Pop-Stack Operators for Torsion Classes and Cambrian Lattices

Let $L$ be a finite lattice with meet operation $\wedge$ and join operation $\vee$. The pop-stack operator $\mathrm{pop}^{\downarrow}_{L}\colon L\to L$ and the dual pop-stack operator $\mathrm{pop}^{\uparrow}_{L}\colon L\to L$ are defined by

$$\mathrm{pop}^{\downarrow}_{L}(x)=x\wedge\left(\bigwedge\{y\mid y{\,\,<\!\!\!\!\cdot\,\,\,}x\}\right)\quad\text{and}\quad\mathrm{pop}^{\uparrow}_{L}(x)=x\vee\left(\bigvee\{y\mid x{\,\,<\!\!\!\!\cdot\,\,\,}y\}\right),$$

where we write $u\lessdot v$ to mean that $u$ is covered by $v$ in $L$. These operators have appeared in various contexts; they serve as both useful tools [AP22, BH, Eno23, ES22, Hana, Müh19, Rea11, Sak] and objects of interest in their own right [CS, CG19, CGP21, Def22b, Def22a, DW23, Hon22, PS19, Ung82]. When the lattice $L$ is understood, we will omit subscripts and simply denote these operators by $\mathrm{pop}^{\downarrow}$ and $\mathrm{pop}^{\uparrow}$.

In [CG19, Def22b, Def22a, Hon22, PS19, Ung82], the pop-stack operator has been considered as a dynamical system. Given a map $f\colon L\to L$ and an element $x\in L$, the forward orbit of $x$ under $f$ is the set

$$\mathrm{Orb}_{f}(x)=\{x,f(x),f^{2}(x),f^{3}(x),\ldots\},$$

where $f^{t}$ is the map obtained by composing $f$ with itself $t$ times. To ease notation, let us write

$$\mathcal{O}_{L}(x)=\mathrm{Orb}_{\mathrm{pop}^{\downarrow}_{L}}(x)$$

for the forward orbit of $x$ under $\mathrm{pop}^{\downarrow}_{L}$. If $t$ is sufficiently large, then $(\mathrm{pop}^{\downarrow}_{L})^{t}(x)$ is equal to the minimal element $\hat{0}$ of $L$ (which is a fixed point of $\mathrm{pop}^{\downarrow}_{L}$). Thus, $\lvert\mathcal{O}_{L}(x)\rvert-1$ is equal to the number of iterations of $\mathrm{pop}^{\downarrow}_{L}$ needed to send $x$ to $\hat{0}$. Given an interesting lattice $L$, one of the primary problems one can consider about its pop-stack operator is that of computing

$$\max_{x\in L}|\mathcal{O}_{L}(x)|.$$

For a fixed $t$, one can also consider the $t$-pop-stack sortable elements of $L$, which are the elements $x\in L$ such that $(\mathrm{pop}^{\downarrow}_{L})^{t}(x)=\hat{0}$.

Defant and Williams [DW23] also found that it is fruitful to study the image of the pop-stack operator when $L$ is a semidistrim lattice; this is because the image of $\mathrm{pop}^{\downarrow}$ has numerous interesting properties, some of which relate to a certain bijective rowmotion operator $\mathrm{row}\colon L\to L$. For example, $\lvert\mathrm{pop}^{\downarrow}(L)\rvert$ and $\lvert\mathrm{pop}^{\uparrow}(L)\rvert$ are both equal to the number of elements $x\in L$ such that $\mathrm{row}(x)\leq x$. The images of $\mathrm{pop}^{\downarrow}$ and $\mathrm{pop}^{\uparrow}$ are also naturally in bijection with the set of facets of a certain simplicial complex called the canonical join complex of $L$.

