# A partial order on antichains of a fixed size

R.M. Green and Tianyuan Xu

###### Abstract.

We introduce a new partial order on the set of all antichains of a fixed size
in a given poset. When applied to minuscule posets, these partial orders give
rise to distributive lattices that appear in the branching rules for minuscule
representations of simply laced complex simple Lie algebras.

###### Key words and phrases:

antichain, distributive lattice, minuscule representation

###### 1991 Mathematics Subject Classification:

Primary: 06A07; Secondary: 05E10, 06A11

## Introduction

This paper introduces a new partial order $\leq_{k}$ on the set
$\mathcal{A}_{k}(P)$ of all antichains of a fixed size $k$ in a given poset
$P$. We assume basic familiarity with poset theory, including the notions of
antichains, order ideals, covering relations, Hasse diagrams, products of
posets, and distributive lattices. These notions can all be found in [10, §3],
whose definitions and notations we will follow. In particular, we write
$a\lessdot_{P}b$ to indicate two elements $a,b$ are in a covering relation in
a poset $P$. We denote the sets of positive integers and nonnegative integers
by ${\mathbb{Z}}_{+}$ and $\mathbb{N}$, respectively, and we write $[n]$ for
the set $\\{1,2,\dots,n\\}$, viewed as a poset with the natural order, for all
$n\in\mathbb{N}$. Unless otherwise stated, all posets in the paper will be
finite.

It is well known that for a (finite) poset $P$, there is a bijection between
the set $J(P)$ of ideals of $P$ and the set $\mathcal{A}(P)$ of antichains of
$P$, given by associating an ideal with its set of maximal elements. The
containment order on $J(P)$ then induces a partial order on $\mathcal{A}(P)$,
which we will denote by $\leq_{J}$, and which we may restrict to the set
$\mathcal{A}_{k}(P)$ for each $k\in\mathbb{N}$. It is a classic result of
Dilworth [2] that when $k$ is the _width_ of $P$, defined as
$\operatorname{\mathrm{width}}(P)=\max\\{\lvert
A\rvert:A\in\mathcal{A}(P)\\}$, the set $\mathcal{A}_{k}(P)$ is a distributive
lattice under the restriction of $\leq_{J}$.

The new partial order $\leq_{k}$ we introduce on $\mathcal{A}_{k}(P)$ is
defined as the reflexive transitive extension of the relation $\prec_{k}$,
where we declare $A\prec_{k}B$ for $A,B\in\mathcal{A}_{k}(P)$ if
$B=A\setminus\\{a\\}\cup\\{b\\}$ for elements $a,b\in P$ such that
$a\lessdot_{P}b$. As we will show in Section 2, the order $\leq_{k}$ is
coarser than the restriction of $\leq_{J}$ in general, and
$\mathcal{A}_{k}(P)$ may not be a distributive lattice under the order
$\leq_{k}$ when $k=\operatorname{\mathrm{width}}(P)$.

The order $\leq_{k}$ has striking properties when applied to the minuscule
posets of [9] and [5, §11.2]. Also known as vertex labellable posets,
minuscule posets have intimate connections to minuscule representations of
simple Lie algebras, and they are the only known examples of irreducible
Gaussian posets. They admit a classification in terms of Dynkin diagrams of
types $ADE$, with the irreducible minuscule posets being the posets of the
forms $[a]\times[b]$,
$J([n]\times[2]),J^{m}([2]\times[2]),J^{2}([2]\times[3])$, and
$J^{3}([2]\times[3])$ where $a,b,n\in{\mathbb{Z}}_{+}$ and $m\in\mathbb{N}$.
The Hasse diagrams of these posets are shown in Figure 1. In Theorem 4.2, We
will determine the posets of form $\mathcal{A}_{k}(P)$ for all minuscule
posets $P$ and show that they are distributive lattices in all cases. In the
special cases where $P=[a]\times[b]$, which correspond to minuscule
representations of type $A$, the poset $\mathcal{A}_{k}(P)$ is naturally
isomorphic to the poset $\mathcal{D}_{k}([a]\times[b])$ of Young diagrams of
Durfee length $k$ that fit into an $a\times b$ box, ordered by inclusion; see
Corollary 3.2. The poset $\mathcal{D}_{k}([a]\times[b])$ encodes information
about the posets of weights appearing in the context of branching rules of
minuscule representations for simple Lie algebras of type $A_{n}$. There are
analogous results for all the other minuscule representations of simply laced
simple Lie algebras, as we explain in Theorem 5.1.

(a) $[a]\times[b]$

(b) $J([n]\times[2])$

(c) $J^{m}([2]\times[2])$

(d) $J^{2}([2]\times[3])$

(e) $J^{3}([2]\times[3])$

Figure 1. Minuscule posets

Our motivation for studying the partial orders $\leq_{k}$ comes from our
earlier work [6] on Kazhdan–Lusztig cells of $\mathbf{a}$-value 2, where
$\mathbf{a}$ is Lusztig’s $\mathbf{a}$-function. For $\mathbf{a}(2)$-finite
Coxeter groups (as defined and described in [6]), every element $w$ of
$\mathbf{a}$-value 2 has an associated heap poset $H$. A key property of $H$
can be summarized as follows: the poset $\mathcal{A}_{2}(H)$ has a minimum
element with respect to $\leq_{2}$, and the ideal generated by the minimal
element determines the left cell of $w$. A similar statement holds for maximal
elements and right cells.

The rest of the paper is organized as follows. We introduce and study two
natural families of posets consisting of integer sequences in Section 1. This
section does not treat the order $\leq_{k}$ directly, but these families of
posets will provide a useful model for the subsequent parts of the paper.
Section 2 introduces the order $\leq_{k}$, compares it with restrictions of
the order $\leq_{J}$, and discusses a few basic examples. Sections 3 and 4
study the poset $\mathcal{A}_{k}(P)$ for minuscule posets of type $A$ and of
all other types, respectively, culminating in the explicit descriptions of all
such posets $\mathcal{A}_{k}(P)$ in Theorem 4.2. Section 5 establishes a
connection between Theorem 4.2 and minuscule representations of simply laced
simple Lie algebras. Finally, we discuss several open questions related to the
order $\leq_{k}$ in Section 6.

##  1\. Two posets of sequences

In this section we study two posets, denoted $\mathcal{C}(n,k)$ and
$\mathcal{S}(a,b)$, that will be especially useful for this paper. In
particular, we will show in Corollary 1.8 that
$J([a]\times[b])\cong\mathcal{S}(a,b)\cong\mathcal{C}(a+b,b)$ for any
$a,b\in\mathbb{N}$.

######  Definition 1.1.

Suppose $k,n\in\mathbb{N}$ and $k\leq n$. Let $\mathcal{C}(n,k)$ be the set of
$k$-subsets of $[n]$, and write each element $\mathbf{x}\in\mathcal{C}(n,k)$
as an increasing sequence $\mathbf{x}=(x_{1},x_{2},\dots,x_{k})$. For elements
$\mathbf{x}=(x_{1},x_{2},\ldots,x_{k})$ and
$\mathbf{y}=(y_{1},y_{2},\ldots,y_{k})$ of $\mathcal{C}(n,k)$, we write
$\mathbf{x}\leq_{\mathcal{C}}\mathbf{y}$ to mean $x_{i}\leq y_{i}$ for all
$1\leq i\leq k$. We define the function
$\rho:\mathcal{C}(n,k)\rightarrow\mathbb{N}$ by

$\rho(\mathbf{x})=x_{1}+x_{2}+\cdots+x_{k}.$

The fact that $\leq_{\mathcal{C}}$ is a partial order follows immediately from
the above definition. Indeed, the order $\leq_{\mathcal{C}}$ is a _Gale order_
in the sense of [3] and [12, Definition 6]: if $A$ and $B$ are $I$-tuples of a
subset $X$, then we have $A\leq B$ in the Gale order if there exists a
bijection $f_{i}:A_{i}\rightarrow B_{i}$ such that $a\leq f_{i}(a)$ for any
$a\in A_{i}$.

