Combinatorics of quasi-hereditary structures

Abstract

A quasi-hereditary algebra is an Artin algebra together with a partial order on its set of isomorphism classes of simple modules which satisfies certain conditions. In this article we investigate all the possible choices that yield quasi-hereditary structures on a given algebra, in particular we introduce and study what we call the poset of quasi-hereditary structures. Our techniques involve certain quiver decompositions and idempotent reductions. For a path algebra of Dynkin type $\mathbb{A}$, we provide a full classification of its quasi-hereditary structures. For types $\mathbb{D}$ and $\mathbb{E}$, we give a counting method for the number of quasi-hereditary structures. In the case of a hereditary incidence algebra, we present a necessary and sufficient condition for its poset of quasi-hereditary structures to be a lattice.

Introduction

Quasi-hereditary algebras were introduced by Scott in [20] as an algebraic axiomatization of the theory of rational representations of semisimple algebraic groups. They were generalized to the concept of highest weight categories soon after in [4] allowing infinitely many simple objects. They form a huge class of algebras and categories which appear in many areas of modern representation theory, including complex semisimple Lie algebras, Schur algebras, algebras of global dimension at most two and many more.

A quasi-hereditary algebra is a pair $(A,(I,◁))$ where A is an Artin algebra and $(I,◁)$ is a partially ordered set indexing the isomorphism classes of simple A-modules which satisfies additional properties involving the filtrations of the projective modules by the so-called standard modules (see Definition 2.11). In particular, an algebra is not intrinsically a quasi-hereditary algebra since the standard modules heavily depend on the partial order $(I,◁)$.

In the early examples of highest weight categories and quasi-hereditary algebras, the partial orders were easy to choose because they were related to the classical combinatorics of weights and roots and we already knew that these partial orderings were related to the representation theory of our examples. However, there are instances of quasi-hereditary algebras where there is no natural choice for the partial ordering and even if there is such a natural choice, one may wonder about the other possible orderings.

To our knowledge, there are two known results in this direction. The first, which is due to Dlab and Ringel, asserts that an Artin algebra is hereditary if and only if it is quasi-hereditary for any total ordering on I (see [7, Theorem 1] for more details). The second, due to Coulembier, says that if an algebra has a simple preserving duality, then it has at most one ‘quasi-hereditary structure’ (see [6, Theorem 2.1.1] for more details, the notion of quasi-hereditary structure is explained below).

The main objective of this article is to continue the systematic study of all the possible choices of partially ordered sets that yield quasi-hereditary structures on a given Artin algebra. We start by known and easy remarks that will allow us to better define our objectives, and then give the foundations for our investigations. The organisation of this article can be summarised chronologically as follows.

The first main issue when one tries to classify all the partial orders ◁ on I such that $(A,(I,◁))$ is a quasi-hereditary algebra is that there are too many of them. However, as it has been proved by Conde (see [5, Proposition 1.4.12]), that if $(A,(I,◁))$ is a quasi-hereditary algebra, then the partial order ◁ is an adapted poset to A in the sense of Dlab and Ringel (see Definition 2.1). The definition of adapted order is rather technical, but it reduces significantly the number of partial orders that we have to consider.

The next step is to realize that we do not want to classify adapted partial orders, but we want to classify equivalence classes for an appropriate equivalence relation. Indeed, as it can be seen in Definition 2.11, the only place where the partial order $(I,◁)$ is involved is in the construction of the standard modules. As a consequence, if $(A,(I,{◁}_1))$ and $(A,(I,{◁}_2))$ are two quasi-hereditary algebras with same set of standard modules, it is then natural to say that the partial orders ${◁}_1$ and ${◁}_2$ are equivalent, and we write ${◁}_1\sim {◁}_2$. We call the equivalence classes of this relation the quasi-hereditary structures on the algebra A, and these are exactly what we want to classify. We denote by $\mathsf{qh}\text{.}\mathsf{str}(A)$ the set of quasi-hereditary structures on the algebra A.

This equivalence relation was introduced by Dlab and Ringel in [8] where they proved that any quasi-hereditary structure contains at least one total ordering. This key fact is the reason why in most of the references the authors usually assume the ordering on I to be total. However, for our purpose, we should not restrict ourselves to total orderings since one equivalence class may contain many different total orderings. For example, when A is a semisimple algebra with n isomorphism classes of simple modules, it has a unique quasi-hereditary structure which contains n! total orderings.