In this article, we take a representation-theoretic perspective and consider a finite-dimensional basic algebra $\Lambda$ over a field $K$. The set of torsion classes (see Section 3.2 for the definition) of finitely-generated (right) $\Lambda$-modules forms a complete lattice [IRTT15] that we denote by $\operatorname{\mathsf{tors}}\Lambda$. Examples of lattices arising in this way include the weak order and Cambrian lattices associated to any finite crystallographic Coxeter group [AHI⁺, Miz14, IT09]. This paper continues a rich tradition of using lattices of torsion classes as a tool for proving new results in both representation theory and in algebraic combinatorics; see, e.g., [DIR⁺23, Eno23, IRRT18, RST21, TW19b, TW19a] and many others.

We focus our attention on the case when $\Lambda$ is $\tau$-tilting finite (in the sense of [DIJ17]); this is equivalent to assuming $\operatorname{\mathsf{tors}}\Lambda$ is finite. Our goal is to study the image and the dynamical properties of the pop-stack operator of $\operatorname{\mathsf{tors}}\Lambda$. While $\mathrm{pop}^{\downarrow}$ has already appeared (sometimes under different names) in the theory of lattices of torsion classes [AP22, BH, Eno23, ES22, Hana, Sak], it has primarily been used as a tool rather than a dynamical operator worthy of its own investigation. Let us remark that the article [BTZ21] initiated the study of dynamical combinatorics of torsion classes by considering rowmotion.

In Section 4, we consider certain pairs of sets of modules called 2-term simple-minded collections and semibrick pairs. 2-term simple-minded collections were first introduced in the special case when $\Lambda$ is a symmetric algebra in [Ric02]; they are special cases of the more general simple-minded collections introduced in [AN09] (under the name spherical collections). Semibrick pairs are a generalization of 2-term simple-minded collections that were introduced in [HI21b] as a tool for studying a generalization of the picture group of [ITW]. As we recall in Section 3.3, there is a bijection between $\operatorname{\mathsf{tors}}\Lambda$ (with $\Lambda$ $\tau$-tilting finite) and the set $\operatorname{2\textnormal{-}\mathsf{smc}}\Lambda$ of 2-term simple-minded collections [Asa20]. By the results of [BCZ19], this bijection encodes information about cover relations and canonical join representations in $\operatorname{\mathsf{tors}}\Lambda$.

The set $\operatorname{2\textnormal{-}\mathsf{smc}}\Lambda$ also comes equipped with a set of mutation operators that categorify the mutation theory of cluster algebras. See [BY13, Section 3.7] and [KY14, Section 7.2] (special cases also appeared earlier in [KS, Section 8.1] and [KQ15, Section 3]). These mutation formulas can also be applied to many of the more general semibrick pairs; see [BH22, HI21a]. Moreover, under the bijection between $\operatorname{2\textnormal{-}\mathsf{smc}}\Lambda$ and $\operatorname{\mathsf{tors}}\Lambda$, the mutation operators give a representation-theoretic formula for how the corresponding 2-term simple-minded collection changes when one traverses a cover relation in $\operatorname{\mathsf{tors}}\Lambda$.

For each semibrick pair $(\mathcal{X},\mathcal{Y})$, there is classically one mutation operator for each module in $\mathcal{X}\cup\mathcal{Y}$. In the present paper, we more generally define a mutation operator for each nonempty subset $\mathcal{X}^{\prime}\subseteq\mathcal{X}$ or $\mathcal{Y}^{\prime}\subseteq\mathcal{Y}$. Our first main result (Theorem 5.1) states that, under the bijection between $\operatorname{\mathsf{tors}}\Lambda$ and $\operatorname{2\textnormal{-}\mathsf{smc}}\Lambda$, applying the pop-stack operator and its dual to a torsion class corresponds to performing certain mutations on the associated 2-term simple-minded collections.