######  Lemma 1.2.

Maintain the notation of Definition 1.1.

  * (i)

The function $\rho$ is a rank function on $\mathcal{C}(n,k)$. In other words,
if $\mathbf{x},\mathbf{y}\in\mathcal{C}(n,k)$ satisfy
$\mathbf{x}\leq_{\mathcal{C}}\mathbf{y}$, then we have
$\rho(\mathbf{x})\leq\rho(\mathbf{y})$, with
$\rho(\mathbf{y})=\rho(\mathbf{x})+1$ if and only if
$\mathbf{x}\lessdot_{\mathcal{C}}\mathbf{y}$.

  * (ii)

Two elements $\mathbf{x},\mathbf{y}\in\mathcal{C}(n,k)$ satisfying
$\mathbf{x}\leq_{\mathcal{C}}\mathbf{y}$ are in a covering relation if and
only if $x$ and $y$ differ only in one element, with the unique elements
$c\in\mathbf{x}\backslash\mathbf{y}$ and $d\in\mathbf{y}\backslash\mathbf{x}$
satisfying $d=c+1$.

###### Proof.

It is immediate from the definition of $\leq_{\mathcal{C}}$ that
$\rho(\mathbf{x})<\rho(\mathbf{y})$ whenever
$\mathbf{x}<_{\mathcal{C}}\mathbf{y}$. This implies that if
$\mathbf{x}<_{\mathcal{C}}\mathbf{y}$ and
$\rho(\mathbf{y})=\rho(\mathbf{x})+1$, then
$\mathbf{x}\lessdot_{\mathcal{C}}\mathbf{y}$.

Suppose that $\mathbf{x}<_{\mathcal{C}}\mathbf{y}$ for elements
$\mathbf{x}=(x_{1},x_{2},\ldots,x_{k})$ and
$\mathbf{y}=(y_{1},y_{2},\ldots,y_{k})$ in $\mathcal{C}(n,k)$. Choose $1\leq
r\leq k$ to be the maximal index satisfying $x_{r}<y_{r}$, and let
$\mathbf{z}=(\mathbf{x}\backslash\\{x_{r}\\})\cup\\{x_{r}+1\\}$. Note that the
set $\mathbf{z}$ still consists of $k$ distinct numbers: if $r=k$ then
$\mathbf{z}$ is obtained from $\mathbf{x}$ by increasing the largest entry in
$\mathbf{x}$, while if $r<k$ then we have $x_{r}+1\leq y_{r}<y_{r+1}=x_{r+1}$.
Note also that regardless of whether $r=k$, we have
$x\leq_{\mathcal{C}}z\leq_{\mathcal{C}}y$ and
$\rho(\mathbf{z})=\rho(\mathbf{x})+1$. It follows that if
$\mathbf{x}\lessdot_{\mathcal{C}}\mathbf{y}$ then we must have
$\mathbf{z}=\mathbf{y}$ and $\rho(\mathbf{y})=\rho(\mathbf{x})+1$, which
proves the converse direction of (i).

Condition (i) implies that $\mathbf{x}\lessdot_{\mathcal{C}}\mathbf{y}$ if and
only if $\rho(\mathbf{y})=\rho(\mathbf{x})+1$. By the definitions of
$\leq_{\mathcal{C}}$ and $\rho$, this happens if and only if $\mathbf{x}$ and
$\mathbf{y}$ satisfy the conditions of (ii), which completes the proof. ∎

######  Definition 1.3.

For $a,b\in\mathbb{N}$, define

$\mathcal{S}(a,b)=\\{(x_{b},\ldots,x_{1})\in{\mathbb{Z}}^{k}:0\leq
x_{b}\leq\cdots\leq x_{1}\leq a\\}.$

If $\mathbf{x}=(x_{b},\ldots,x_{1})$ and $\mathbf{y}=(y_{b},\ldots,y_{1})$ are
elements of $\mathcal{S}(a,b)$, we write
$\mathbf{x}\leq_{\mathcal{S}}\mathbf{y}$ to mean that $x_{i}\leq y_{i}$ for
all $1\leq i\leq b$.

######  Lemma 1.4.

Maintain the notation of Definition 1.3.

  * (i)

The relation $\leq_{\mathcal{S}}$ is a partial order on $\mathcal{S}(a,b)$.

  * (ii)

The function $h:\mathcal{S}(a,b)\rightarrow\mathcal{C}(a+b,b)$ given by

$h:(x_{b},\ldots,x_{1})\mapsto(x_{b}+1,\ x_{b-1}+2,\ \ldots,\ x_{i}+(b+1-i),\
\ldots,\ x_{1}+b)$

is an isomorphism of posets.

  * (iii)

The covering relations in $\mathcal{S}(a,b)$ are precisely those of the form

$(x_{b},\ldots,x_{1})<(y_{b},\ldots,y_{1})$

where for some $i$ with $1\leq i\leq b$, we have $y_{i}=x_{i}+1$, and
$y_{t}=x_{t}$ for all $t\neq i$.

###### Proof.

Parts (i) and (ii) follow from the definitions; (iii) follows by combining
(ii) with Lemma 1.2 (ii). ∎

We now discuss two applications of the posets $\mathcal{C}(n,k)$ and
$\mathcal{S}(a,b)$. First, we show that they model posets of the form
$J([a]\times[b])$. We recall the fact that given two posets $P$, $Q$ and two
elements $(p,q)$ and $(p^{\prime},q^{\prime})$ in the product poset $P\times
Q$, we have $(p,q)\lessdot_{P\times Q}(p^{\prime},q^{\prime})$ if and only if
we have $p=p,q\lessdot_{Q}q$ or have $p\lessdot_{P}p^{\prime},q=q^{\prime}$.
We will use this fact without further comment in the rest of the paper. We
will also think of the poset $[a]\times[b]$ as embedded in the lattice
${\mathbb{Z}}_{+}^{2}:=\\{(x,y):x,y\in{\mathbb{Z}}_{+}\\}$ in the plane
${\mathbb{R}}^{2}$, so that a point $(i_{1},j_{1})$ is smaller than another
point $(i_{2},j_{2})$ if and only if $(i_{1},j_{1})$ lies weakly to the
southwest of $(i_{2},j_{2})$ in ${\mathbb{Z}}_{+}^{2}$.

######  Lemma 1.5.

Let $P=[a]\times[b]$. Then two points $(i_{1},j_{1})$ and $(i_{2},j_{2})$ are
incomparable in $P$ if and only if either (a) $i_{1}<i_{2}$ and $j_{1}>j_{2}$,
or (b) $i_{1}>i_{2}$ and $j_{1}<j_{2}$. Any antichain in $P$ can be written
uniquely in the form $A=\\{(x_{1},y_{1}),\ldots,(x_{k},y_{k})\\}$, where
$x_{1}<\ldots<x_{k}$ and $y_{1}>\ldots>y_{k}$.

###### Proof.

The first statement is immediate; the second one follows by induction. ∎

######  Definition 1.6.

Let $a,b\in\mathbb{N}$ and let $P=[a]\times[b]$. We define
$f:J(P)\rightarrow{\mathbb{Z}}^{b}$ by

$f(I)=(m^{I}_{b},\ldots,m^{I}_{1}),$

where $m^{I}_{j}$ is the maximal integer $i\in[a]$ such that $(i,j)\in I$, or
zero if no such integer exists.

######  Proposition 1.7.

The map $f$ of Definition 1.6 gives an isomorphism of posets
$f:J([a]\times[b])\rightarrow\mathcal{S}(a,b)$.

