## When (under what conditions) is the dual of an atomistic lattice also atomistic?

According to
https://en.wikipedia.org/wiki/Geometric_lattice#Duality
“the property of being atomistic is not preserved by order-reversal”

But it simply makes that statement, without going on to enumerate the conditions under which the property is preserved. So, what are the necessary&sufficient conditions for a given atomistic lattice such that its order-dual is also atomistic? (Just a reference would be fine&dandy — I couldn’t google one myself.)

## co.combinatorics – Reference for a combinatoric problem about ‘the center of gravity of some lattice points is still a lattice point’

Fix natural numbers $$d$$ and $$k$$, here $$kge 2$$. Say $$n$$ is $$(k,d)-good$$, if for any $$n$$ given $$d$$-dimensional lattice points $$P_1,P_2, … ,P_n$$, we can always choose $$k$$ points $$P_{i_1}, … ,P_{i_k}$$ of them such that the center of gravity $$frac{P_{i_1}+ … +P_{i_k}}{k}$$of them is still a lattice point. The least such $$n$$ is denoted by $$n=n(k,d)$$.

For example, $$n(2,d)=2^d+1$$, $$n(3,1)=5$$, $$n(3,2)=9$$.

I am going to research on this question, so I want to know is there any results/references for it. I will be grateful for your help.

## graphs – How to quickly determine whether a poset is a lattice?

Recently I encountered an interesting problem while studying discrete mathematics:

Give the pseudo code to judge whether a poset $$(S,preceq)$$ is a lattice, and analyze the time complexity of the algorithm.

I am an algorithm beginner, and I am not familiar with various advanced algorithms. I have no idea about this problem at present, but I have some thoughts as follows:

• Partially ordered sets can be transformed into Hasse diagrams, the algorithm to solve this problem may be a graph-related algorithm.
• To determine whether a poset is a lattice, each pair $${a,b}$$ in the poset must be considered, so will it be a graph traversal algorithm?

Could you provide some specific ideas for this problem? Or even further, could you give the corresponding pseudo code?

## representation theory – Each Weyl group orbit in the character lattice of \$V\$ contains exactly one dominant weight

Let $$V = mathbb{C}^3 otimes mathbb{C}^3$$ be a representation of $$G = SL_3(mathbb{C})$$.
The weights of this representation is the set of $$varepsilon_i + varepsilon_j$$ for $$i, j = 1, 2, 3$$, where $$varepsilon_i$$ takes $$text{diag}(h_1, h_2, h_3) in mathfrak{h}$$ to $$h_i$$.

The Weyl group is $$W = { 1, s_1, s_2, s_1s_2, s_2s_1, s_1s_2s_1 }$$.
For a simple root $$alpha_i$$, the coroot $$h_i$$ is simply the matrix $$E_{ii} – E_{i+1, i+1}$$.
Then the pairing $$langle varepsilon_j, h_i rangle alpha_i = begin{cases} -alpha_i & text{ if } i = j, \ alpha_i & text{ if } i = j -1, \ 0 & text{ else }. end{cases}$$
By the defining equation of root reflections, $$s_i (beta) = beta – langle beta, h_i rangle alpha_i$$ for $$beta in mathfrak{h}^*$$, we have $$s_i(varepsilon_j) = begin{cases} varepsilon_j-alpha_i & text{ if } i = j, \ varepsilon_j + alpha_i & text{ if } i = j -1, \ varepsilon_j & text{ else }. end{cases}$$
Using this last part, the $$W$$-orbit of the weight $$varepsilon_1 + varepsilon_2$$ is the set $${ varepsilon_1 + varepsilon_2, varepsilon_1 + varepsilon_3, varepsilon_2 + varepsilon_3 }$$.

