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?

Thank you in advance for your help!

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.

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

enter image description here

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
$$0=x_1<x_2<dots<x_v qquad text{and} qquad y_1>y_2>cdots>y_{v-1}>y_v=0.$$
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 $nge 2$?