###### Proof.

Let $I\in J([a]\times[b])$. For each $j\in[n-1]$, the fact that
$(m^{I}_{j+1},j+1)\in I$ implies that $(m^{I}_{j+1},j)\in I$ and thus
$m^{I}_{j}\geq m^{I}_{j+1}$; therefore we have $0\leq m_{b}^{I}\leq...\leq
m_{1}^{I}\leq a$. By the definition of the integers $m_{j}^{I}$, we also have
$I=\\{(i,j):1\leq j\leq b,1\leq i\leq m_{j}^{I}\\}$. Conversely, if
$(x_{b},\ldots,x_{1})$ is an element of $\mathcal{S}(a,b)$, then the set
$I^{\prime}=\\{(i,j):1\leq j\leq b,\ 1\leq i\leq x_{i}\\}$ is an element of
$J([a]\times[b])$. It follows that $f$ is a bijection between
$J([a]\times[b])$ and $\mathcal{S}(a,b)$.

Let $I_{1},I_{2}\in J([a]\times[b])$. It follows from the definition of the
ordering $\leq_{\mathcal{S}}$ that we have $I_{1}\subseteq I_{2}$ if and only
if $f(I_{1})\leq_{\mathcal{S}}f(I_{2})$. This implies that $f$ is an
isomorphism of posets, completing the proof. ∎

######  Corollary 1.8.

Let $a,b\in\mathbb{N}$. Then
$J([a]\times[b])\cong\mathcal{S}(a,b)\cong\mathcal{C}(a+b,b)$ as posets.

###### Proof.

This follows immediately from Proposition 1.7 and Lemma 1.4 (ii). ∎

######  Remark 1.9.

Given two posets $P$ and $Q$, the map $P\times Q\rightarrow Q\times
P,(p,q)\mapsto(q,p)$ is clearly a poset isomorphism. Therefore Corollary 1.8
implies that $\mathcal{C}(a+b,b)\cong J([a]\times[b])\cong
J([b]\times[a])\cong\mathcal{C}(a+b,a)$ as posets for all $a,b\in\mathbb{N}$,
which may be viewed as a lift of the numerical equality
$\binom{a+b}{b}=\binom{a+b}{a}$ of binomial coefficients.

######  Corollary 1.10.

Posets of the form $\mathcal{C}(n,k)$ and $\mathcal{S}(a,b)$ are distributive
lattices.

###### Proof.

This follows from Corollary 1.8 since $J(P)$ is a distributive lattice for any
finite poset $P$. ∎

The second application of the poset $\mathcal{C}(n,k)$ and $\mathcal{S}(a,b)$
concerns Young diagrams or, equivalently, Ferrers diagrams (see [10, §1.7]).
Using the French notation, we may conveniently view Ferrers diagrams to be
finite ideals of the infinite poset ${\mathbb{Z}}_{+}^{2}$. The _Durfee
length_ of a Ferrers diagram $D$ is the largest integer $k\in\mathbb{N}$ such
that $[k]\times[k]\subseteq D$, i.e., the side length of the largest square
grid $S$ that fits inside $D$; this largest square $S$ is called the _Durfee
square_ of $D$.

######  Definition 1.11.

For any $a,b,k\in\mathbb{N}$ with $k\leq\min(a,b)$, we define
$\mathcal{D}_{k}(a,b)$ to be the poset of all Ferrers diagrams with Durfee
length $k$ contained in the set $[a]\times[b]$, ordered by set containment.

######  Proposition 1.12.

If $a,b,k\in\mathbb{N}$ satisfy $k\leq\min(a,b)$, then we have
$\mathcal{D}_{k}([a]\times[b])\cong\mathcal{C}(a,k)\times\mathcal{C}(b,k)$ as
posets.

###### Proof.

In light of Corollary 1.8 and Remark 1.9, it suffices to find an isomorphism
from $\mathcal{D}_{k}([a]\times[b])$ to $J([k]\times[b-k])\times
J([a-k]\times[k])$. To find such a map, note that each diagram
$D\in\mathcal{D}_{k}([a]\times[b])$ can be canonically decomposed into (and
recovered from) three parts: its Durfee square $S=[k]\times[k]$; the part
$I_{1}=\\{(i,j)\in D:j>k\\}$ above $S$; and the part $I_{2}=\\{(i,j)\in
D:i>k\\}$ to the right of $S$. Shifting $I_{1}$ down and $I_{2}$ to the left
by $k$, we obtain ideals $I^{\prime}_{1}=\\{(i,j-k):(i,j)\in I_{1}\\}$ and
$I^{\prime}_{2}=\\{(i-k,j):(i,j)\in I_{2}\\}$ in the grids $[k]\times[b-k]$
and $[a-k]\times[k]$, respectively. It follows that we have a bijection
$\varphi:\mathcal{D}_{k}([a]\times[b])\rightarrow J([k]\times[b-k])\times
J([a-k]\times[k]),D\mapsto(I^{\prime}_{1},I^{\prime}_{2})$. Both $\phi$ and
its obvious inverse map (stacking $I_{1}^{\prime}$ on top of and
$I_{2}^{\prime}$ to the right of $S$) clearly preserve containment, so $\phi$
is a poset isomorphism, as desired. ∎

##  2\. Partial orders on antichains

Recall from the introduction that the containment order on ideals of $P$
induces an order $\leq_{J}$ on the antichains of $P$. Note that $\leq_{J}$ can
be described without reference to ideals as follows: for any
$A,B\in\mathcal{A}(P)$, we have

$A\leq_{J}B\quad\Longleftrightarrow\quad\forall\,a\in A,\exists\,b\in
B:a\leq_{P}b.$

Also recall the following definition of the order $\leq_{k}$. We will study
the relationship between $\leq_{k}$ and $\leq_{J}$ in this section.

######  Definition 2.1.

Let $P$ be a poset, let $\mathcal{A}_{k}(P)$ be the set of antichains of $P$
of cardinality $k$, and let $A,B\in\mathcal{A}_{k}(P)$. We write $A\prec_{k}B$
if $A\backslash B=\\{a\\}$ and $B\backslash A=\\{b\\}$ are both singleton sets
with the property that $a<_{P}b$. The relation $\leq_{k}$ on
$\mathcal{A}_{k}(P)$ is defined to be the reflexive transitive extension of
$\prec_{k}$.

######  Proposition 2.2.

Let $P$ be a finite poset, let $k\in\mathbb{N}$, and let $\mathcal{A}_{k}(P)$
be the set of all antichains of $P$ of cardinality $k$.

  * (i)

The relation $\leq_{k}$ of Definition 2.1 is a partial order on
$\mathcal{A}_{k}(P)$.

  * (ii)

The restriction of the partial order $\leq_{J}$ to $\mathcal{A}_{k}(P)$
refines the order $\leq_{k}$.

  * (iii)

If $A,B\in\mathcal{A}_{k}(P)$ satisfy $A\leq_{k}B$, then the elements of
$A=\\{a_{1},a_{2},\ldots,a_{k}\\}$ and $B=\\{b_{1},b_{2},\ldots,b_{k}\\}$ can
be ordered in such a way that $a_{i}\leq_{P}b_{i}$ for all $1\leq i\leq k$.

  * (iv)

Two elements $A,B\in\mathcal{A}_{k}(P)$ are in a covering relation
$A\lessdot_{\mathcal{A}_{k}(P)}B$ if and only both (1) $A\prec_{k}B$ and (2)
the unique elements $a\in A\backslash B$ and $b\in B\backslash A$ satisfy
$a\lessdot_{P}b$.

###### Proof.

It follows from the definition of $\prec$ that if $A,B\in\mathcal{A}_{k}(P)$
satisfy $A\prec_{k}B$, then we have $A\leq_{J}B$. The antisymmetry of
$\leq_{k}$ now follows from the antisymmetry of $\leq_{J}$, and this proves
(i) and (ii).