In Section 5.2, we characterize the preimages of a prescribed torsion class under $\mathrm{pop}^{\downarrow}_{\operatorname{\mathsf{tors}}\Lambda}$ and $\mathrm{pop}^{\uparrow}_{\operatorname{\mathsf{tors}}\Lambda}$. As corollaries, we obtain descriptions of the $1$-pop-stack sortable and the $2$-pop-stack sortable elements of $\operatorname{\mathsf{tors}}\Lambda$ (Corollaries 5.4 and 5.5).

Let $W$ be a finite irreducible Coxeter group, and let $\mathrm{Weak}(W)$ denote the (right) weak order on $W$. Given a Coxeter element $c$ of $W$, one can construct the $c$-Cambrian lattice $\mathrm{Camb}_{c}$, which is the sublattice of $\mathrm{Weak}(W)$ consisting of Reading’s $c$-sortable elements [Rea06, Rea07]. (The $c$-Cambrian lattice is also a lattice quotient of $\mathrm{Weak}(W)$.) When $W$ is crystallographic, we can realize $\mathrm{Camb}_{c}$ as the lattice of torsion classes of the tensor algebra $KQ_{c}$ associated to the weighted Dynkin quiver associated to $c$ [IT09].

Two very special Cambrian lattices are the Tamari lattice and the type-B Tamari lattice; the images of the pop-stack operators on these lattices were characterized and enumerated in [Hon22] and [CS], respectively. On the other hand, analogous results for the Cambrian lattice $\mathrm{Camb}_{c^{\times}}$ associated to a type-A bipartite Coxeter element $c^{\times}$ have remained elusive: a conjectural enumeration of the image was formulated by Defant and Williams [DW23], while a characterization of the image was conjectured by Choi and Sun [CS].

It turns out that our representation-theoretic perspective is quite useful for understanding the image of $\mathrm{pop}^{\downarrow}_{\mathrm{Camb}_{c}}$. In Theorem 5.10, we provide a representation-theoretic characterization of the image of $\mathrm{pop}^{\downarrow}_{\operatorname{\mathsf{tors}}\Lambda}$ whenever $\Lambda$ is hereditary. When the Coxeter group $W$ is crystallographic, the tensor algebra $KQ_{c}$ is hereditary, and we can reformulate our description of the image of the pop-stack operator in purely lattice-theoretic and Coxeter-theoretic terms. We then check directly (by hand and by computer) that these lattice-theoretic and Coxeter-theoretic descriptions still holds when $W$ is not crystallographic.

Our Coxeter-theoretic description of the image of $\mathrm{pop}^{\downarrow}_{\mathrm{Camb}_{c}}$ is surprisingly simple: we will prove (Theorem 7.8) that a $c$-sortable element $w\in\mathrm{Camb}_{c}$ is in the image of $\mathrm{pop}^{\downarrow}_{\mathrm{Camb}_{c}}$ if and only if the right descents of $w$ all commute and $w$ has no left inversions in common with $c^{-1}$.

To state our purely lattice-theoretic description of the image of $\mathrm{pop}^{\downarrow}_{\mathrm{Camb}_{c}}$, let us write $s_{1},\ldots,s_{n}$ for the simple reflections of $W$; these are the atoms of $\mathrm{Camb}_{c}$. For $1\leq i\leq n$, let $p_{i}$ be the unique maximal element of the set

$$\{x\in\mathrm{Camb}_{c}\mid s_{i}\leq x\text{ and }s_{j}\not\leq x\text{ for all }s_{j}\in S\setminus\{s_{i}\}\}.$$

We will prove that an element $w\in\mathrm{Camb}_{c}$ is in the image of $\mathrm{pop}^{\downarrow}_{\mathrm{Camb}_{c}}$ if and only if the interval $[\mathrm{pop}^{\downarrow}_{\mathrm{Camb}_{c}}(w),w]$ (in $\mathrm{Camb}_{c}$) is Boolean and $p_{i}\not\leq w$ for all $1\leq i\leq n$. When $W$ is crystallographic, the elements $p_{1},\ldots,p_{n}$ correspond in a natural way to the projective indecomposable modules of the tensor algebra $KQ_{c}$; this is one reason why our representation-theoretic perspective was so useful for discovering and proving our description of the image of $\mathrm{pop}^{\downarrow}_{\mathrm{Camb}_{c}}$.

