2.1 Lattice-theoretic preliminaries

Proofs and additional details for the material reviewed here can be found in [Reference ReadingRea16a, §§ 9–5 and 9–6].

For any poset $P$ , we say that $x$ covers $y$ , and we write $x\gtrdot y$ , if $x>y$ , and there is no $z\in P$ such that $x>z>y$ . We represent $P$ by its Hasse quiver $\mathsf{Hasse}(P)=(P,\mathsf{Hasse}_{1}(P))$ , whose vertex set is $P$ , and whose arrow set $\mathsf{Hasse}_{1}(P)$ consists of all arrows $v\rightarrow w$ where $v$ covers  $w$ .

Given a subset $S$ of $P$ , if there is a unique smallest element which is greater than or equal to all elements in $S$ , then this least upper bound is called the join of $S$ and denoted $\vee S$ . Similarly, if there is a unique largest element in $P$ that is less than or equal to all elements in $S$ , then this element is called the meet of $S$ and denoted $\bigwedge S$ . A lattice $L$ is a poset in which every pair $a,b$ of elements in $L$ has both a meet $a\wedge b$ and a join $a\vee b$ , and a complete lattice $L$ is a poset in which every subset $S$ of $L$ has both a meet and a join. (Every finite subset of a lattice $L$ has both a meet and a join, but an infinite lattice fails to be complete if it has some infinite subset without a meet or without a join.)

We restrict our attention to finite lattices in this paper. Some of the assertions made here for finite lattices hold for infinite lattice as well, but some do not.

An element $j$ of a finite lattice $L$ is called join-irreducible, whenever $j=a\vee b$ for some $a,b\in L$ , either $a=j$ or $b=j$ or both, and $j$ is not the minimum element of  $L$ . Equivalently, $j$ is join-irreducible if and only if it covers exactly one element of $L$ . We write $j_{\ast }$ for the unique element covered by a join-irreducible element $j$ . Dually, a meet-irreducible element of $L$ is an element $m$ that is covered by a unique element $m^{\ast }$ . The set of join-irreducible (respectively, meet-irreducible) elements of $L$ is denoted $\mathsf{j}\text{-}\mathsf{Irr}\,L$ (respectively, $\mathsf{m}\text{-}\mathsf{Irr}\,L$ ).

A (lattice) congruence on a lattice $L$ is an equivalence relation $\unicode[STIX]{x1D6E9}$ having the property that the $\unicode[STIX]{x1D6E9}$ -class of $a\vee b$ depends only on the $\unicode[STIX]{x1D6E9}$ -class of $a$ and the $\unicode[STIX]{x1D6E9}$ -class of $b$ , and having the same property for meets. Given a congruence $\unicode[STIX]{x1D6E9}$ on $L$ , the set $L/\unicode[STIX]{x1D6E9}$ of $\unicode[STIX]{x1D6E9}$ -classes has a well-defined meet and join operation, making $L/\unicode[STIX]{x1D6E9}$ a lattice called the quotient of $L$ modulo  $\unicode[STIX]{x1D6E9}$ .

The set of all equivalence relations on a given set $L$ forms a lattice, where the meet of two relations is given by the intersection of relations and the join of two relations is given by the transitive closure of union of relations. When $L$ is a lattice, the set $\mathsf{Con}(L)$ consisting of congruences of $L$ is a sublattice of the lattice of equivalence relations. Furthermore, $\mathsf{Con}(L)$ is a distributive lattice. We denote by $\mathsf{Con}_{\mathsf{JI}}(L)$ the set of all join-irreducible congruences. As mentioned in the introduction, we have a surjective map $\mathsf{Hasse}_{1}(L)\rightarrow \mathsf{Con}_{\mathsf{JI}}(L)$ sending an arrow $x\rightarrow y$ to $\mathsf{con}(x,y)$ . Here $\mathsf{con}(x,y)$ is the meet, in $\mathsf{Con}(L)$ , of all congruences with $x\equiv y$ . A congruence $\unicode[STIX]{x1D6E9}$ on a finite lattice $L$ is determined completely by the set of cover relations $x\gtrdot y$ in $L$ such that $x\equiv y$ modulo  $\unicode[STIX]{x1D6E9}$ . It is also determined uniquely by the set of join-irreducible elements $j$ in $L$ such that $j\equiv j_{\ast }$ modulo $\unicode[STIX]{x1D6E9}$ , and thus we have an injective map $\mathsf{Con}(L)\rightarrow 2^{\mathsf{j}\text{-}\mathsf{Irr}(L)}$ .