If $A,B\in\mathcal{A}_{k}(P)$ satisfy $A\prec_{k}B$, then (iii) follows by the
definition of $\prec$. The general case of (iii) follows by induction.

To prove (iv), assume first that $A\lessdot_{\mathcal{A}_{k}(P)}B$. By the
definition of $\leq_{k}$, we must have $A\prec_{k}B$. We then have
$A=C\cup\\{a\\}$ and $B=C\cup\\{b\\}$, where $C=A\cap
B\in\mathcal{A}_{k-1}(P)$ and $a,b\in P$ satisfy $a<_{P}b$. Suppose that $x\in
P$ satisfies $a<_{P}x<_{P}b$. In this case, we have
$C^{\prime}:=C\cup\\{x\\}\in\mathcal{A}_{k}(P)$ for the following reason: we
cannot have $x<c$ for any $c\in C$ because $a<x<c$ and $A$ is an antichain,
and we cannot have $c<x$ for any $c\in C$ because $c<x<b$ and $B$ is an
antichain. It follows that $A<_{k}C^{\prime}<_{k}B$, which is a contradiction,
therefore we have $a\lessdot_{P}b$.

Conversely, assume that $A,B\in\mathcal{A}_{k}(P)$ satisfy $A\prec_{k}B$, and
also that $a\lessdot_{P}b$, where $C=A\cap
B=\\{c_{1},c_{2},\ldots,c_{k-1}\\}$, $A=C\cup\\{a\\}$, and $B=C\cup\\{b\\}$.
Write $A=\\{a_{1},a_{2},\ldots,a_{k}\\}$, where $a_{i}=c_{i}$ for $i<k$ and
$a_{k}=a$, and $B=\\{b_{1},b_{2},\ldots,b_{k}\\}$, where $b_{i}=c_{i}$ for
$i<k$ and $b_{k}=b$. Suppose for a contradiction that there exists
$X=\\{x_{1},x_{2},\ldots,x_{k}\\}\in\mathcal{A}_{k}(P)$ such that $A<_{k}X$
and $X<_{k}B$. It follows from (iii) that there are permutations $\sigma$ and
$\tau$ of $\\{1,2,\ldots,k\\}$ such that for all $1\leq i\leq k$, we have
$a_{i}\leq_{P}x_{\sigma(i)}$ and $x_{i}\leq_{P}b_{\tau(i)}$, which implies
that $a_{i}\leq_{P}b_{\tau(\sigma(i))}$. Because $A$ and $B$ are antichains,
we must have $\tau(\sigma(i))=i$ for all $1\leq i<k$, and this implies that
$\tau=\sigma^{-1}$. By relabelling $X$ if necessary, we may assume that
$\sigma$ and $\tau$ are both the identity permutation, and that
$X=C\cup\\{x_{k}\\}$, where $a=a_{k}<_{P}x_{k}<_{P}b_{k}=b$. This contradicts
the hypothesis that $a\lessdot_{P}b$, and (iv) follows. ∎

######  Remark 2.3.

  * (i)

It is immediate from the definitions that $\mathcal{A}_{k}(P)$ is nonempty if
and only if $0\leq k\leq\operatorname{\mathrm{width}}(P)$, that
$\mathcal{A}_{0}(P)$ is the singleton poset, and that $\mathcal{A}_{1}(P)$ is
canonically isomorphic to $P$ itself.

  * (ii)

Dilworth [2, Theorem 2.1] proved that $\mathcal{A}_{k}(P)$ is a distributive
lattice under the partial order $\leq_{J}$ for
$k=\operatorname{\mathrm{width}}(P)$. In general, the same set
$\mathcal{A}_{k}(P)$ is not a distributive lattice under the partial order
$\leq_{k}$, because $\leq_{k}$ may be strictly coarser than $\leq_{J}$.
However, for some important classes of examples, the partial orders $\leq_{J}$
and $\leq_{k}$ on the maximal antichains of $P$ are identical. Examples of
such posets include the heaps of fully commutative elements (in the sense of
[11]) in finite Coxeter groups, and more generally in star reducible Coxeter
groups (in the sense of [4]).

  * (iii)

For some choices of $P$, all the nonempty posets $\mathcal{A}_{k}(P)$ are
distributive lattices under the order $\leq_{k}$; see Theorem 4.2, for
example. By part (i), a necessary condition for this to happen is for $P$
itself to be a distributive lattice. However, this is not a sufficient
condition, as Example 2.4 (ii) shows.

######  Example 2.4.

  * (i)

Let $P=\\{a,b,c,d,e\\}$ be the poset with covering relations $a<c$, $b<c$,
$c<d$, and $c<e$. Then $\operatorname{\mathrm{width}}(P)=2$, and the set
$\mathcal{A}_{2}(P)$ consists of the two elements $\\{a,b\\}$ and $\\{d,e\\}$,
which are comparable in the partial order $\leq_{J}$. Proposition 2.2 (iv)
shows that no covering relations exist between elements of
$\mathcal{A}_{2}(P)$. This proves that $\\{a,b\\}$ and $\\{d,e\\}$ are not
comparable under $\leq_{2}$, even though their elements can be ordered as in
Proposition 2.2 (iii). It follows that the converse of Proposition 2.2 (iii)
does not hold, and that $\leq_{2}$ strictly coarsens $\leq_{J}$. It also
follows that $(\mathcal{A}_{k}(P),\leq_{2})$ is not a distributive lattice,
because it has no maximum or minimum element.

  * (ii)

Let $P=\\{a,b,c\\}$ be an antichain with three elements, so that $J(P)$ is a
distributive lattice with 8 elements. In this case, $\mathcal{A}_{2}(J(P))$
has 9 elements, including three maximal and three minimal elements. As in part
(i), $\mathcal{A}_{2}(J(P))$ is not a distributive lattice, because it has no
maximum or minimum element.

##  3\. Minuscule posets of type $A$

In this and the next two sections, we will focus on posets of the form
$\mathcal{A}_{k}(P)$ where $P$ is a minuscule poset. We start with the posets
$P=[a]\times[b]$, which correspond to minuscule representations of type $A$.
The main result of the section is the following proposition.

######  Proposition 3.1.

If $a,b,k\in\mathbb{N}$ satisfy $k\leq\min(a,b)$, then we have
$\mathcal{A}_{k}([a]\times[b])\cong\mathcal{C}(a,k)\times\mathcal{C}(b,k)$ as
posets.

###### Proof.

Lemma 1.5 implies that any antichain of $P$ of size $k$ can be written
uniquely as

$A=\\{(x_{1},y_{1}),\ (x_{2},y_{2}),\ \ldots,\ (x_{k},y_{k})\\},$

where $1\leq x_{1}<x_{2}<\cdots<x_{k}\leq a$ and $b\geq
y_{1}>y_{2}>\cdots>y_{k}\geq 1$. It follows that there is a function
$\phi:\mathcal{A}_{k}(P)\rightarrow\mathcal{C}(a,k)\times\mathcal{C}(b,k)$
given by

$\phi(A)=(\phi_{1}(A),\phi_{2}(A))=((x_{1},x_{2},\ldots,x_{k}),\
(y_{k},y_{k-1},\ldots,y_{1})).$

Furthermore, $\phi$ is a bijection, because the assignment

$((x_{1},x_{2},\dots,x_{k}),(y_{k},y_{k-1},\dots,y_{1}))\mapsto\\{(x_{1},y_{1}),\dots,(x_{k},y_{k})\\}$

takes $\mathcal{C}(a,k)\times\mathcal{C}(b,k)$ to $\mathcal{A}_{k}(P)$ by
Lemma 1.5 and is clearly a two-sided inverse of $\phi$.