Let $(A,(I,◁))$ be a quasi-hereditary algebra with set of standard and costandard modules denoted by Δ and ∇. We define subsets $\mathsf{Dec}(◁)$ and $\mathsf{Inc}(◁)$ of $I^2$ as follows:

$$\mathsf{Dec}(◁):=\{(i,j)\in I^2|[\mathrm{\Delta }(j):S(i)]\ne 0\},\mathsf{Inc}(◁):=\{(i,j)\in I^2|[\mathrm{\nabla }(j):S(i)]\ne 0\}.$$

Then, we show in Lemma 2.8 that the transitive cover ${◁}_m$ of the relation $(\mathsf{Dec}(◁)\cup \mathsf{Inc}(◁))$ is equivalent to ◁ and we prove in Proposition 2.9 that it is the unique minimal element (with respect to the inclusion of relations) in the quasi-hereditary structure containing ◁. These partial orders were already considered by Coulembier under the name of essential orderings (see [6, Definition 1.2.5]). Here, we call them minimal adapted orders. The relations $\mathsf{Dec}(◁)$ and $\mathsf{Inc}(◁)$ are called the decreasing and increasing relations of ◁. These names have been chosen to highlight the relationship of our work and the work of Châtel, Pilaud and Pons on the weak order of integer posets (see [2]). It turns out that the combinatorics of the quasi-hereditary structures on an equioriented quiver of type $\mathbb{A}$ is the same as their combinatorics of the “Tamari order element poset” (see Section 4). Moreover, in the case of a quiver algebra, these names are particularly relevant since an increasing relation follows the paths of the quiver, and a decreasing one goes against them.

In addition to standard and costandard modules, a quasi-hereditary algebra has a distinguished tilting module called the characteristic tilting module (see Proposition 2.16). This leads to the following easy proposition.

Proposition 1.1 Lemma 2.18

Let ${◁}_1$ and ${◁}_2$ be two partial orders on I such that $(A,(I,{◁}_1))$ and $(A,(I,{◁}_2))$ are quasi-hereditary algebras. Then the following statements are equivalent.

• ${◁}_1\sim {◁}_2$.

• ${({◁}_1)}_m={({◁}_2)}_m$.

• $T_1\cong T_2$, where $T_i$ is the characteristic tilting module of $(A,(I,{◁}_i))$ for $i=1,2$.

It is now clear that we should see the set of quasi-hereditary structures on A as a subposet of its poset of tilting modules in the sense of Happel and Unger (see [11], [15] for more details).

The poset of tilting modules is most of the time infinite, but the poset of quasi-hereditary structures is always finite since any quasi-hereditary structure contains at least one total order. In Example 2.26 we show that the weak order on the set of permutations on n letters can be realized as the poset of quasi-hereditary structures on the path algebra of a suitable orientation of a complete graph. The Tamari lattices are obtained by considering path algebras of an equioriented orientation of a type $\mathbb{A}$ Dynkin diagram. More precisely, we prove the following result in Section 4.

Theorem 1.2 Theorem 4.7

Let $n\in \mathbb{N}$, and $A_n$ be an equioriented quiver of type $\mathbb{A}$ with n vertices, and ${\mathit{\Lambda }}_n=\mathbf{k}{A}_n$. Then, there are explicit bijections between

• Minimal adapted posets to ${\mathit{\Lambda }}_n$,

• Binary trees with n vertices,

• Isomorphism classes of tilting modules over ${\mathit{\Lambda }}_n$.

Moreover, the poset of quasi-hereditary structures on ${\mathit{\Lambda }}_n$ is isomorphic to the Tamari lattice.

This implies that the number of quasi-hereditary structures is given by the n-th Catalan number $c_n=\frac{1}{n+1}\left(\begin{array}{c}2n\\ n\end{array}\right)$. Furthermore, any tilting module for this algebra is a characteristic tilting module.

The understanding of the other orientations of a type $\mathbb{A}$ quiver is obtained as a consequence of a more general argument involving sources and sinks of a quiver.

A (iterated) deconcatenation of a quiver Q is a disjoint union $Q^1\bigsqcup \dots \bigsqcup Q^{\ell }$ of full subquivers $Q^i$ of Q satisfying certain properties, see Subsection 3.1 for details. The quivers $Q^i$ are obtained by deconcatenating Q at sources or sinks. Then in Section 3 we prove the following result.

Theorem 1.3 Theorem 3.7

Let $Q^1\bigsqcup Q^2\bigsqcup \dots \bigsqcup Q^{\ell }$ be an iterated deconcatenation of Q. Let A be a factor algebra of $\mathbf{k}Q$ modulo some admissible ideal and $A^i:=A/〈e_u|u\in Q_0\setminus Q_0^i〉$. Then we have an isomorphism of posets

$$\mathsf{qh}\mathit{\text{.}}\mathsf{str}(A)⟶\prod _{i=1}^{\ell }\mathsf{qh}\mathit{\text{.}}\mathsf{str}(A^i).$$

Under the isomorphism, we have an explicit construction of the characteristic tilting A-modules in terms of the characteristic tilting $A^i$-modules, see Theorem 3.14.

A quiver Q whose underlying graph is of type $\mathbb{A}$ has an iterated deconcatenation $Q^1\bigsqcup Q^2\bigsqcup \dots \bigsqcup Q^{\ell }$ such that each $Q^i$ is an equioriented quiver of type $\mathbb{A}$. Therefore as a corollary we obtain a classification of the quasi-hereditary structures on the path algebras of type $\mathbb{A}$.