The map from cover relations $x\gtrdot y$ to join-irreducible congruences is typically not one-to-one. The restriction of the map to cover relations of the form $j\gtrdot j_{\ast }$ is also surjective onto join-irreducible congruences, but may still fail to be one-to-one. A lattice is called congruence uniform if the map $j\mapsto \mathsf{con}(j,j_{\ast })$ is injective (and thus a bijection) from join-irreducible elements to join-irreducible congruences and the map $m\rightarrow \mathsf{con}(m^{\ast },m)$ is injective (and thus a bijection) from meet-irreducible elements to join-irreducible congruences. A finite congruence uniform lattice is always semidistributive. This means that if $x\vee y=x\vee z$ then $x\vee (y\wedge z)=x\vee y$ and if $x\wedge y=x\wedge z$ then $x\wedge (y\vee z)=x\wedge y$ .

Since $\mathsf{Con}(L)$ is a finite distributive lattice, the Fundamental Theorem of Finite Distributive Lattices says that its elements are naturally identified with order ideals in the subposet $\mathsf{Con}_{\mathsf{JI}}(L)$ of $\mathsf{Con}(L)$ . When $L$ is congruence uniform, the subposet $\mathsf{Con}_{\mathsf{JI}}(L)$ induces a partial order on the join-irreducible elements of $L$ , which we call the forcing order. A congruence $\unicode[STIX]{x1D6E9}\in \mathsf{Con}(L)$ corresponds to the order ideal in $\mathsf{Con}_{\mathsf{JI}}(L)$ consisting of those join-irreducible congruences below $\unicode[STIX]{x1D6E9}$ in $\mathsf{Con}(L)$ (i.e. finer than $\unicode[STIX]{x1D6E9}$ as equivalence relations). These are the join-irreducible congruences $\mathsf{con}(j,j_{\ast })$ such that $j\equiv j_{\ast }$ modulo $\unicode[STIX]{x1D6E9}$ . The forcing order on join-irreducible elements sets $j\leqslant j^{\prime }$ if and only if $j\equiv j_{\ast }$ modulo $\mathsf{con}(j^{\prime },j_{\ast }^{\prime })$ .

As mentioned above, each cover relation $x\gtrdot y$ in a finite lattice defines a join-irreducible congruence of  $L$ . In a finite congruence uniform lattice  $L$ , each join-irreducible congruence is $\mathsf{con}(j,j_{\ast })$ for a unique join-irreducible element $j$ of $L$ . The map $\mathsf{Hasse}_{1}(L)\rightarrow \mathsf{j}\text{-}\mathsf{Irr}(L)$ sending the arrow $x\rightarrow y$ to the unique $j$ with $\mathsf{con}(j,j_{\ast })=\mathsf{con}(x,y)$ is called the join-irreducible labelling of  $L$ . Each join-irreducible congruence is also $\mathsf{con}(m^{\ast },m)$ for a unique meet-irreducible element $m$ , and the map $\mathsf{Hasse}_{1}(L)\rightarrow \mathsf{m}\text{-}\mathsf{Irr}(L)$ sending $x\rightarrow y$ to the unique $m$ with $\mathsf{con}(m^{\ast },m)=\mathsf{con}(x,y)$ is called the meet-irreducible labelling of  $L$ . These labellings are described explicitly as follows.

The following proposition is [Reference ReadingRea16a, Propositions 9–5.20]. Since that proposition’s proof is left to an exercise, we give a proof here.