We claim that $\phi$ is an isomorphism of posets. To see this, it is enough to
prove that $\phi$ respects covering relations. Let $A\in\mathcal{A}_{k}(P)$,
and write $A=\\{(x_{1},y_{1}),\ (x_{2},y_{2}),\ \ldots,\ (x_{k},y_{k})\\}$,
where $x_{1}<x_{2}<\cdots<x_{k}$ and $y_{1}>y_{2}>\cdots>y_{k}$.

Suppose that we have $A^{\prime}\lessdot_{\mathcal{A}_{k}(P)}A$ for some
$A^{\prime}\in\mathcal{A}_{k}(P)$. Proposition 2.2 (iv) implies that
$A^{\prime}$ can be obtained from $A$ by replacing one of the $(x_{i},y_{i})$
by $(x^{\prime}_{i},y^{\prime}_{i})$, where
$(x^{\prime}_{i},y^{\prime}_{i})\lessdot_{P}(x_{i},y_{i})$. This condition
means that we either have (a) $x_{i}^{\prime}=x_{i}$ and
$y_{i}^{\prime}=y_{i}-1$, or (b) $x_{i}^{\prime}=x_{i}-1$,
$y_{i}^{\prime}=y_{i}$. In case (a), we have
$\phi_{1}(A^{\prime})=\phi_{1}(A)$ and
$x_{1}<x_{2}<\ldots<x_{i-1}<x_{i}^{\prime}<x_{i+1}<\ldots<x_{k}$, which
implies that $y_{1}>y_{2}>\ldots>y_{i-1}>y_{i}^{\prime}>y_{i+1}>\ldots>y_{k}$
by Lemma 1.5. Since $y_{i}^{\prime}=y_{i}-1$, it follows that
$\phi_{2}(A^{\prime})<\phi_{2}(A)$ is a covering relation in
$\mathcal{C}(b,k)$. Similarly, in case (b), we have
$\phi_{2}(A^{\prime})=\phi_{2}(A)$ and $\phi_{1}(A^{\prime})<\phi_{1}(A)$ is a
covering relation in $\mathcal{C}(a,k)$. In either case,
$\phi(A^{\prime})<\phi(A)$ is a covering relation
$\mathcal{C}(a,k)\times\mathcal{C}(b,k)$.

Conversely, suppose that
$(X^{\prime},Y^{\prime})\lessdot_{\mathcal{C}(a,k)\times\mathcal{C}(b,k)}(X,Y)$.
This means that we either have (a) $Y^{\prime}=Y$ and
$X^{\prime}\lessdot_{\mathcal{C}(a,k)}X$, or (b) $X^{\prime}=X$ and
$Y^{\prime}\lessdot_{\mathcal{C}(b,k)}Y$. Suppose that we are in case (a).
Lemma 1.2 (ii) implies that if we write $X=\\{x_{1},x_{2},\ldots,x_{k}\\}$
with $x_{1}<x_{2}<\cdots<x_{k}$, then there exists $i$ such that we have
$1\leq i\leq k$, $X^{\prime}=X\backslash\\{x_{i}\\}\cup\\{x^{\prime}_{i}\\}$,
$x^{\prime}_{i}=x_{i}-1$, and

$x_{1}<x_{2}<\cdots<x_{i-1}<x^{\prime}_{i}<x_{i+1}<\cdots<x_{k}.$

It follows that $A=\\{(x_{1},y_{1}),\ (x_{2},y_{2}),\ldots,(x_{k},y_{k})\\}$
and
$A^{\prime}=(A\backslash\\{(x_{i},y_{i})\\})\cup\\{(x^{\prime}_{i},y_{i})\\}$
are antichains in $\mathcal{A}_{k}(P)$, and that
$(x^{\prime}_{i},y_{i})\lessdot_{P}(x_{i},y_{i})$. Proposition 2.2 (iv) then
implies that $A^{\prime}\lessdot_{\mathcal{A}_{k}(P)}A$. This completes the
proof of case (a). Case (b) is proved using an analogous argument, which
completes the proof of the proposition. ∎

######  Corollary 3.2.

If $a,b,k\in\mathbb{N}$ satisfy $k\leq\min(a,b)$, then we have
$\mathcal{A}_{k}([a]\times[b])\cong\mathcal{D}_{k}([a]\times[b])$ as posets.

###### Proof.

This follows immediately from propositions 1.12 and 3.1. ∎

##  4\. Other minuscule posets

We now investigate the posets $\mathcal{A}_{k}(P)$ for minuscule posets of
other types. We first deal with the infinite family $P_{n}=J([n]\times[2])$,
which correspond to the spin representations of type $D$ and satisfy
$\operatorname{\mathrm{width}}(P_{n})=\lfloor n+2/2\rfloor$.

######  Proposition 4.1.

If $n,k\in\mathbb{N}$ satisfy $0\leq k\leq\lfloor n+2/2\rfloor$, then we have
isomorphisms of posets

$\mathcal{A}_{k}(J([n]\times[2]))\cong\mathcal{A}_{k}(\mathcal{C}(n+2,2))\cong\mathcal{C}(n+2,2k).$

###### Proof.

Since $J([n]\times[2])\cong\mathcal{C}(n+2,2)$ by Corollary 1.8, it suffices
to prove $\mathcal{A}_{k}(\mathcal{C}(n+2,2))\cong\mathcal{C}(n+2,2k)$. Let
$P=\mathcal{C}(n+2,2)$ and write every element of $P$ in the form $(x,y)$
where $x<y$ as in Definition 1.1. Then we may view $P$ as embedded in the
lattice $\mathbb{N}^{2}$ as we did the poset $[a]\times[b]$ immediately before
Lemma 1.5, because two elements $p_{1}=(x_{1},y_{1})$ and
$p_{2}=(x_{2},y_{2})$ in $P$ satisfy $p_{1}\leq_{\mathcal{C}}p_{2}$ if and
only if $x_{1}\leq x_{2}$ and $y_{1}\leq y_{2}$. This implies that we can
apply Lemma 1.5 to $P$.

Every antichain of $P$ of size $k$ can be written uniquely as

$A=\\{(x_{1},y_{1}),\ (x_{2},y_{2}),\ \ldots,\ (x_{k},y_{k})\\},$

where $x_{1}<x_{2}<\cdots<x_{k}$. Lemma 1.5 now implies that
$y_{1}>y_{2}>\cdots>y_{k}$. Furthermore, by assumption we have $x_{k}<y_{k}$,
therefore we have

$x_{1}<x_{2}<\dots<x_{k}<y_{k}<\dots<y_{2}<y_{1}.$

It follows that there is a function
$\phi:\mathcal{A}_{k}(P)\rightarrow\mathcal{C}(n+2,2k)$ given by

$\phi(A)=(x_{1},x_{2},\ldots,x_{k},y_{k},\ldots,y_{2},y_{1}).$

Furthermore, $\phi$ is a bijection, because the assignment

$(x_{1},x_{2},\dots,x_{k},y_{k},y_{k-1},\dots,y_{1})\mapsto\\{(x_{1},y_{1}),\dots,(x_{k},y_{k})\\}$

takes $\mathcal{C}(n+2,2k)$ to $\mathcal{A}_{k}(P)$ by Lemma 1.5 and is
clearly a two-sided inverse of $\phi$.

We claim that $\phi$ is an isomorphism of posets. To see this, it is enough to
prove that $\phi$ respects covering relations. Let $A\in\mathcal{A}_{k}(P)$,
and write $A=\\{(x_{1},y_{1}),\ (x_{2},y_{2}),\ \ldots,\ (x_{k},y_{k})\\}$,
where $x_{1}<x_{2}<\cdots<x_{k}$ and $y_{1}>y_{2}>\cdots>y_{k}$.