Along the way to proving our characterization of the image of $\mathrm{pop}^{\downarrow}_{\mathrm{Camb}_{c}}$, we prove (as part of Theorem 7.6) that for $w\in\mathrm{Camb}_{c}$, the right descents of $w$ all commute with each other if and only if the interval $[\mathrm{pop}^{\downarrow}_{\mathrm{Camb}_{c}}(w),w]$ of $\mathrm{Camb}_{c}$ is equal to the interval $[\mathrm{pop}^{\downarrow}_{\mathrm{Weak}(W)}(w),w]$ of $\mathrm{Weak}(W)$. By combining this surprising result with our characterization of the image of $\mathrm{pop}^{\downarrow}_{\mathrm{Camb}_{c}}$, we obtain an equally surprising dynamical corollary (Corollary 7.10); namely, for $w\in\mathrm{Camb}_{c}$, we have

(1.1)

$$(\mathrm{pop}^{\downarrow}_{\mathrm{Weak}(W)})^{t}(\mathrm{pop}^{\downarrow}_{\mathrm{Camb}_{c}}(w))=(\mathrm{pop}^{\downarrow}_{\mathrm{Camb}_{c}})^{t+1}(w)$$

for all $t\geq 0$.

When $W$ is of type $A_{n}$ and $c=c^{\times}$ is a bipartite Coxeter element, our description of the image of $\mathrm{pop}^{\downarrow}_{\mathrm{Camb}_{c}}$ resolves the aforementioned conjecture of Choi and Sun [CS]. We will also construct a bijection from the image of $\mathrm{pop}^{\downarrow}_{\mathrm{Camb}_{c^{\times}}}$ to a certain set of Motzkin paths (Theorem 8.10); this allows us to resolve the aforementioned enumerative conjecture of Defant and Williams [DW23]. This result, the bijective proof of which utilizes the combinatorics of arc diagrams, provides an exact enumeration of the facets of the canonical join complex of a bipartite type-A Cambrian lattice.

Finally, we turn our attention to forward orbits under $\mathrm{pop}^{\downarrow}_{L}$ when $L$ is a lattice quotient of the weak order on a finite irreducible Coxeter group $W$. In this setting, we first show that $\max_{x\in L}\lvert\mathcal{O}_{L}(x)\rvert\leq h$, where $h$ is the Coxeter number of $W$. We then prove that this inequality is actually an equality when $L=\mathrm{Camb}_{c}$ for some Coxeter element $c$ of $W$. To do so, we utilize the combinatorial properties of the $c$-sorting word for the long element of $W$ to construct an element $\mathbf{z}_{c}\in\mathrm{Camb}_{c}$ such that $\lvert\mathcal{O}_{\mathrm{Camb}_{c}}(\mathbf{z}_{c})\rvert=h$.

Sections 2 and 3 provide necessary background on posets and representation theory, respectively. In Section 4, we discuss the theory of mutation of semibrick pairs, and we extend the theory to allow mutation at multiple bricks simultaneously; the proof of one of the main results of this section (Theorem 4.6) is postponed until Appendix A. Section 5 provides a representation-theoretic description of the pop-stack operator on $\operatorname{\mathsf{tors}}\Lambda$, describes preimages of a torsion class under $\mathrm{pop}^{\downarrow}_{\operatorname{\mathsf{tors}}\Lambda}$, and characterizes the image of $\mathrm{pop}^{\downarrow}_{\operatorname{\mathsf{tors}}\Lambda}$. Beginning in Section 6, we fixate on Cambrian lattices. Section 6 provides background on Coxeter groups, root systems, and Cambrian lattices (including their realizations as lattices of torsion classes). Section 7 is devoted to characterizing the image of the pop-stack operator on an arbitrary Cambrian lattice and deducing Equation 1.1. In Section 8, we provide explicit combinatorial characterizations and enumerations of the images of pop-stack operators on Cambrian lattices of type A. In Section 9, we prove our results concerning the maximum sizes of forward orbits under the pop-stack operators on lattice quotients of the weak order. Section 10 collects several ideas for future work.