We summarize some of what we know about join-irreducible elements, meet-irreducible elements, and congruences in a finite congruence uniform lattice in Figure 4. If $L$ is a finite congruence uniform lattice with Hasse quiver $\mathsf{Hasse}(L)$ , join-irreducible elements $\mathsf{j}\text{-}\mathsf{Irr}(L)$ , meet-irreducible elements $\mathsf{m}\text{-}\mathsf{Irr}(L)$ , and join-irreducible congruences $\mathsf{Con}_{\mathsf{JI}}(L)$ , then the diagram in Figure 4 commutes and the maps are bijections or surjections as marked with tildes ‘ ${\sim}$ ’ or double-headed arrows.

Figure 4.

Join-irreducible elements, meet-irreducible elements, and congruences in a finite congruence uniform lattice.

Let $x$ be an element of a finite lattice $L$ . The expression $x=\vee S$ is the canonical join representation of $x$ if no proper subset of $S$ joins to $x$ and if every join-representation $x=\vee T$ has the property that for all $s\in S$ , there exists $t\in T$ with $s\leqslant t$ . An element $x$ may fail to have a canonical join representation, but the canonical join representation of $x$ is unique if it exists. The canonical meet representation is defined dually. The semidistributive property of a finite lattice  $L$ , described above, is equivalent to the property that every element of $L$ has a canonical join representation and a canonical meet representation. A finite congruence uniform lattice is in particular semidistributive, and canonical join and meet representations can be described in terms of the join-irreducible labelling and meet-irreducible labelling as follows.

The following proposition is [Reference ReadingRea16a, Proposition 9–5.30]. Since that proposition’s proof is also left to an exercise, we give a proof here.

A polygon in a finite lattice $L$ is an interval $[x,y]$ such that $\{z\in L:x<z<y\}$ consists of two disjoint non-empty chains. (Thus the Hasse quiver of $[x,y]$ is a cycle with one source and one sink.) The lattice $L$ is polygonal if the following two conditions hold: first, if distinct elements $y_{1}$ and $y_{2}$ both cover an element $x$ , then $[x,y_{1}\vee y_{2}]$ is a polygon; and second, if an element $y$ covers distinct elements $x_{1}$ and $x_{2}$ , then $[x_{1}\wedge x_{2},y]$ is a polygon.

If $L$ is a polygonal lattice, then we define a quiver $\mathsf{FPoly}(L)$ whose set of vertices is $\mathsf{Hasse}_{1}(L)$ , with arrows defined in every polygon $P$ of $L$ as follows. The two arrows into the bottom element of $P$ are called bottom arrows, while the two arrows from the top element of $P$ are called top arrows, and all other arrows of $P$ are called side arrows. Every bottom arrow of $P$ has an arrow (in $\mathsf{FPoly}(L)$ ) to the opposite top arrow in $P$ (i.e. the top arrow in the opposite chain) and has an arrow to every side arrow in  $P$ . Every top arrow of $P$ has an arrow to the opposite bottom arrow and every side arrow in  $P$ . For example, a square and hexagon in $L$ would contribute to $\mathsf{FPoly}(L)$ as indicated below.

The following is [Reference ReadingRea16a, Theorem 9–6.5].

When $L$ is also congruence uniform, the quiver $\mathsf{FPoly}(L)$ is closely related to the forcing order on join-irreducible elements of $L$ , as described in the following corollary, which is immediate by combining Theorem 2.3 with facts about finite congruence uniform lattices already given in this section. We say that a quiver $Q$ is strongly connected if given any two vertices $x$ and $y$ of the quiver, there exists a path from $x$ to $y$ and a path from $y$ to  $x$ . A strongly connected component of $Q$ is a set of vertices of $Q$ that is maximal with respect to inducing a strongly connected subquiver.

Informally, the corollary says that all forcing in a finite polygonal, congruence uniform lattice comes from forcing in polygons. Indeed, the notation $\mathsf{FPoly}(L)$ suggests the phrase ‘forcing in polygons.’