Suppose that we have $A^{\prime}\lessdot_{\mathcal{A}_{k}(P)}A$ for some
$A^{\prime}\in\mathcal{A}_{k}(P)$. Proposition 2.2 (iv) implies that
$A^{\prime}$ can be obtained from $A$ by replacing one of the $(x_{i},y_{i})$
by $(x^{\prime}_{i},y^{\prime}_{i})$, where
$(x^{\prime}_{i},y^{\prime}_{i})\lessdot_{P}(x_{i},y_{i})$. By Lemma 1.2 (ii),
it follows that that we either have (a) $x_{i}^{\prime}=x_{i}$ and
$y_{i}^{\prime}=y_{i}-1$, or (b) $x_{i}^{\prime}=x_{i}-1$,
$y_{i}^{\prime}=y_{i}$. In the first case, we have
$x_{1}<\dots<x_{i}^{\prime}=x_{i}<\dots<x_{k}$, so that we must have

$x_{1}<\dots<x_{i}^{\prime}<\dots<x_{k}<y_{k}<\dots<y_{i}^{\prime}<\dots<y_{1}$

by the argument from the second paragraph of this proof. It follows that

$\displaystyle\varphi(A^{\prime})$ $\displaystyle=$
$\displaystyle(x_{1},\dots,x_{i},\dots,x_{k},y_{k},\dots,y_{i}^{\prime},\dots,y_{1})$
$\displaystyle<$
$\displaystyle(x_{1},\dots,x_{i},\dots,x_{k},y_{k},\dots,y_{i},\dots,y_{1})$
$\displaystyle=$ $\displaystyle\varphi(A)$

in $\mathcal{C}(n+2,2k)$. Moreoever, since $y_{i}^{\prime}=y_{i}-1$, the
relation $\varphi(A^{\prime})<\varphi(A)$ is a covering relation in
$\mathcal{C}(n+2,2k)$ by Lemma 1.2 (ii). Similarly, a symmetric argument shows
that in the second case we have

$\displaystyle\varphi(A^{\prime})$ $\displaystyle=$
$\displaystyle(x_{1},\dots,x_{i}^{\prime},\dots,x_{k},y_{k},\dots,y_{i},\dots,y_{1})$
$\displaystyle\lessdot$
$\displaystyle(x_{1},\dots,x_{i},\dots,x_{k},y_{k},\dots,y_{i},\dots,y_{1})$
$\displaystyle=$ $\displaystyle\varphi(A)$

in $\mathcal{C}(n+2,2k)$.

Conversely, suppose that $\varphi(A^{\prime})\lessdot_{\mathcal{C}}\varphi(A)$
for some $A^{\prime}\in\mathcal{A}_{k}(P)$. Lemma 1.2 (ii) implies that one of
the following conditions must hold for some $i\in[k]$:

  1. (1)

we have
$\varphi(A^{\prime})=(x_{1},\dots,x_{i}^{\prime}=x_{i}-1,\dots,x_{k},y_{k},\dots,y_{1})$;

  2. (2)

we have
$\varphi(A)=(x_{1},\dots,x_{k},y_{k},\dots,y_{i}^{\prime}=y_{i}-1,\dots,y_{1})$.

In the first case, we can use the inverse of $\varphi$ mentioned earlier to
recover $A^{\prime}$ as the set

$A^{\prime}=\\{(x_{j},y_{j}):j\in[k],j\neq
i\\}\cup\\{(x^{\prime}_{i},y_{i})\\},$

where $(x^{\prime}_{i},y_{i})\lessdot_{P}(x_{i},y_{i})$ by Lemma 1.2 (ii),
therefore $A^{\prime}\lessdot_{\mathcal{A}_{k}(P)}A$ by Proposition 2.2 (iv).
A similar argument shows that $A^{\prime}\lessdot_{\mathcal{A}_{k}(P)}A$ in
the second case, and we are done. ∎

######  Theorem 4.2.

Let $P$ be an irreducible minuscule poset, let $k\in\mathbb{N}$, and suppose
$k\leq\operatorname{\mathrm{width}}(P)$.

  * (i)

If $P\cong[a]\times[b]$, then we have
$\mathcal{A}_{k}(P)\cong\mathcal{C}(a,k)\times\mathcal{C}(b,k)$ as posets.

  * (ii)

If $P\cong J([n]\times[2])$, then we have
$\mathcal{A}_{k}(P)\cong\mathcal{A}_{k}(\mathcal{C}(n+2,2))\cong\mathcal{C}(n+2,2k)$
as posets.

  * (iii)

If $P\cong J^{m}([2]\times[2])$ where $m\in\mathbb{N}$, then
$\mathcal{A}_{k}(P)$ is a singleton if $k\in\\{0,2\\}$, and
$\mathcal{A}_{k}(P)$ is isomorphic to $P$ if $k=1$.

  * (iv)

If $P\cong J^{2}([2]\times[3])$, then $\mathcal{A}_{k}(P)$ is a singleton if
$k=0$, $\mathcal{A}_{k}(P)$ is isomorphic to $P$ if $k=1$, and
$\mathcal{A}_{k}(P)$ is isomorphic to $J^{3}([2]\times[2])$ if $k=2$.

  * (v)

If $P\cong J^{3}([2]\times[3])$, then $\mathcal{A}_{k}(P)$ is a singleton if
$k=0$ or $k=3$, and $\mathcal{A}_{k}(P)$ is isomorphic to $P$ if $k=1$ or
$k=2$.

  * (vi)

The poset $\mathcal{A}_{k}(P)$ is a distributive lattice under the order
$\leq_{k}$.

###### Proof.

Part (i) is a restatement of Proposition 3.1, and part (ii) is a restatement
of Proposition 4.1.

A straightforward induction on $m$ shows that the Hasse diagram of the poset
$P_{m}=J^{m}([2]\times[2])$ for $m\in\mathbb{N}$ is as shown in Figure 1 (c).
In particular, the width of $P_{m}$ is 2 for all $m\geq 0$, which is achieved
by a unique antichain of size 2. Part (iii) then follows from Remark 2.3 (i).

The cases $k=0$ and $k=1$ of parts (iv) and (v) are again covered by Remark
2.3 (i). Direct computation shows that if $P=J^{2}([2]\times[3])$, then the
width of $P$ is $2$ and $\mathcal{A}_{2}(P)$ is isomorphic to the poset
$J^{3}([2]\times[2])$, which completes the proof of (iv). Direct computation
also shows that if $P=J^{3}([2]\times[3])$, then the width of $P$ is 3, that
$\mathcal{A}_{3}(P)$ is the singleton poset, and that $\mathcal{A}_{2}(P)$ is
isomorphic to $P$; see Figure 2. This completes the proof of (v).

Recall that the singleton poset, posets of the form $J(P)$ where $P$ is a
finite poset, and products of distributive lattices are distributive lattices,
as are posets of the form $\mathcal{C}(n,k)$ (by Corollary 1.10). By (i)–(v),
every poset of the form $\mathcal{A}_{k}(P)$ where $P$ is a minuscule poset
has one of the forms described above, and (vi) follows. ∎

$a$$b$$c$$d$$e$$f$$g$$h$$i$$j$$k$$l$$m$$n$$o$$p$$q$$r$$s$$t$$u$$v$$w$$x$$y$$z$$\theta$
(a)

$\cong$ (b)

$ef$$eh$$gh$$ih$$ij$$kh$$kj$$mh$$kl$$mj$$ko$$ml$$no$$mo$$mn$$po$$mq$$pq$$ms$$ps$$mu$$rs$$pu$$ru$$tu$$vu$$vw$
(c)

Figure 2. Isomorphism between $P=J^{3}([2]\times[3])$ and
$\mathcal{A}_{2}(P)$, with each element
$\\{\alpha,\beta\\}\in\mathcal{A}_{2}(P)$ written as $\alpha\beta$.