By a subcategory of $\operatorname{\mathsf{mod}}\Lambda$, we will always mean a subcategory that is full and closed under isomorphisms. For a subcategory $\mathcal{C}\subseteq\operatorname{\mathsf{mod}}\Lambda$, we define several additional subcategories as follows:

• •

  $\operatorname{\mathsf{add}}(\mathcal{C})$ is the subcategory of $\operatorname{\mathsf{mod}}\Lambda$ consisting of all direct summands of finite direct sums of the objects in $\mathcal{C}$.

• •

  $\operatorname{\mathsf{Gen}}(\mathcal{C})$ is the subcategory of $\operatorname{\mathsf{mod}}\Lambda$ consisting of all objects that can be written as quotients of objects in $\operatorname{\mathsf{add}}(\mathcal{C})$.

• •

  $\operatorname{\mathsf{Cogen}}(\mathcal{C})$ is the subcategory of $\operatorname{\mathsf{mod}}\Lambda$ consisting of all objects that can be written as subobjects of objects in $\operatorname{\mathsf{add}}(\mathcal{C})$.

• •

  $\operatorname{\mathsf{Filt}}(\mathcal{C})$ is the subcategory of $\operatorname{\mathsf{mod}}\Lambda$ consisting of all objects that can be written as iterated extensions of objects in $\mathcal{C}$; that is, $M\in\operatorname{\mathsf{Filt}}(\mathcal{C})$ if and only if there exists a chain

  $$0=M_{0}\subsetneq M_{1}\subsetneq\cdots\subsetneq M_{k}=M$$

  of modules such that each factor $M_{i}/M_{i-1}$ is in $\operatorname{\mathsf{add}}(\mathcal{C})$.

• •

  $\mathsf{T}(\mathcal{C})=\operatorname{\mathsf{Filt}}(\operatorname{\mathsf{Gen}}(\mathcal{C}))$.

• •

  $\mathcal{C}^{\perp}=\mathcal{C}^{\perp_{0}}=\{Y\in\operatorname{\mathsf{mod}}\Lambda\mid\mathrm{Hom}_{\Lambda}(-,Y)|_{\mathcal{C}}=0\}$.

• •

  $\prescript{\perp}{}{\mathcal{C}}=\prescript{\perp_{0}}{}{\mathcal{C}}=\{Y\in\operatorname{\mathsf{mod}}\Lambda\mid\mathrm{Hom}_{\Lambda}(Y,-)|_{\mathcal{C}}=0\}$.

• •

  $\mathcal{C}^{\perp_{1}}=\{Y\in\operatorname{\mathsf{mod}}\Lambda\mid\mathrm{Ext}^{1}_{\Lambda}(-,Y)|_{\mathcal{C}}=0\}$

• •

  $\prescript{\perp_{1}}{}{\mathcal{C}}=\{Y\in\operatorname{\mathsf{mod}}\Lambda\mid\mathrm{Ext}^{1}_{\Lambda}(Y,-)|_{\mathcal{C}}=0\}$.

• •

  $\mathcal{C}^{\perp_{0,1}}=\mathcal{C}^{\perp}\cap\mathcal{C}^{\perp_{1}}$ and $\prescript{\perp_{0,1}}{}{\mathcal{C}}=\prescript{\perp}{}{\mathcal{C}}\cap\prescript{\perp_{1}}{}{\mathcal{C}}$.

We also apply the above definitions to the objects of $\operatorname{\mathsf{mod}}\Lambda$ by considering (the isomorphism class of) a given object as a subcategory.

