Compatible Algebras with Straightening Laws on Distributive Lattices

Abstract

We characterize the finite distributive lattices on which there exists a unique compatible algebra with straightening laws.

1. Introduction

Let P be a finite partially ordered set (poset for short) and $\mathcal{I}\left(P\right)$ the distributive lattice of the poset ideals of $P.$ A subset $\alpha$ of P is a poset ideal of P if it satisfies the following condition: for every $x\in \alpha$ and $y\in P$, if $y\le x$, then $y\in \alpha .$ By a famous theorem of Birkhoff [1], for every finite distributive lattice L, there exists a unique subposet P of L such that $L\cong \mathcal{I}\left(P\right).$ The order polytope $\mathcal{O}\left(P\right)$ and the chain polytope $\mathcal{C}\left(P\right)$ were introduced in [2]. In [3], it was shown that the toric ring $K\left[\mathcal{O}\right(P\left)\right]$ over a field K is an algebra with straightening laws (ASL in brief) on the distributive lattice $\mathcal{I}\left(P\right)$ over the field $K.$ In [4], it was shown that the ring $K\left[\mathcal{C}\right(P\left)\right]$ associated with the chain polytope shares the same property.

Let $S=K[x_1,\dots ,x_n,t]$ be the polynomial ring over a field K and ${\left\{w_{\alpha }\right\}}_{\alpha \in \mathcal{I}\left(P\right)}$ be an arbitrary set of monomials in $x_1,\dots ,x_n$ indexed by $\mathcal{I}\left(P\right)$. Let $K\left[\mathsf{\Omega }\right]\subset S$ be the toric ring generated over K by the set of monomials $\mathsf{\Omega }={\left\{{\omega }_{\alpha }\right\}}_{\alpha \in \mathcal{I}\left(P\right)}$ where ${\omega }_{\alpha }=w_{\alpha }t$ for all $\alpha \in \mathcal{I}\left(P\right).$ Clearly, $K\left[\mathsf{\Omega }\right]$ is a graded algebra if we set $deg\left({\omega }_{\alpha }\right)=1$ for all $\alpha \in \mathcal{I}\left(P\right).$ Let $\phi :\mathcal{I}\left(P\right)\to K\left[\mathsf{\Omega }\right]$ be the injective map defined by $\phi \left(\alpha \right)={\omega }_{\alpha }$ for all $\alpha \in \mathcal{I}\left(P\right).$ Assume that $K\left[\mathsf{\Omega }\right]$ is an ASL on $\mathcal{I}\left(P\right)$ over $K.$ According to [4], $K\left[\mathsf{\Omega }\right]$ is a compatible ASL if each of its straightening relations is of the form $\phi \left(\alpha \right)\phi \left({\alpha }^{\prime }\right)=\phi \left(\beta \right)\phi \left({\beta }^{\prime }\right)$ with $\beta \subseteq \alpha \cap {\alpha }^{\prime }$ and ${\beta }^{\prime }\supseteq \alpha \cup {\alpha }^{\prime },$ where $\alpha ,{\alpha }^{\prime }$ are incomparable elements in $\mathcal{I}\left(P\right).$ If $K\left[\mathsf{\Omega }\right]$ and $K\left[{\mathsf{\Omega }}^{\prime }\right]$ are compatible ASL on $\mathcal{I}\left(P\right)$ over $K,$ we identify them if they have the same straightening relations. In this case, we write $K\left[\mathsf{\Omega }\right]\equiv K\left[{\mathsf{\Omega }}^{\prime }\right].$

In ([4], Question 5.1), Hibi and Li asked the following questions:

(a)

Given a finite poset P, find all possible compatible algebras with straightening laws on $\mathcal{I}\left(P\right)$ over $K.$

(b)

For which posets P, does there exist a unique compatible ASL on $\mathcal{I}\left(P\right)$ over $K?$

In this note, we give a complete answer to question (b). Namely, we prove the following:

Theorem 1. 

Let P be a finite poset. Then, the following statements are equivalent:

(i)

There exists a unique compatible ASL on$\mathcal{I}\left(P\right)$over$K.$

(ii)

$K\left[\mathcal{O}\left(P\right)\right]\equiv K\left[\mathcal{C}\left(P\right)\right]\equiv K\left[\mathcal{C}\left(P^{*}\right)\right],$where$P^{*}$denotes the dual poset of$P.$

(iii)

Each connected component of P is a chain, that is, P is a direct sum of chains.