We define another quiver $\mathsf{SFPoly}(L)$ , again with the set of vertices given by $\mathsf{Hasse}_{1}(L)$ , but with strictly fewer arrows than $\mathsf{FPoly}(L)$ . The notation $\mathsf{SFPoly}(L)$ suggests the phrase ‘strong forcing in polygons.’ In $\mathsf{SFPoly}(L)$ , every bottom arrow of a polygon $P$ has an arrow only to the opposite top arrow in $P$ and every top arrow of $P$ has an arrow only to the opposite bottom arrow in  $P$ . Thus from the square and hexagon as labelled above, $\mathsf{SFPoly}(L)$ gets arrows as indicated below:

$$\begin{eqnarray}\begin{array}{@{}ccccc@{}}a\leftrightarrow d & b\leftrightarrow c & \qquad & e\leftrightarrow j & g\leftrightarrow h.\end{array}\end{eqnarray}$$

Strong forcing in polygons controls whether two Hasse arrows in a finite polygonal, congruence uniform lattice determine the same congruence, as described in the following corollary.

2.2 Weak order preliminaries

Fix a simply-laced Dynkin type (i.e., one of $A_{n}$ for $n\geqslant 1$ , $D_{n}$ for $n\geqslant 4$ , $E_{6}$ , $E_{7}$ , or $E_{8}$ ). Let $W$ be the finite Weyl group of that type. For background on Weyl groups, see [Reference Björner and BrentiBB05].

Let $S=\{s_{1},\ldots ,s_{n}\}$ be the set of simple reflections of $W$ . By definition, any element $w$ of $W$ can be written as a product of the simple reflections. Such an expression for $w$ of minimal length is called reduced. The length of a reduced expression is the length of $w$ , denoted $\ell (w)$ .

The (right) weak order on $W$ is the partial order with $u\geqslant v$ if $\ell (u)=\ell (v)+\ell (v^{-1}u)$ . We write $\mathsf{Hasse}(W)$ for the Hasse quiver of the weak order on $W$ . The arrows of $\mathsf{Hasse}(W)$ are all arrows $ws_{i}\rightarrow w$ such that $w\in W$ , $s_{i}\in S$ , and $\ell (ws_{i})>\ell (w)$ , or equivalently $\ell (ws_{i})=\ell (w)+1$ . For $w\in W$ , we denote by $w^{+}$ (respectively, $w^{-}$ ) the set of arrows in $\mathsf{Hasse}(W)$ starting (respectively, ending) at  $w$ .

The weak order on $W$ is a finite lattice. In particular, it has a maximal element, denoted $w_{0}$ and often called the longest element of  $W$ . For our purposes, the most important properties of the weak order are the following.

2.3 Algebraic preliminaries

Fix a base field $k$ , and let $A$ be a finite-dimensional $k$ -algebra. We write $\mathsf{mod}\,A$ for the finite-dimensional left $A$ -modules. We denote by $\unicode[STIX]{x1D70F}$ and $\unicode[STIX]{x1D70F}^{-}$ the Auslander–Reiten translations of  $A$ . They give mutually inverse bijections between isomorphism classes of indecomposable non-projective $A$ -modules and those of indecomposable non-injective $A$ -modules.

We say that a full subcategory $\mathscr{T}$ of $\mathsf{mod}\,A$ is a torsion class if it is closed under factors, isomorphisms and extensions. Torsion-free classes are defined dually. We write $\mathsf{tors}\,A$ for the torsion classes of $A$ , and $\mathsf{torf}\,A$ for its torsion-free classes. We view $\mathsf{tors}\,A$ and $\mathsf{torf}\,A$ as posets under the inclusion order. Then we have an anti-isomorphism given by $\mathscr{T}\mapsto \mathscr{T}^{\bot }=\{X\in \mathsf{mod}\,A\mid \operatorname{Hom}_{A}(\mathscr{T},X)=0\}$

$$\begin{eqnarray}\mathsf{tors}\,A\rightarrow \mathsf{torf}\,A,\end{eqnarray}$$

whose inverse is given by $\mathscr{F}\mapsto ^{\bot }\mathscr{F}=\{X\in \mathsf{mod}\,A\mid \operatorname{Hom}_{A}(X,\mathscr{F})=0\}$ .