Dominant weights are non-negative integral linear combinations of the fundamental weights, $$mvarpi_1 + n varpi_2$$.
Expanding this out in terms of the $$varepsilon_i$$, I get $$m varpi_1 + n varpi_2 = frac{1}{3} left( 2(m+n) varepsilon_1 + (2n-m) varepsilon_2 – (m+n) varepsilon_3 right).$$
Equating coefficients shows that none of the set of weights $${ varepsilon_1 + varepsilon_2, varepsilon_1 + varepsilon_3, varepsilon_2 + varepsilon_3 }$$ are dominant weights, contradicting what I am supposed to show.
What have I done wrong?

## Convergence of double sum on the lattice

I am working with a commutator $$T$$ acting on the lattice $$ell^2(mathbb{Z}^2;mathbb{C})$$, the function space made up by the basis elements
begin{align}left|vec{x}right>,:,mathbb{Z}^2&rightarrow mathbb{C}\ vec{y}&mapsto delta_{vec{x},vec{y}} end{align}
I am studying a paper in which the authors claim a particular (double) sum is convergent. It is the following,
$$begin{equation} C_Nsum_{r_1in mathbb{Z}}sum_{x_1in mathbb{Z}}left(1+frac{1}{2}left(|x_1+r_1|+|x_1|right)right)^{-N}.qquad (1) end{equation}$$
Here, $$N$$ can be any real number, to each $$N$$ a certain constant $$C_N$$ is fit. In the paper, the authors claim this sum to be convergent, but I have a hard time realising why that is the case. At first thought, for fixed $$r_1’in mathbb{Z}$$, e.g. for $$N=2$$, the inner sum will look very much like the sum $$sum_{x_1in mathbb{Z}}frac{1}{1+|x_1|^2}$$, which I now converge to a finite number. But if I sum over this finite number an infinite number of times, obviously it will not converge. Is it possible to pick a proper $$N$$ such that the double sum in (1) does indeed converge?

## co.combinatorics – Okada-Schur functions and the Martin boundary of the Young-Fibonacci lattice

This question is related to three earlier posts addressing properties of the Young-Fibonacci lattice $$Bbb{YF}$$, namely:

Differential posets, the Plancherel state \$varphi_mathrm{P}\$, and minimality

Young-Fibonacci tableaux, content, and the Okada algebra

Infinite tridiagonal matrices and a special class of totally positive sequences

According to the results of Goodman and Kerov, the parameter space $$Omega$$ (as a set) of the Martin Boundary $$E$$ of the Young-Fibonacci lattice $$Bbb{YF}$$
consists of: (1) an outlier point $$mathrm{P}$$ together with (2) all pairs $$(w,beta)$$ where $$0 < beta leq 1$$
is a real parameter and $$w = cdots a_4 a_3 a_2 a_1$$ is an infinite fibonacci word with
$$2$$‘s occurring at positions $$dots, d_4, d_3, d_2, d_1$$ when read from right to left
such that $$sum_{i geq 1} 1/d_i < infty$$. Note that the position
of $$2$$ in a fibonacci word of the form $$w = u2v$$ is $$1 + |v|$$ where $$|v|$$
denotes the sum of the digits of the suffix $$v$$, otherwise called
the length of $$v$$. The reader should consult Goodman and Kerov’s paper for a description of $$Omega$$‘s topology.

To each $$omega in Omega$$ the corresponding point $$varphi_omega in E$$,
is a non-negative, normalised harmonic function on $$Bbb{YF}$$. Under this correspondence $$varphi_mathrm{P}$$
is the Plancherel state, i.e. for $$u in Bbb{YF}$$

$$begin{equation} label{plancherel-measure} varphi_mathrm{P}(u) := {1 over {, |u|!}} , mathrm{dim}big( emptyset, u big) end{equation}$$

where $$dim(u,v)$$ denotes the number of saturated chains $$(u_0 lhd cdots lhd , u_n)$$ in $$Bbb{YF}$$ starting at $$u_0 = u$$ and ending at $$u_n =v$$.
Recently Vsevolod Evtushevsky (see arXiv:2012.07447 and arXiv:2012.08107)
has announced a proof showing that the Martin Boundary $$E$$ coincides with its minimal boundary. If this is true, then any positive, normalised harmonic function
$$varphi: Bbb{YF} longrightarrow Bbb{R}$$ should be expressed as

$$begin{equation} varphi(u) = int_Omega dM_varphi(omega) , varphi_omega(u) end{equation}$$

where $$dM_varphi$$ is
a measure (morally a boundary condition) uniquely determined by $$varphi$$.

