar
  • 137
1 vote
1 answer
152 views

Submodular measures on the hypercube

By the hypercube I mean the lattice formed by all n-bit strings ordered by pointwise inequality. For example, $000 \leq 110$, $010 \leq 110$, $110$ and $001$ are not comparable. Further we have the ...
Erik Aas's user avatar
  • 406
1 vote
1 answer
172 views

Minimal volume of fundamental domains of lattices

Consider a full rank integer lattice in $\mathbb{R}^n$. Let $v_1$ be the shortest non-zero vector in the lattice, $v_2$ be the shortest one among those not parallel to $v_1$, $v_3$ be the shortest one ...
Yuhang Liu's user avatar
1 vote
0 answers
51 views

Coarse-graining a hypergraph

$\DeclareMathOperator{\poly}{\mathrm{poly}}$I have asked this question on math.SE here, but couldn't get a satisfactory answer. I have also asked a related question on math overflow here, but haven't ...
Pranay Gorantla's user avatar
1 vote
0 answers
68 views

Finding a particular kind of basis of subgroup of a lattice generated by non-negative part

For $\mathbf v=(v_1,\ldots,v_n)\in \mathbb Z^n$, let $\operatorname{supp}(\mathbf v):=\{j: v_j \ne 0\}$. For a subset $X$ of $\mathbb Z^n$, define $\operatorname{supp}(X):=\bigcup_{\mathbf v \in X} \...
uno's user avatar
  • 412
1 vote
0 answers
124 views

Number of lattice points in a structural symmetric convex body

Let $f$ is a convex symmetric function on the interval $[-a,a]$, i.e., $f(-x)=f(x)$ for $\forall \, x\in [-a,a]$. Then we consider a $n$-dimensional convex body in Euclidean space \begin{equation} \...
RyanChan's user avatar
  • 550
1 vote
1 answer
371 views

Basis of cone lattice

I only want to know whether a construction that I use appears in literature and maybe has a name already. Let $V$ be a $\mathbb Q$ vector space of dimension $d\in\mathbb N$. A subset $C\subset V$ is ...
user avatar
1 vote
0 answers
229 views

Geometric interpretation of k-th power of first n natural numbers and summation using Pick's theorem

I want to know is there any interesting properties of this approach or generalization to find $S_k(n)=1^k+2^k+3^k+\cdots+n^k$ by using Pick's Theorem $S=i+\tfrac{b}{2}-1$, where $i$-number of ...
serg_1's user avatar
  • 19
0 votes
2 answers
611 views

Is a lattice of convex sets distributive?

Is a lattice of convex sets in $R^2$ distributive?
pyetras's user avatar
  • 11
0 votes
1 answer
241 views

Constructing a smooth lattice from a discrete one.

I have the standard lattice L defined over partitions of $1\ldots n$ under the split-merge relation. I also have an antimonotone function from L to R that's submodular, and so gives me a metric on L ...
Suresh Venkat's user avatar
0 votes
0 answers
110 views

Optimal covering trails in 3 and 4 dimensions

A couple of years ago, I constructively solved (inside the $AABB$ $[0,3]$ X $[0,3]$ X ... X $[0,3]$) the $k$-dimensional generalization of the infamous Nine-Dot Problem by S. Loyd (see Cyclopedia of ...
Marco Ripà's user avatar
  • 1,451
0 votes
0 answers
113 views

Question regarding contiguous forms

I read about contiguous forms in Achill Scürmann's thesis on positive quadratic forms. I am wondering about one aspect of the Voronoi algorithm presented in there, that enumerates all arithmetically ...
Kap's user avatar
  • 149

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