Now let $\mathcal{C}\subseteq\operatorname{\mathsf{mod}}\Lambda$ be an arbitrary subcategory and $M\in\operatorname{\mathsf{mod}}\Lambda$. A minimal left $\mathcal{C}$-approximation of $M$ is a morphism $g_{M,\mathcal{C}}\colon M\rightarrow C$ with $C\in\mathcal{C}$ such that

1. (i)

   the induced map $(g_{M,\mathcal{C}})^{*}\colon\mathrm{Hom}_{\Lambda}(C,D)\rightarrow\mathrm{Hom}_{\Lambda}(M,D)$ is surjective for all $D\in\mathcal{C}$ and

2. (ii)

   if $f\colon C\rightarrow C$ is such that $f\circ g_{M,\mathcal{C}}=g_{M,\mathcal{C}}$, then $f$ is an isomorphism.

Dually, a minimal right $\mathcal{C}$-approximation of $M$ is a morphism $g_{\mathcal{C},M}\colon C\rightarrow M$ with $C\in\mathcal{C}$ such that

1. (i*)

   the induced map $(g_{\mathcal{C},M})_{*}\colon\mathrm{Hom}_{\Lambda}(D,C)\rightarrow\mathrm{Hom}_{\Lambda}(D,M)$ is surjective for all $D\in\mathcal{C}$ and

2. (ii*)

   if $f\colon C\rightarrow C$ is such that $g_{\mathcal{C},M}\circ f=g_{\mathcal{C},M}$, then $f$ is an isomorphism.

It is well known that minimal left and right $\mathcal{C}$-approximations are unique up to isomorphism whenever they exist. If every object of $\operatorname{\mathsf{mod}}\Lambda$ admits both a minimal left $\mathcal{C}$-approximation and a minimal right $\mathcal{C}$-approximation, then $\mathcal{C}$ is said to be functorially finite. Almost all of the subcategories we consider in this paper will be functorially finite.

A pair $(\mathcal{T},\mathcal{F})$ of subcategories of $\operatorname{\mathsf{mod}}\Lambda$ is called a torsion pair if

1. (i)

   $\mathrm{Hom}_{\Lambda}(T,F)=0$ for all $T\in\mathcal{T}$ and $F\in\mathcal{F}$ and

2. (ii)

   every $M\in\operatorname{\mathsf{mod}}\Lambda$ is the middle term of an exact sequence $t_{\mathcal{T}}M\hookrightarrow M\twoheadrightarrow f_{\mathcal{F}}M$ with $t_{\mathcal{T}}M\in\mathcal{T}$ and $f_{\mathcal{F}}M\in\mathcal{F}$.

In this case, we say that $\mathcal{T}$ is a torsion class and that $\mathcal{F}$ is a torsion-free class. We denote by $\operatorname{\mathsf{tors}}\Lambda$ and $\operatorname{\mathsf{torsf}}\Lambda$ the sets of torsion classes and torsion-free classes in $\operatorname{\mathsf{mod}}\Lambda$, respectively. Torsion pairs were introduced by Dickson in [Dic66] as a generalization of the torsion and torsion-free abelian groups. We refer to the expository article [Tho21] for background information on torsion pairs, including formal statements for the facts mentioned in this section.

It is well known that a subcategory $\mathcal{T}\subseteq\operatorname{\mathsf{mod}}\Lambda$ (resp. $\mathcal{F}\subseteq\operatorname{\mathsf{mod}}\Lambda$) is a torsion class (resp. torsion-free class) if and only if it is closed under extensions and quotients (resp. extensions and submodules). Moreover, every torsion class determines a unique torsion pair and vice versa via the associations $\mathcal{T}\mapsto(\mathcal{T},\mathcal{T}^{\perp})$ and $(\mathcal{T},\mathcal{F})\mapsto\mathcal{T}$. Similarly, every torsion-free class determines a unique torsion pair and vice versa via the associations $\mathcal{F}\mapsto(\prescript{\perp}{}{\mathcal{F}},\mathcal{F})$ and $(\mathcal{T},\mathcal{F})\mapsto\mathcal{F}$.