There is an alternative supply of normalised harmonic functions on the Young-Fibonacci lattice: I’ll call them Okada-Schur functions,
but strictly speaking they are commutative versions of the polynomials defined in two non-commutative variables as introduced by Okada (Goodman and Kerov call them clone symmetric functions):

Let $${bf y} = (y_1, y_2, y_3, dots)$$ be a sequence of real numbers.
The Okada-Schur function $$sigma_{bf y}: Bbb{YF} longrightarrow Bbb{R}$$
associated to the sequence $${bf y}$$ is defined recursively (with respect to length) by

$$begin{equation} sigma_{bf y}(u) = left{ begin{array}{ll} T_k ({bf y}) & text{if u = 1^k for some k geq 0} \ \ S_k big({{bf y} + |v|} big) cdot sigma_{bf y}(v) & text{if u=1^k2v for some k geq 0} end{array} right. end{equation}$$

where

$$begin{equation} T_ell ({bf y}) = det underbrace{begin{pmatrix} 1 & y_1 & 0 & cdots\ 1 & 1 & y_2 &\ 0 & 1 & 1 & \ vdots & & & ddots end{pmatrix}}_{text{ell times ell tridiagonal matrix}} quad S_{ell -1} ({bf y}) = det underbrace{begin{pmatrix} y_1 & y_2 & 0 & cdots\ 1 & 1 & y_3 &\ 0 & 1 & 1 & \ vdots & & & ddots end{pmatrix}}_{text{ell times ell tridiagonal matrix}} end{equation}$$

for integers $$ell geq 1$$ and
where we employ the notation $${bf y} + r:= (y_{1+r}, , y_{2+r}, , y_{3+r}, , dots)$$ for any integer $$r geq 0$$.
Okada proved that $$sigma_{bf y}$$ is a normalised harmonic function for any infinite sequence $${bf y}$$. It is not
clear to me what are necessary and sufficient conditions for $$sigma_{bf y}$$ to be positive, let alone non-negative for that matter.
There is, however, at least one case when $$sigma_{bf y}$$ is
positive, namely:

Remark 1:
Let $${bf y} = big({1 over 2}, {1 over 3}, {1 over 4}, dots big)$$
then $$sigma_{bf y} = varphi_mathrm{P}$$ (i.e. the Plancherel state).

Furthermore

Remark 2: If $${bf y} = (y_1, y_2, y_3, dots)$$ is any sequence of real numbers for which $$sigma_{bf y}$$ is positive
then $$sigma_{{bf y} + r}$$ will be positive for any integer $$r geq 0$$. (This basically follows from some of the observations made in Infinite tridiagonal matrices and a special class of totally positive sequences).

Question 1: Suppose $${bf y}$$ is a sequence such that $$sigma_{bf y}$$
is positive. Under which circumstances will $$sigma_{bf y} in Omega$$ ?
My guess is only when $${bf y} = big({1 over 2}, {1 over 3}, {1 over 4}, dots big)$$ but what about $${bf y} = big({1 over {r+2}}, {1 over {r+3}}, {1 over {r+4}}, dots big)$$ for $$r geq 1$$ ?

Question 2: Suppose instead that $$sigma_{bf y}$$ is positive but $$sigma_{bf y} notin Omega$$. What is the unique measure $$dM_{bf y}$$ on the Martin boundary $$Omega$$ for which

$$begin{equation} sigma_{bf y}(u) = int_Omega dM_{bf y}(omega) , varphi_omega(u) end{equation}$$

Again, what about $${bf y} = big({1 over {r+2}}, {1 over {r+3}}, {1 over {r+4}}, dots big)$$ for $$r geq 1$$ ?

thanks, ines.