We recall that a torsion class $\mathscr{T}$ of $\mathsf{mod}\,A$ is functorially finite if there exists $M\in \mathsf{mod}\,A$ such that $\mathscr{T}=\mathsf{Fac}\,M$ , where $\mathsf{Fac}\,M$ is the full subcategory of $\mathsf{mod}\,A$ consisting of factor modules of finite direct sums of copies of $M$ [Reference Auslander and SmaloAS81]. We denote by $\mathsf{f}\text{-}\mathsf{tors}\,A$ (respectively, $\mathsf{f}\text{-}\mathsf{torf}\,A$ ) the set of all functorially finite torsion (respectively, torsionfree) classes in $\mathsf{mod}\,A$ . We view $\mathsf{f}\text{-}\mathsf{tors}\,A$ also as a poset under inclusion. The above anti-isomorphism restricts to an anti-isomorphism $\mathsf{f}\text{-}\mathsf{tors}\,A\rightarrow \mathsf{f}\text{-}\mathsf{torf}\,A$  [Reference SmaløSma84].

There is a bijection between $\mathsf{f}\text{-}\mathsf{tors}\,A$ and a certain class of $A$ -modules. Recall that a module $M\in \mathsf{mod}\,A$ is called $\unicode[STIX]{x1D70F}$ -rigid if $\operatorname{Hom}_{A}(M,\unicode[STIX]{x1D70F}M)=0$ , and $\unicode[STIX]{x1D70F}^{-}$ -rigid if $\operatorname{Hom}_{A}(\unicode[STIX]{x1D70F}^{-}M,M)=0$ . A module $M\in \mathsf{mod}\,A$ is called $\unicode[STIX]{x1D70F}$ -tilting if it is $\unicode[STIX]{x1D70F}$ -rigid and $|M|=|A|$ holds, where $|M|$ is the number of non-isomorphic indecomposable direct summands of $M$ . A module $M\in \mathsf{mod}\,A$ is called support $\unicode[STIX]{x1D70F}$ -tilting if there exists an idempotent $e$ of $A$ such that $M$ is a $\unicode[STIX]{x1D70F}$ -tilting $(A/\langle e\rangle )$ -module. We denote by $\mathsf{s}\unicode[STIX]{x1D70F}\text{-}\mathsf{tilt}\,A$ the set of isomorphism classes of basic support $\unicode[STIX]{x1D70F}$ -tilting $A$ -modules, and by $\mathsf{i}\unicode[STIX]{x1D70F}\text{-}\mathsf{rigid}\,A$ (respectively, $\mathsf{i}\unicode[STIX]{x1D70F}^{{-}}\text{-}\mathsf{rigid}\,A$ ) the set of isomorphism classes of indecomposable $\unicode[STIX]{x1D70F}$ -rigid (respectively, $\unicode[STIX]{x1D70F}^{-}$ -rigid) $A$ -modules. (See [Reference Adachi, Iyama and ReitenAIR14] for more background on these notions.) By [Reference Adachi, Iyama and ReitenAIR14, 2.7], we have a surjection $\{\unicode[STIX]{x1D70F}\text{-}\text{rigid}~A\text{-}\text{modules}\}\rightarrow \mathsf{f}\text{-}\mathsf{tors}\,A$ given by $M\mapsto \mathsf{Fac}\,M$ , which induces a bijection

$$\begin{eqnarray}\mathsf{s}\unicode[STIX]{x1D70F}\text{-}\mathsf{tilt}\,A\xrightarrow[{}]{{\sim}}\mathsf{f}\text{-}\mathsf{tors}\,A.\end{eqnarray}$$

Recall that $A$ is $\unicode[STIX]{x1D70F}$ -tilting finite if $\mathsf{s}\unicode[STIX]{x1D70F}\text{-}\mathsf{tilt}\,A$ is a finite set, or equivalently, $\mathsf{i}\unicode[STIX]{x1D70F}\text{-}\mathsf{rigid}\,A$ is a finite set. It is shown in [Reference Demonet, Iyama and JassoDIJ15] and [Reference Iyama, Reiten, Thomas and TodorovIRTT15, 1.2] that the following conditions are equivalent:

• ∙ $A$ is $\unicode[STIX]{x1D70F}$ -tilting finite;

• ∙ $\mathsf{f}\text{-}\mathsf{tors}\,A$ is a finite set;

• ∙ $\mathsf{f}\text{-}\mathsf{tors}\,A$ (respectively, $\mathsf{f}\text{-}\mathsf{torf}\,A$ ) forms a complete lattice;

• ∙ $\mathsf{f}\text{-}\mathsf{tors}\,A=\mathsf{tors}\,A$ .

Via the bijection between $\mathsf{f}\text{-}\mathsf{tors}\,A$ and $\mathsf{s}\unicode[STIX]{x1D70F}\text{-}\mathsf{tilt}\,A$ , we obtain a partial order on $\mathsf{s}\unicode[STIX]{x1D70F}\text{-}\mathsf{tilt}\,A$ . We refer to the poset on $\mathsf{s}\unicode[STIX]{x1D70F}\text{-}\mathsf{tilt}\,A$ as generation order. The arrows of the Hasse diagram of this poset are mutations (see [Reference Adachi, Iyama and ReitenAIR14, Theorem 0.6]).

Using $\unicode[STIX]{x1D70F}$ -tilting theory, we have the following description of join-irreducible elements in $\mathsf{tors}\,A$ .

Theorem 2.7. Let $A$ be a finite-dimensional $k$ -algebra which is $\unicode[STIX]{x1D70F}$ -tilting finite. Then we have a bijection given by $M\mapsto \mathsf{Fac}\,M$

$$\begin{eqnarray}\mathsf{i}\unicode[STIX]{x1D70F}\text{-}\mathsf{rigid}\,A\rightarrow \mathsf{j}\text{-}\mathsf{Irr}(\mathsf{tors}\,A).\end{eqnarray}$$

The inverse map is given by $\mathscr{T}\mapsto N$ , where $M\in \mathsf{s}\unicode[STIX]{x1D70F}\text{-}\mathsf{tilt}\,A$ satisfies $\mathsf{Fac}\,M=\mathscr{T}$ and $N$ is a unique indecomposable direct summand of $M$ satisfying $\mathsf{Fac}\,N=\mathsf{Fac}\,M$ .

For a complete lattice $L$ , we denote by $\mathsf{j}\text{-}\mathsf{Irr}^{\mathsf{c}}\,L$ the set of completely join-irreducible elements, that is, elements $a\in L$ such that $a=\vee S$ for a subset $S$ of $L$ implies $a\in S$ . If we drop the $\unicode[STIX]{x1D70F}$ -tilting finiteness assumption on $A$ , then we still have a bijection $\mathsf{i}\unicode[STIX]{x1D70F}\text{-}\mathsf{rigid}\,A\rightarrow \mathsf{f}\text{-}\mathsf{tors}\,A\cap \mathsf{j}\text{-}\mathsf{Irr}^{\mathsf{c}}(\mathsf{tors}\,A)$ given by $M\mapsto \mathsf{Fac}\,M$ . The proof is the same, we only need to use [Reference Demonet, Iyama and JassoDIJ15, Theorem 3.1]. Note that completely join-irreducible elements in $\mathsf{tors}\,A$ are not necessarily functorially finite. For example, consider $\mathsf{mod}\,kQ$ for a Kronecker quiver  $Q$ . Then all preinjective modules together with one tube form such a torsion class.

We denote by $\mathsf{brick}\,A$ the set of isomorphism classes of bricks of $A$ . A full subcategory $\mathscr{W}$ of $\mathsf{mod}\,A$ is called wide if it is closed under kernels, cokernels and extensions. In this case, $\mathscr{W}$ forms an abelian category and the inclusion functor $\mathscr{W}\rightarrow \mathsf{mod}\,A$ is exact. A wide subcategory $\mathscr{W}$ is called local if it contains a unique simple object up to isomorphism. We denote by $\mathsf{l}\text{-}\mathsf{wide}\,A$ the set of local wide subcategories of $\mathsf{mod}\,A$ . We have the following easy observation.