##  5\. Connection to minuscule representations

Theorem 4.2 is interesting partly because for each minuscule poset $P$, the
posets $\mathcal{A}_{k}(P)$ appear as the weight posets for a suitable
restriction of a minuscule representation of a simple Lie algebra. To be more
precise, let ${\mathfrak{g}}$ be a simply laced simple Lie algebra over
${\mathbb{C}}$ of rank $n$ with Serre generators $e_{i},f_{i},h_{i}$ $(1\leq
i\leq n)$, labelled in the convention of [7, Chapter 4]. Let $\omega_{p}$ be a
fundamental weight for ${\mathfrak{g}}$ that is minuscule, and let $L_{p}$ be
the irreducible ${\mathfrak{g}}$-module with highest weight $\omega_{p}$. Then
the possible values of $p$ and the poset of weights in $L_{p}$ are known; see,
for example, Theorem 5.1.5 and Theorem 8.3.10 (v) of [5]. In particular,
minuscule fundamental weights exist in all simply laced types except type
$E_{8}$, and the weight poset of $L_{p}$ is of the form $J(P)$ for a unique
minuscule poset $P$. We summarize these known results in Table 1 below.

Type |  $p$ | Isomorphism Type of $P$
---|---|---
$A_{n}$ |  $k(k\in[n])$ | $[k]\times[n+1-k]$
$D_{n}$ |  $1$ | $J^{n-3}([2]\times[2])$
$D_{n}$ |  $n-1$ or $n$ | $J([n-2]\times[2])$
$E_{6}$ | 1 or 5 | $J^{2}([2]\times[3])$
$E_{7}$ | 6 | $J^{3}([2]\times[3])$
Table 1. Minuscule representations of simply laced simple Lie algebras

Let ${\mathfrak{k}}$ be the subalgebra of ${\mathfrak{g}}$ generated by the
set $\\{e_{i},f_{i},h_{i}:1\leq i\leq n,i\neq p\\}$ and consider the
restriction $L_{P}\\!\downarrow_{\mathfrak{k}}$ of the module $L_{p}$ to
${\mathfrak{k}}$. Then the decomposition of
$L_{P}\\!\downarrow_{\mathfrak{k}}$ into simple components is described in [5,
Proposition 8.2.9 (iv)], in a uniform way. The main result of this section is
that the weight posets of the simple components are exactly the posets
$\mathcal{A}_{k}(P)$:

######  Theorem 5.1.

Let ${\mathfrak{g}},p,L_{p},{\mathfrak{k}},P$ and
$L_{P}\\!\downarrow_{\mathfrak{k}}$ be as described above, with the poset of
the weights of $L_{p}$ being isomorphic to $J(P)$. Then the
${\mathfrak{k}}$-module $L_{P}\\!\downarrow_{\mathfrak{k}}$ decomposes into a
direct sum

$L_{P}\\!\downarrow_{\mathfrak{k}}\cong\bigoplus_{i=0}^{k}V_{i}$

where $k=\operatorname{\mathrm{width}}(P)$ and $V_{i}$ is a simple module of
${\mathfrak{k}}$ whose poset of weights is isomorphic to $\mathcal{A}_{i}(P)$
for each integer $0\leq i\leq k$.

###### Proof.

We will prove the theorem by case discussion and use the descriptions of the
weight posets of the simple summands of $L_{P}\\!\downarrow_{\mathfrak{k}}$
from [5, Proposition 8.2.9 (iv)]. We denote by $L_{A,n,p}$ the type $A_{n}$
minuscule representation with highest weight $\omega_{p}$ for $1\leq p\leq n$,
and define $L_{A,n,p}$ to be the trivial module if $p\in\\{0,n+1\\}$.

Suppose that ${\mathfrak{g}}$ is of type $A_{n}$, $p\in[n]$, and
$P=[p]\times[n+1-p]$. Then the weight poset of each direct summand of
$L_{P}\\!\downarrow_{\mathfrak{k}}$ is given by the “ideals of $U_{p,i}$
containing $D_{p,i-1}$” under the containment order. After translating
notation, this is the poset $\mathcal{D}_{i}([p]\times[n+1-p])$ of the Ferrers
diagrams of Durfee length $i$, ordered by containment. The proof in type
$A_{n}$ is now completed by Corollary 3.2.

Suppose that ${\mathfrak{g}}$ is of type $D_{n}$, $p=1$, and
$P=J^{n-3}([2]\times[2])$. Then $L_{p}$ is the natural representation of
dimension $2n$ in type $D_{n}$. It is known (see [5, Exercise 8.2.16 (iii)])
that ${\mathfrak{k}}$ is of type $D_{n-1}$ and that $L_{\omega_{p}}$ restricts
to the direct sum of three ${\mathfrak{k}}$-modules: two copies of the trivial
representation, and one copy of the natural representation of
${\mathfrak{k}}$. The proof in this case is completed by Theorem 4.2 (iii).

Suppose that ${\mathfrak{g}}$ is of type $D_{n}$, $p\in\\{n-1,n\\}$, and
$P=J([n-2]\times[2])$. Then $L_{p}$ is one of the half spin representations in
type $D_{n}$ of highest weights $\omega_{n-1}$ and $\omega_{n}$. These two
representations are interchanged by an automorphism of the Dynkin diagram, so
it suffices to consider the case of $\omega_{n}$. By Proposition 4.1 and
Corollary 1.8, we have

$\mathcal{A}_{k}(P)\cong\mathcal{C}(n,2k)\cong J([n-2k]\times[2k]),$

which is the weight poset of $L_{A,n-1,2k}$. It is known (see [5, Exercise
8.2.15]) that ${\mathfrak{k}}$ is of type $A_{n-1}$ in this case, and we have
an isomorphism of ${\mathfrak{k}}$-modules

$L_{\omega_{n}}\\!\\!\downarrow_{{\mathfrak{k}}}\cong\bigoplus_{i=0}^{\lfloor
n/2\rfloor}L_{A,n-1,2i}.$

This completes the proof in this case.

Suppose that ${\mathfrak{g}}$ is of type $E_{6}$, $p\in\\{1,5\\}$, and
$P=J^{2}([2]\times[3])$. Then $L_{p}$ is one of the two 27-dimensional
minuscule representations in type $E_{6}$. It is known (see [5, Exercise
8.2.17]) that ${\mathfrak{k}}$ is of type $D_{5}$ and that $L_{\omega_{p}}$
restricts to the direct sum of three ${\mathfrak{k}}$-modules: the trivial
module; one of the 16-dimensional half-spin representations, whose weight
poset is $J^{2}([2]\times[3])$; and the 10-dimensional natural representation,
whose weight poset is $J^{3}([2]\times[2])$. The proof is completed by Theorem
4.2 (iv).

Finally, suppose that ${\mathfrak{g}}$ is of type $E_{7}$, $p=6$, and
$P=J^{2}([2]\times[3])$. Then $L_{p}$ is the 56-dimensional simple
representation in type $E_{7}$. It is known (see [5, Exercise 8.2.18]) that
${\mathfrak{k}}$ is of type $E_{6}$ and that $L_{\omega_{p}}$ restricts to the
direct sum of four ${\mathfrak{k}}$-modules: two copies of the trivial module,
and one copy of each of the two 27-dimensional minuscule representations, each
of whose weight posets is $J^{3}([2]\times[3])$. The proof is completed by
Theorem 4.2 (v), and we are done.∎

######  Remark 5.2.

The uniformly described direct sum decomposition of
$L_{p}\\!\\!\downarrow_{\mathfrak{k}}$ of [5, Proposition 8.2.9 (iv)], depends
heavily on a labelling of the poset $P$ by vertices of the Dynkin diagram. It
is remarkable that the description of Theorem 5.1 relies only on the structure
of the underlying poset $P$.