## linear algebra – Why \$mathbb Q^n\$ is not a lattice of \$mathbb R^n\$?

We can show that $$mathbb Z^n$$, additive subgroup of $$mathbb R^n$$, is a lattice and intuitively see that it might not be possible to generate $$mathbb Q^n$$ as integral multiple of $$m$$, $$mleq n$$, linearly independent vectors in $$mathbb R^n$$. But can you give a proof, I am having difficulty grasping the definition of lattices itself.

## reference request – Cylindric partitions for lattice paths with a weight of binomial form

In Cylindric partitions prop.1, Gessel and Krattenthaler prove a formula for lattice paths on a cylinder

In our particular problem, we again have paths $$((P_{1},k_{1}),…,(P_{r},k_{r}))$$ and the weights do satisfy $$w(Se)=w(e)$$ for all edges, but for the weight of the final-step we have
$$Large w_{last}(x,y,k):=left{begin{matrix} (-1)^{y-x}binom{k}{y-x}1_{kgeq y-xgeq 0},&text{ if } kgeq 0 \ (-1)^{k}binom{x-y-1}{|k|-1}1_{xgeq y+|k|}, &text{ if } k< 0 end{matrix}right.$$
meaning that for each ith path the weight on the final edge $$P_{i}(N_{i}-1)to P_{i}(N_{i}):=S^{k_{i}}v_{j}$$ is given by
$$w(P_{i}(N_{i}-1), S^{k_{i}}v_{j}):=w_{last}(P_{i}(N_{i}-1), S^{k_{i}}v_{j},k_{i}).$$

I am trying to see if there is any extension of prop.1 for weights as in our particular problem. The existing proof from 1 fails at the step of $$P_{1}$$ and $$SP_{r}$$ intersecting because when we insert $$k’_{r}=k_{1}-1$$ and $$k_{1}’=k_{r}+1$$, the weights of the last step change i.e. the involution $$phi$$ is no longer “weight-preserving”.

I am thinking that one possible remedy is to find some prefactor like the prefactor $$z^{ak_{s}^{2}/2-v_{t}k_{s}}$$ in prop.1 that will turn the involution $$phi$$ into “weight-preserving”.

Have you seen any situations with path-weights in binomial form? How about situations where one has to find a prefactor in order to make the involution $$phi$$ be “weight-preserving”?

## co.combinatorics – Anchor sets for lattice polygons

Suppose $$V={(x_1,y_1), (x_2,y_2),dots,(x_v,y_v)}$$ is a vertex set of lattice points satisfying
$$0=x_1y_2>cdots>y_{v-1}>y_v=0.$$
Construct the polygonal region $$DsubsetBbb{R}^2$$ with boundary $$partial D$$ along the closed path
$$(0,0)rightarrow (x_1,y_1)rightarrow(x_2,y_2)rightarrowcdots(x_{v-1},y_{v-1})rightarrow(x_v,y_v)rightarrow(0,0).$$
Assume $$mathcal{D}=mathbb{R}_{geq0}^2-D$$ is a convex domain.

QUESTION. Can you determine the smallest positive integer $$ell$$ and a finite set $$Ssubsetmathbb{Z}_{geq0}^2$$ of lattice points containing $$V$$ such that each of the lattice points $$(x,y)inmathcal{D}$$ can be reached from $$(x_j,y_j)in S$$ after performing a lattice-path-move $$N$$ (north bound) and/or $$E$$ (east bound), for some $$jin{1,2,dots,vert Svert}$$; that is, if $$(s)={1,2,dots,s}$$ and $$s=vert Svert$$, then
$$(x,y)inmathcal{D} qquad Longrightarrow qquad exists a,binBbb{Z}_{geq0},,, exists jin(s): (x_j+a,y_j+b)=(x,y),,?$$

## geometry – Can a n by n square lattice grid be linear projection of vertex of some high dimensional convex polytope?

Given a $$n times n$$ square lattice grid. can it come from some linear projection of vertex of a high dimensional convex polytope?

E.g. $$n=2$$, it is obviously possible, it would be a cube projected on one face. but is this generally true for all $$nge 2$$?