An answer to question (a) seems to be quite difficult. In ([4], Example 5.2), it was observed that, if one considers the poset $P=\{a,b,c,d,\mathbf{e}\}$ with $a<c<\mathbf{e}$ and $b<c<d,$ then there exist nine compatible ASL structures on $\mathcal{I}\left(P\right)$ over $K,$ while if one considers $P=\{a,b,c,d\}$ with $a<c$, $b<c,b<d$, then there are three compatible ASL structures on $\mathcal{I}\left(P\right)$ over $K,$ namely, $K\left[\mathcal{O}\right(P\left)\right],$ $K\left[\mathcal{C}\right(P\left)\right]$, and $K\left[\mathcal{C}\left(P^{*}\right)\right].$

2. Order Polytopes, Chain Polytopes, and Their Associated Toric Rings

Let $P=\{p_1,\dots ,p_n\}$ be a finite poset. For the basic terminology regarding posets used in this paper, we refer to [1] and ([5], Chapter 3). The order polytope $\mathcal{O}\left(P\right)$ is defined as

$$\mathcal{O}\left(P\right)=\{(x_1,\dots ,x_n)\in {\mathbb{R}}^n:0\le x_i\le 1,1\le i\le n,\mathrm{and}{x}_i\ge x_j\mathrm{if}{p}_i\le p_j\mathrm{in}P\}.$$

In ([2], Corollary 1.3), it was shown that the vertices of $\mathcal{O}\left(P\right)$ are ${\sum }_{p_i\in \alpha }{\mathbf{e}}_i,\alpha \in \mathcal{I}\left(P\right).$ Here, ${\mathbf{e}}_i$ denotes the unit coordinate vector in ${\mathbb{R}}^n.$ If $\alpha =\varnothing ,$ then the corresponding vertex in $\mathcal{O}\left(P\right)$ is the origin of ${\mathbb{R}}^n.$

The chain polytope $\mathcal{C}\left(P\right)$ is defined as

$$\mathcal{C}\left(P\right)=\{(x_1,\dots ,x_n)\in {\mathbb{R}}^n:x_i\ge 0,1\le i\le n,$$

$$x_{i_1}+\cdots +x_{i_r}\le 1\mathrm{if}{p}_{i_1}<\cdots <p_{i_r}\mathrm{is}\mathrm{a}\mathrm{maximal}\mathrm{chain}\mathrm{in}P\}.$$

In ([2], Theorem 2.2), it was proved that the vertices of $\mathcal{C}\left(P\right)$ are ${\sum }_{p_i\in A}{\mathbf{e}}_i$, where A is an antichain in $P.$ Recall that an antichain in P is a subset of P such that any two distinct elements in the subset are incomparable. Since every poset ideal is uniquely determined by its antichain of maximal elements, it follows that $\mathcal{O}\left(P\right)$ and $\mathcal{C}\left(P\right)$ have the same number of vertices. However, as it was observed in [2], $\mathcal{O}\left(P\right)$ and $\mathcal{C}\left(P\right)$ need not have the same number of i-dimensional faces for $i>0.$ Therefore, in general, they are not combinatorial equivalent. Combinatorially, equivalence of order and chain polytopes are studied in [6].

The Toric Rings $K\left[\mathcal{O}\right(P\left)\right]$ and $K\left[\mathcal{C}\right(P\left)\right]$

To each subset $W\subset P$, we attach the squarefree monomial $u_W\in K[x_1,\dots ,x_n],$ $u_W={\prod }_{p_i\in W}{x}_i.$ If $W=\varnothing ,$ then $u_W=1.$ The toric ring $K\left[\mathcal{O}\right(P\left)\right]$, known as the Hibi ring associated with the distributive lattice $\mathcal{I}\left(P\right),$ is generated over K by all the monomials $u_{\alpha }t\in S$, where $\alpha \in \mathcal{I}\left(P\right).$ The toric ring $K\left[\mathcal{C}\right(P\left)\right]$ is generated by all the monomials $u_{A}t$ where A is an antichain in $P.$ In addition, as we have already mentioned in the Introduction, both rings are algebras with straightening laws on $\mathcal{I}\left(P\right)$ over $K.$

We recall the definition of an ASL as it was introduced in [7]. For a quick introduction to this topic, we refer to [7] and ([8], Chapter XIII). Algebras with straightening laws turned out to be useful tools in studying determinantal rings. Let K be a field, $R={⨁}_{i\ge 0}{R}_i$ with $R_0=K$ be a graded K-algebra, H a finite poset, and $\phi :H\to R$ an injective map which maps each $\alpha \in H$ to a homogeneous element $\phi \left(\alpha \right)\in R$ with $deg\phi \left(\alpha \right)\ge 1$. A standard monomial in R is a product $\phi \left({\alpha }_1\right)\phi \left({\alpha }_2\right)\cdots \phi \left({\alpha }_k\right)$ where ${\alpha }_1\le {\alpha }_2\le \cdots \le {\alpha }_k$ in $H.$

Definition 1. 