##  6\. Concluding remarks

We discuss a few open problems concerning the partial order $\leq_{k}$ in this
section. Firstly, it would be interesting to have a conceptual, case-free
proof of Theorem 5.1. Secondly, apart from the minuscule poset setting, it may
be interesting to study the structure of posets of the form
$\mathcal{A}_{k}(\Phi^{+})$ where $\Phi^{+}$ is a root poset, i.e., the poset
of positive roots of a Weyl group $W$. Thirdly, if $W$ has rank $r$, then the
so-called Narayana numbers $\lvert\mathcal{A}_{k}(\Phi^{+})\rvert$ are
symmetric in the sense that
$\lvert\mathcal{A}_{k}(\Phi^{+})\rvert=\lvert\mathcal{A}_{r-k}(\Phi^{+})\rvert$
for any $0\leq k\leq r$ (see [1]), so it would also be interesting to know
whether this symmetry can be realized by a poset isomorphism between
$\mathcal{A}_{k}(\Phi^{+})$ and $\mathcal{A}_{r-k}(\Phi^{+})$.

Bijections realizing the symmetry
$\lvert\mathcal{A}_{k}(\Phi^{+})\rvert=\lvert\mathcal{A}_{r-k}(\Phi^{+})\rvert$
have been studied before. For root poset $\Phi^{+}$ of type $A_{n-1}\,(n\geq
2)$, Panyushev constructed in [8] an involution $*$ on the set
$\mathcal{A}(\Phi^{+})$ of all antichains of $\Phi^{+}$ with certain “natural”
properties, one of which is that it restricts to bijections between
$\mathcal{A}_{k}(\Phi^{+})$ and $\mathcal{A}_{n-1-k}(\Phi^{+})$. If we write
$\Phi^{+}=\\{\epsilon_{i}-\epsilon_{j}:1\leq i<j\leq
n\\}\subseteq{\mathbb{R}}^{n}$ and write $[i,j]$ for each element
$\epsilon_{i}-\epsilon_{j}\in\Phi^{+}$, then for each antichain
$A=\\{[i_{1},j_{1}],\dots,[i_{k},j_{k}]\\}$ in $\mathcal{A}_{k}(\Phi^{+})$,
the antichain $A^{*}$ is the unique element in $\mathcal{A}_{n-1-k}$
consisting of elements
$[i^{\prime}_{1},j^{\prime}_{1}],\dots,[i^{\prime}_{n-1-k},j^{\prime}_{n-1-k}]$
where

$\\{i^{\prime}_{1},\dots,i^{\prime}_{n-1-k}\\}=\\{1,2,\dots,n-1\\}\setminus\\{j_{1}-1,\dots,j_{k}-1\\},$
$\\{j^{\prime}_{1},\dots,j^{\prime}_{n-1-k}\\}=\\{2,3,\dots,n\\}\setminus\\{i_{1}+1,\dots,i_{k}+1\\}$

as sets and

$i^{\prime}_{1}<\dots<i^{\prime}_{n-1-k},\quad
j^{\prime}_{1}<\dots<j^{\prime}_{n-1-k}.$

Using Proposition 2.2 (iv) and the fact that $[i,j]\lessdot[l,m]$ in
$\Phi^{+}$ if and only if either $i=l,m=j+1$ or $l=i-1,j=m$, it is
straightforward to verify that if $A^{\prime}\lessdot A$ for some antichain
$A^{\prime}\in\mathcal{A}_{k}(\Phi^{+})$, then $A^{\prime*}\lessdot A^{*}$ in
$\mathcal{A}_{n-1-k}(\Phi^{+})$. Since the map $*$ is an involution, it
follows that it is indeed a poset isomorphism between
$\mathcal{A}_{k}(\Phi^{+})$ and $\mathcal{A}_{n-1-k}(\Phi^{+})$.

For root posets of arbitrary types, Panyushev also proposed in [8] a program
of finding an involution on $\mathcal{A}(\Phi^{+})$ satisfying the natural
properties of $*$; these properties are summarized in Conjecture 2.11 of [1].
In the latter paper, Defant and Hopkins prove that for root systems of types
$A,B,C$ and $D$, a so-called rowvacuation operator satisfies Panyushev’s
desired properties and recovers the map $*$ in type $A$. However, we note that
while rowvacuation provides bijections between $\mathcal{A}_{k}(\Phi^{+})$ and
$\mathcal{A}_{r-k}(\Phi^{+})$, it does not give a poset isomorphism between
these posets in types $B,C$ and $D$. To see this, recall that $\Phi^{+}$ is a
ranked poset. Let $R$ be the rank of $\Phi^{+}$ and let $\Phi^{+}_{i}$ be the
antichain in $\Phi^{+}$ consisting of all elements of rank $i$ for each $0\leq
i\leq R$. Then in types $B,C$ and $D$, both $\Phi^{+}_{R}$ and
$\Phi^{+}_{R-1}$ are singletons satisfying
$\Phi^{+}_{R-1}\lessdot\Phi^{+}_{R}$ in $\mathcal{A}_{1}(\Phi^{+})$. On the
other hand, rowvacuation sends $\Phi^{+}_{R-1}$ and $\Phi^{+}_{R}$ to
$\Phi^{+}_{2}$ and $\Phi^{+}_{1}$, respectively, by Proposition 2.9 of [1],
yet $\Phi^{+}_{2}$ and $\Phi^{+}_{1}$ are elements of
$\mathcal{A}_{r-1}(\Phi^{+})$ that are not in a covering relation.

## Acknowledgements

We thank Darij Grinberg for telling us about Gale orders. We also thank Hugh
Thomas for helpful discussions.

## References

  * [1] C. Defant and S. Hopkins.  Symmetry of Narayana numbers and rowvacuation of root posets.  Forum Math. Sigma, 9:Paper No. e53, 24, 2021.
  * [2] R.P. Dilworth.  Some combinatorial problems on partially ordered sets.  In Proc. Sympos. Appl. Math., Vol. 10, pages 85–90. American Mathematical Society, 1960.
  * [3] D. Gale.  Optimal assignments in an ordered set: An application of matroid theory.  J. Combinatorial Theory, 4:176–180, 1968.
  * [4] R.M. Green.  Star reducible Coxeter groups.  Glasg. Math. J., 48(3):583–609, 2006.
  * [5] R.M. Green.  Combinatorics of minuscule representations, volume 199.  Cambridge University Press, 2013.
  * [6] R.M. Green and Tianyuan Xu.  Kazhdan–Lusztig cells of a-value 2 in a(2)-finite Coxeter systems.  Algebraic Combinatorics, 6(3):727–772, 2023.
  * [7] V.G. Kac.  Infinite-dimensional Lie algebras.  Cambridge University Press, 1990.
  * [8] D.I. Panyushev.  ad-nilpotent ideals of a Borel subalgebra: generators and duality.  J. Algebra, 274(2):822–846, 2004.
  * [9] R.A. Proctor.  A Dynkin diagram classification theorem arising from a combinatorial problem.  Adv. in Math., 62(2):103–117, 1986.
  * [10] R.P. Stanley.  Enumerative combinatorics. Vol. 1, volume 49 of Cambridge Studies in Advanced Mathematics.  Cambridge University Press, Cambridge, 1997.  With a foreword by Gian-Carlo Rota, Corrected reprint of the 1986 original.
  * [11] J.R. Stembridge.  On the fully commutative elements of Coxeter groups.  J. Algebraic Combin., 5(4):353–385, 1996.
  * [12] A. Vince.  The greedy algorithm and Coxeter matroids.  J. of Algebraic Combin., 11(2):155–178, 2000.