Proposition 2.8. Let $A$ be a finite-dimensional $k$ -algebra. Then we have a bijection

$$\begin{eqnarray}\mathsf{brick}\,A\rightarrow \mathsf{l}\text{-}\mathsf{wide}\,A\quad \text{given by}~S\mapsto \mathsf{Filt}\,S,\end{eqnarray}$$

where $\mathsf{Filt}\,S$ consists of $A$ -modules $X$ which have a filtration $X=X_{0}\supset X_{1}\supset \cdots \supset X_{\ell -1}\supset X_{\ell }=0$ with $\ell \geqslant 0$ such that $X_{i}/X_{i+1}\simeq S$ for any $0\leqslant i<\ell$ .

2.4 Preliminaries on preprojective algebras

Let $\unicode[STIX]{x1D6F1}=\unicode[STIX]{x1D6F1}(W)$ be a preprojective algebra of the same Dynkin type as $W$ . To construct $\unicode[STIX]{x1D6F1}$ , take the Dynkin diagram and replace each edge by a pair of opposite arrows $a,a^{\ast }$ to obtain the quiver $\overline{Q}$ . Then $\unicode[STIX]{x1D6F1}(W)$ is the path algebra of $\overline{Q}$ modulo the ideal generated by $\sum _{a}(aa^{\ast }-a^{\ast }a)$ . It is a finite-dimensional self-injective algebra. We write $S_{i}$ for the simple module corresponding to vertex  $i$ .

We let $e_{i}$ be the idempotent corresponding to the vertex $i$ . Let $I_{i}$ be the two-sided ideal $\unicode[STIX]{x1D6F1}(1-e_{i})\unicode[STIX]{x1D6F1}$ . It is maximal as a left ideal and as a right ideal. For each $w\in W$ , we take a reduced word $w=s_{i_{1}}\ldots s_{i_{k}}$ , and we define

$$\begin{eqnarray}I(w)=I_{i_{1}}\ldots I_{i_{k}},\quad \mathsf{T}(w):=\mathsf{Fac}\,I(w)\quad \text{and}\quad \mathsf{F}(w):=\mathsf{Sub}(\unicode[STIX]{x1D6F1}/I(w)).\end{eqnarray}$$

Here $\mathsf{Sub}\,X$ refers to the subcategory consisting of subobjects of direct sums of copies of  $X$ .

The following result due to Mizuno is fundamental.

We remark that Mizuno uses right modules, while we use left modules, and therefore his results need to be suitably translated to account for this difference. (In particular, this has the effect that he uses left weak order while we use right weak order.) We also remark here that [Reference Adachi, Iyama and ReitenAIR14] uses right modules, but writes $I_{w}$ for the ideal which we would refer to as $I(w^{-1})$ .

Thus $I(e)=\unicode[STIX]{x1D6F1}$ gives the maximum torsion class $\mathsf{T}(e)=\mathsf{mod}\,\unicode[STIX]{x1D6F1}$ and $I(w_{0})=0$ gives the minimum torsion class $\mathsf{T}(w_{0})=\{0\}$ , where $e\in W$ is the identity and $w_{0}\in W$ is the longest element.

The following is an easy consequence of (2.1).

We prepare the following results on indecomposable $\unicode[STIX]{x1D70F}$ -rigid $\unicode[STIX]{x1D6F1}$ -modules.

We show a few examples. Following the usual convention, we describe $\unicode[STIX]{x1D6F1}$ -modules $X$ in terms of their composition factors. Thus the numbers $i$ give a basis of the $k$ -vector space $e_{i}X$ , and the action of arrows is shown by the relative position of these numbers. For example, $\begin{smallmatrix}1 & & \\ & 2 & \\ & & 3\end{smallmatrix}$ refers to the indecomposable $\unicode[STIX]{x1D6F1}$ -module with composition factors $S_{1},S_{2},S_{3}$ from top to bottom.