The K-algebra R is called an algebra with straightening laws on H over K if the following conditions hold:

(1) 

The set of standard monomials is a K–basis of$R;$

(2) 

If$\alpha ,\beta \in H$are incomparable and if$\phi \left(\alpha \right)\phi \left(\beta \right)=\sum c_i\phi \left({\gamma }_{i1}\right)\dots \phi \left({\gamma }_{i{k}_i}\right),$where$c_i\in K\backslash \left\{0\right\}$and${\gamma }_{i1}\le \dots \le {\gamma }_{i{k}_i},$is the unique expression of$\phi \left(\alpha \right)\phi \left(\beta \right)$as a linear combination of standard monomials, then${\gamma }_{i1}\le \alpha ,\beta$for all$i.$

The above relations $\phi \left(\alpha \right)\phi \left(\beta \right)=\sum c_i\phi \left({\gamma }_{i1}\right)\dots \phi \left({\gamma }_{i{k}_i}\right)$ are called the straightening relations of R and they generate all the relations of of $R.$

Let us go back to the toric rings $K\left[\mathcal{O}\right(P\left)\right]$ and $K\left[\mathcal{C}\right(P\left)\right].$

One considers $\phi :\mathcal{I}\left(P\right)\to K\left[\mathcal{O}\right(P\left)\right]$ defined by $\phi \left(\alpha \right)=u_{\alpha }t$ for every $\alpha \in \mathcal{I}\left(P\right).$ As it was proved by Hibi in [3], $K\left[\mathcal{O}\right(P\left)\right]$ is an ASL on $\mathcal{I}\left(P\right)$ over K with the straightening relations $\phi \left(\alpha \right)\phi \left(\beta \right)=\phi (\alpha \cap \beta )\phi (\alpha \cup \beta ),$ where $\alpha ,\beta$ are incomparable elements in $\mathcal{I}\left(P\right).$

On the other hand, one defines $\psi :\mathcal{I}\left(P\right)\to K\left[\mathcal{C}\right(P\left)\right]$ by setting $\psi \left(\alpha \right)=u_{max\alpha }t$ for all $\alpha \in \mathcal{I}\left(P\right)$ where $max\alpha$ denotes the set of the maximal elements in $\alpha .$ Note that, for every $\alpha \in \mathcal{I}\left(P\right),$ $max\alpha$ is an antichain in P and each antichain $A\subset P$ determines a unique ideal $\alpha \in \mathcal{I}\left(P\right),$ namely, the poset ideal generated by $A.$ Therefore, $\psi$ is an injective well defined map and by ([4], Theorem 3.1), the ring $K\left[\mathcal{C}\right(P\left)\right]$ is an ASL on $\mathcal{I}\left(P\right)$ over K with the straightening relations

$$\psi \left(\alpha \right)\psi \left(\beta \right)=\psi (\alpha \ast \beta )\psi (\alpha \cup \beta ),$$

where $\alpha \ast \beta$ is the poset ideal of P generated by $max(\alpha \cap \beta )\cap (max\alpha \cup max\beta ).$

We observe that one may also consider $K\left[\mathcal{C}\left(P^{*}\right)\right]$ as an ASL on $\mathcal{I}\left(P\right)$, where $P^{*}$ is the dual poset of $P.$ We may define $\delta :\mathcal{I}\left(P\right)\to K\left[C\left(P^{*}\right)\right]$ by $\delta \left(\alpha \right)=u_{min\overline{\alpha }}t$ for $\alpha \in \mathcal{I}\left(P\right),$ where $min\overline{\alpha }$ is the set of minimal elements in $\overline{\alpha }$ and $\overline{\alpha }$ is the filter $P\backslash \alpha$ of $P.$ We recall that a filter $\gamma$ in P (or dual order ideal) is a subset of P with the property that for every $p\in \gamma$ and every $q\in P$ with $q\ge p,$ we have $q\in \gamma .$ Thus, a filter in P is simply a poset ideal in the dual poset $P^{*}.$ The ring $K\left[C\left(P^{*}\right)\right]$ is an ASL on $\mathcal{I}\left(P\right)$ over K as well with the straightening relations

$$\delta \left(\alpha \right)\delta \left(\beta \right)=\delta (\alpha \cap \beta )\delta (\alpha \circ \beta )$$

for incomparable elements $\alpha ,\beta \in \mathcal{I}\left(P\right),$ where $\alpha \circ \beta$ is the poset ideal of P which is the complement in P of the filter generated by $min(\overline{\alpha }\cap \overline{\beta })\cap (min\overline{\alpha }\cup min\overline{\beta }).$ Let us also observe that all the algebras $K\left[\mathcal{O}\right(P\left)\right],K\left[\mathcal{C}\right(P\left)\right]$, and $K\left[\mathcal{C}\left(P^{*}\right)\right]$ are compatible algebras with straightening laws.