Theorem 1.4 Theorem 4.9, Theorem 4.10

Let Q be a quiver whose underlying graph is of type $\mathbb{A}$. Let $Q^1\bigsqcup Q^2\bigsqcup \dots \bigsqcup Q^{\ell }$ be an iterated deconcatenation of Q such that each $Q^i$ is an equioriented quiver of type ${\mathbb{A}}_{n_i}$ for some $n_i\in {\mathbb{Z}}_{\ge 1}$. Then we have an isomorphism of posets

$$\mathsf{qh}\mathit{\text{.}}\mathsf{str}(\mathbf{k}Q)⟶\prod _{i=1}^{\ell }\mathsf{qh}\mathit{\text{.}}\mathsf{str}({\mathit{\Lambda }}_{n_i}).$$

Moreover, if ◁ is a minimal adapted order to $\mathbf{k}Q$ and $T={⨁}_{i\in I}T(i)$ is the characteristic tilting module associated to ◁, then for vertices $i,j\in Q_0$, a simple $\mathbf{k}Q$-module associated to j is a composition factor of $T(i)$ if and only if $j◁i$ holds.

In Section 5 we count the number of quasi-hereditary structures on the path algebras of Dynkin types $\mathbb{D}$ and $\mathbb{E}$. For the quivers of type $\mathbb{D}$, by Theorem 1.3 and the duality argument of Lemma 2.22, it is enough to consider only two orientations of ${\mathbb{D}}_n$, with $n\ge 4$, that is, the quivers $D_1$ and $D_2$ defined below.

Theorem 1.5 Lemma 5.8, Lemma 5.9

Let $n\ge 3$. Then,

• $|\mathsf{qh}\mathit{\text{.}}\mathsf{str}(\mathbf{k}{D}_1)|=2c_n-3c_{n-1}$,

• $|\mathsf{qh}\mathit{\text{.}}\mathsf{str}(\mathbf{k}{D}_2)|=3c_{n-1}-c_{n-2}$,

where $c_n$ is the n-th Catalan number and

The type $\mathbb{E}$ case is explained in Example 5.10.

So far, in all our examples, we have found a lattice of quasi-hereditary structures: in the complete graph case, we obtained the weak order and for quivers of type $\mathbb{A}$ we obtained products of Tamari lattices. As it can be seen in Proposition 6.11, this is no longer true in the case of affine quivers of type $\mathbb{A}$. The question of understanding which Artin algebras have a lattice of quasi-hereditary structures is both natural and very intriguing. We have the feeling that this will only happen in a few cases but our progress is rather modest: in Section 6 we prove the following result.

Theorem 1.6 Theorem 6.1

Let Q be a finite acyclic quiver whose underlying graph is a tree. Then $\mathsf{qh}\mathit{\text{.}}\mathsf{str}(\mathbf{k}Q)$ is a lattice if and only if Q does not have the following quiver as a subquiver for any $n\ge 4$:

When it is a lattice, we give an explicit description of the partial orders which represent a meet or a join of given two quasi-hereditary structures, see Subsection 6.3.

As an immediate corollary of this theorem, we have the following one.

Corollary 1.7

For a Dynkin quiver Q, $\mathsf{qh}\mathit{\text{.}}\mathsf{str}(\mathbf{k}Q)$ is a lattice.

Curiously, Theorem 1.6 does not seem to have a nice generalization to the setting of finite acyclic quivers. However, it seems to generalize to the setting of incidence algebras of a finite poset. We denote by $Z_n$ the partially ordered set whose Hasse diagram is a zig-zag orientation of an affine quiver of type ${\mathbb{A}}_n$ (see diagram 6.2) and we propose the following conjecture.

Conjecture 1.8

Let $(P,\le )$ be a finite poset. Then, the poset of quasi-hereditary structures on the incidence algebra of $(P,\le )$ is a lattice if and only if $Z_n$ is not isomorphic to a full subposet of $(P,\le )$ for any $n\ge 4$.

Actually, the setting of incidence algebras is not only a good setting for a generalization of our result, it also simplifies our arguments. So we prove Theorem 1.6 as a corollary of the following theorem.

Theorem 1.9 Theorem 6.5

Let $(P,\le )$ be a finite poset. We assume that the incidence algebra $A(P)$ of $(P,\le )$ is hereditary. Then, the poset of quasi-hereditary structures on $A(P)$ is a lattice if and only if $Z_n$ is not isomorphic to a full subposet of $(P,\le )$ for any $n\ge 4$.

The relationship between Theorem 1.6 and Theorem 1.9 is explained in Remark 6.9.

Experiments were carried out using the GAP-package QPA [14] and SageMath [19]. The code is available online [9].

Notation and conventions

Throughout this paper, k denotes a field of arbitrary characteristic. A subcategory always means a full subcategory which is closed under isomorphisms. An Artin algebra is an R-algebra A such that R is a commutative Artinian ring and A is finitely generated as an R-module. For an Artin algebra, we usually deal with finitely generated right modules. We denote by mod A the category of finitely generated right A-modules. For an element or subset X of A, we denote by $〈X〉$ the two-sided ideal of A generated by X. For a quiver Q, let $\mathbf{k}Q$ be the path algebra of Q. For two arrows $\alpha ,\beta$ of Q, if the terminating vertex of α equals the starting vertex of β, then αβ indicates the composite of α with β. An order means a partially ordered set.