Example 2.12. The left picture in Figure 5 shows the weak order on permutations in $S_{3}$ (the Weyl group of type  $A_{2}$ ). The right picture in Figure 5 shows the set $\mathsf{s}\unicode[STIX]{x1D70F}\text{-}\mathsf{tilt}\,\unicode[STIX]{x1D6F1}$ for $\unicode[STIX]{x1D6F1}$ of type $A_{2}$ , arranged in generation order, where bars mean direct sums. We show in grey (red online) the meet-irreducible elements of $W$ and the join-irreducible elements of $\mathsf{s}\unicode[STIX]{x1D70F}\text{-}\mathsf{tilt}\,\unicode[STIX]{x1D6F1}$ . The two pictures are arranged so that the map $w\rightarrow I(w)$ is accomplished by a translation of the page.

Figure 5.

(Colour online) The weak order and $\mathsf{s}\unicode[STIX]{x1D70F}\text{-}\mathsf{tilt}\,\unicode[STIX]{x1D6F1}$ in type $A_{2}$ .

Example 2.13. Figure 6 shows the weak order on permutations in $S_{4}$ (the Weyl group of type  $A_{3}$ ) and $\mathsf{s}\unicode[STIX]{x1D70F}\text{-}\mathsf{tilt}\,\unicode[STIX]{x1D6F1}$ for $\unicode[STIX]{x1D6F1}$ of type $A_{3}$ , with the same conventions as described in Example 2.12.

Figure 6.

(Colour online) The weak order and $\mathsf{s}\unicode[STIX]{x1D70F}\text{-}\mathsf{tilt}\,\unicode[STIX]{x1D6F1}$ in type $A_{3}$ .

Let $\widehat{\unicode[STIX]{x1D6F1}}$ be the preprojective algebra of the extended Dynkin type corresponding to $\unicode[STIX]{x1D6F1}$ . We identify $\unicode[STIX]{x1D6F1}$ -modules with $\widehat{\unicode[STIX]{x1D6F1}}$ -modules annihilated by $e_{0}$ , where $e_{0}$ is an idempotent of $\widehat{\unicode[STIX]{x1D6F1}}$ satisfying $\unicode[STIX]{x1D6F1}=\widehat{\unicode[STIX]{x1D6F1}}/\langle e_{0}\rangle$ . Then the category $\mathsf{mod}\,\unicode[STIX]{x1D6F1}$ can be regarded as a full subcategory of $\mathsf{fd}\,\widehat{\unicode[STIX]{x1D6F1}}$ which is closed under extensions. Therefore we have

(2.2) $$\begin{eqnarray}\operatorname{Hom}_{\unicode[STIX]{x1D6F1}}(X,Y)=\operatorname{Hom}_{\widehat{\unicode[STIX]{x1D6F1}}}(X,Y)\quad \text{and}\quad \operatorname{Ext}_{\unicode[STIX]{x1D6F1}}^{1}(X,Y)=\operatorname{Ext}_{\widehat{\unicode[STIX]{x1D6F1}}}^{1}(X,Y)\end{eqnarray}$$

for any $X,Y\in \mathsf{mod}\,\unicode[STIX]{x1D6F1}$ . On the other hand, $\widehat{\unicode[STIX]{x1D6F1}}$ is a 2-Calabi–Yau algebra [Reference KellerKel08, § 4.2] and hence there exists a functorial isomorphism

(2.3) $$\begin{eqnarray}\operatorname{Hom}_{\mathsf{D}^{\operatorname{b}}(\mathsf{fd}\,\widehat{\unicode[STIX]{x1D6F1}})}(X,Y)\simeq D\operatorname{Hom}_{\mathsf{D}^{\operatorname{b}}(\mathsf{fd}\,\widehat{\unicode[STIX]{x1D6F1}})}(Y,X[2])\end{eqnarray}$$

for any $X,Y\in \mathsf{D}^{\operatorname{b}}(\mathsf{fd}\,\widehat{\unicode[STIX]{x1D6F1}})$ [Reference KellerKel08, 4.1], where $D$ is the $k$ -dual.