3. Proof of Theorem 1

We clearly have (i) ⇒ (ii). Let us now prove (ii) ⇒ (iii). By hypothesis, the straightening relations of $K\left[\mathcal{O}\right(P\left)\right],K\left[\mathcal{C}\right(P\left)\right],$ and $K\left[\mathcal{C}\left(P^{*}\right)\right]$ coincide. Therefore, we must have

$$\alpha \cap \beta =\alpha \ast \beta \mathrm{and}\overline{\alpha }\cap \overline{\beta }=\overline{\alpha \circ \beta }$$

(1)

for all $\alpha ,\beta$ incomparable elements in $\mathcal{I}\left(P\right).$ From the second equality in (1), it follows that $\overline{\alpha }\cap \overline{\beta }$ is the filter of P generated by $min(\overline{\alpha }\cap \overline{\beta })\cap (min\overline{\alpha }\cup min\overline{\beta }).$ Assume that there exists two incomparable elements $p,p^{\prime }\in P$ such that there exists $q\in P$ with $q>p$ and $q>p^{\prime }.$ Consider $\overline{\alpha }$ the filter generated by p and $\overline{\beta }$ the filter generated by $p^{\prime }.$ Then, $min(\overline{\alpha }\cap \overline{\beta })\cap (min\overline{\alpha }\cup min\overline{\beta })=\varnothing$, but obviously, $\overline{\alpha }\cap \overline{\beta }\ne \varnothing .$ This shows that, for any two incomparable elements $p,p^{\prime }\in P,$ there is no upper bound for p and $p^{\prime }.$

Similarly, by using the first equality in Equation (1), we derive that, for any two incomparable elements $p,p^{\prime }\in P,$ there is no lower bound for p and $p^{\prime }$. This shows that every connected component of the poset P is a chain.

Finally, we prove (iii) ⇒ (i). Let P be a poset such that all its connected components are chains and assume that the cardinality of P is equal to $n.$ Let ${\left\{{\omega }_{\alpha }\right\}}_{\alpha \in \mathcal{I}\left(P\right)}$ be the generators of $K\left[\mathsf{\Omega }\right]\subset S$ and assume that the straightening relations of $K\left[\mathsf{\Omega }\right]$ are $\phi \left(\alpha \right)\phi \left({\alpha }^{\prime }\right)=\phi \left(\beta \right)\phi \left({\beta }^{\prime }\right)$ where $\beta \subseteq \alpha \cap {\alpha }^{\prime },$ ${\beta }^{\prime }\supseteq \alpha \cup {\alpha }^{\prime },$ and $\alpha ,{\alpha }^{\prime }$ are incomparable elements in $\mathcal{I}\left(P\right).$ We have to show that, for all $\alpha ,{\alpha }^{\prime }$ incomparable elements in $\mathcal{I}\left(P\right),$ we have $\beta =\alpha \cap {\alpha }^{\prime }$ and ${\beta }^{\prime }=\alpha \cup {\alpha }^{\prime }.$

We proceed by induction on

$$k=n-(rank(\alpha \cup {\alpha }^{\prime })-rank(\alpha \cap {\alpha }^{\prime })).$$

Let us recall that, if $\gamma \in \mathcal{I}\left(P\right),$ then $rank\gamma$ denotes the rank of the subposet of $\mathcal{I}\left(P\right)$ consisting of all elements $\delta \in \mathcal{I}\left(P\right)$ with $\delta \subseteq \gamma .$

If $k=0,$ that is, $rank(\alpha \cup {\alpha }^{\prime })-rank(\alpha \cap {\alpha }^{\prime })=n$, then $\alpha \cup {\alpha }^{\prime }=P$ and $\alpha \cap {\alpha }^{\prime }=\varnothing ,$ thus $\beta =\alpha \cap {\alpha }^{\prime }$ and ${\beta }^{\prime }=\alpha \cup {\alpha }^{\prime }.$ Assume that the desired conclusion is true for $rank(\alpha \cup {\alpha }^{\prime })-rank(\alpha \cap {\alpha }^{\prime })=n-k$ with $k\ge 0.$ Let us choose now $\alpha ,{\alpha }^{\prime }$ incomparable in $\mathcal{I}\left(P\right)$ such that $rank(\alpha \cup {\alpha }^{\prime })-rank(\alpha \cap {\alpha }^{\prime })=n-k-1$ and assume that we have a straightening relation $\phi \left(\alpha \right)\phi \left({\alpha }^{\prime }\right)=\phi \left(\beta \right)\phi \left({\beta }^{\prime }\right)$ with $\beta ⊊\alpha \cap {\alpha }^{\prime }$ or ${\beta }^{\prime }⊋\alpha \cup {\alpha }^{\prime }.$ By duality, we may reduce to considering ${\beta }^{\prime }⊋\alpha \cup {\alpha }^{\prime }.$ In other words, in $K\left[\mathsf{\Omega }\right],$ we have

$${\omega }_{\alpha }{\omega }_{{\alpha }^{\prime }}={\omega }_{\beta }{\omega }_{{\beta }^{\prime }},\mathrm{with}\beta \subseteq \alpha \cap {\alpha }^{\prime }\mathrm{and}{\beta }^{\prime }⊋\alpha \cup {\alpha }^{\prime }.$$

As P is a direct sum of chains, we may find $p\in max(\alpha \cup {\alpha }^{\prime })$ and $q\in {\beta }^{\prime }\backslash (\alpha \cup {\alpha }^{\prime })$ such that q covers p in P, that is, $q>p$ and there is no other element $q^{\prime }$ in P with $q>q^{\prime }>p.$ Without loss of generality, we may assume that $p\in {\alpha }^{\prime }.$ Let ${\alpha }_1$ be the poset ideal of P generated by ${\alpha }^{\prime }\cup \left\{q\right\}.$ As all the connected components of P are chains, we have ${\alpha }_1={\alpha }^{\prime }\cup \left\{q\right\}$ since there are no other elements in P which are smaller than q except those that are on the same chain as p and q, which are in ${\alpha }^{\prime }.$ Moreover, by the choice of $q,$ we have

$${\alpha }_1\subseteq {\beta }^{\prime }\mathrm{and}\alpha \cap {\alpha }_1=\alpha \cap ({\alpha }^{\prime }\cup \left\{q\right\})=\alpha \cap {\alpha }^{\prime }.$$

On the other hand,

$$rank(\alpha \cup {\alpha }_1)-rank(\alpha \cap {\alpha }_1)=rank(\alpha \cup {\alpha }^{\prime }\cup \left\{q\right\})-rank(\alpha \cap {\alpha }^{\prime })$$

$$=rank(\alpha \cup {\alpha }^{\prime })+1-rank(\alpha \cap {\alpha }^{\prime })=n-k.$$

By the inductive hypothesis, it follows that $\phi \left(\alpha \right)\phi \left({\alpha }_1\right)=\phi (\alpha \cap {\alpha }_1)\phi (\alpha \cup {\alpha }_1),$ or, equivalently, in $K\left[\mathsf{\Omega }\right]$ we have the equality ${\omega }_{\alpha }{\omega }_{{\alpha }_1}={\omega }_{\alpha \cap {\alpha }_1}{\omega }_{\alpha \cup {\alpha }_1}.$ Thus, we have obtained the following equalities in $K\left[\mathsf{\Omega }\right]:$

$${\omega }_{\alpha }{\omega }_{{\alpha }^{\prime }}={\omega }_{\beta }{\omega }_{{\beta }^{\prime }}\mathrm{and}{\omega }_{\alpha }{\omega }_{{\alpha }_1}={\omega }_{\alpha \cap {\alpha }^{\prime }}{\omega }_{\alpha \cup {\alpha }_1}.$$

This implies that

$${\omega }_{\alpha \cap {\alpha }^{\prime }}{\omega }_{{\alpha }^{\prime }}{\omega }_{\alpha \cup {\alpha }_1}={\omega }_{\beta }{\omega }_{{\alpha }_1}{\omega }_{{\beta }^{\prime }}.$$

(2)

In addition, we have:

$$\alpha \cap {\alpha }^{\prime }\subset {\alpha }^{\prime }\subset \alpha \cup {\alpha }_1\mathrm{and}\beta \subset \alpha \cap {\alpha }^{\prime }=\alpha \cap {\alpha }_1\subset {\alpha }_1\subset {\beta }^{\prime }.$$

This implies that the monomials in Equation (2) are distinct standard monomials in $K\left[\mathsf{\Omega }\right],$ which is in contradiction to the condition that the standard monomials form a K-basis in $K\left[\mathsf{\Omega }\right].$ Therefore, our proof is completed.