For $\mathcal{C}\subseteq\operatorname{\mathsf{mod}}\Lambda$ an arbitrary subcategory, we have that $\mathsf{T}(\mathcal{C})$ is the smallest torsion class containing $\mathcal{C}$; the corresponding torsion pair is $(\mathsf{T}(\mathcal{C}),\mathcal{C}^{\perp})$. Dually, $\operatorname{\mathsf{Filt}}(\operatorname{\mathsf{Cogen}}(\mathcal{C}))$ is the smallest torsion-free class containing $\mathcal{C}$, and the corresponding torsion pair is $(\prescript{\perp}{}{\mathcal{C}},\operatorname{\mathsf{Filt}}(\operatorname{\mathsf{Cogen}}(\mathcal{C})))$.

We consider $\operatorname{\mathsf{tors}}\Lambda$ and $\operatorname{\mathsf{torsf}}\Lambda$ as posets under the containment relation. Each of these is a complete lattice whose meet operation is given by intersection and whose join operation is given by $\bigvee\mathcal{A}=\operatorname{\mathsf{Filt}}\left(\bigcup_{\mathcal{T}\in\mathcal{A}}\mathcal{T}\right)$. The associations $\mathcal{T}\mapsto\mathcal{T}^{\perp}$ and $\mathcal{F}\mapsto\prescript{\perp}{}{\mathcal{F}}$ thus give lattice anti-isomorphisms between $\operatorname{\mathsf{tors}}\Lambda$ and $\operatorname{\mathsf{torsf}}\Lambda$. In particular, the join operation in $\operatorname{\mathsf{tors}}\Lambda$ can be rewritten as ${\mathcal{T}\vee\mathcal{U}=\prescript{\perp}{}{(}{\mathcal{T}^{\perp}\cap\mathcal{U}^{\perp}})}$. The maximum element of $\operatorname{\mathsf{tors}}\Lambda$ is $\hat{1}=\operatorname{\mathsf{mod}}\Lambda$ (the torsion class consisting of all modules), and the minimum element is $\hat{0}=0$ (the torsion class consisting only of the zero module). The atoms of $\operatorname{\mathsf{tors}}\Lambda$ are precisely those torsion classes of the form $\operatorname{\mathsf{Filt}}(S)$ for $S\in\operatorname{\mathsf{mod}}\Lambda$ a simple module.

In this paper, we restrict our attention to algebras $\Lambda$ for which the lattice $\operatorname{\mathsf{tors}}\Lambda$ is finite. These are precisely the $\tau$-tilting finite algebras introduced in [DIJ17]. They are also precisely the algebras $\Lambda$ for which every torsion class in $\operatorname{\mathsf{mod}}\Lambda$ is functorially finite. A related class is that of representation-finite algebras, which are characterized by the property that $\operatorname{\mathsf{mod}}\Lambda$ contains only finitely many indecomposable modules (up to isomorphism). If $\Lambda$ is hereditary, then it is $\tau$-tilting finite if and only if it is representation-finite. For $\Lambda$ not hereditary, representation-finiteness implies $\tau$-tilting finiteness, but not vice versa. Another special class is that of silting discrete algebras [AM17]; an algebra $\Lambda$ belongs to this class when every algebra that is derived equivalent to $\Lambda$ (including $\Lambda$ itself) is $\tau$-tilting finite. All representation-finite hereditary algebras are silting discrete.

From now on, $\Lambda$ will always denote a $\tau$-tilting finite algebra. While many of the results of the following subsections admit generalizations to arbitrary finite-dimensional algebras, this allows us to streamline the exposition.

Recall that a module $X\in\operatorname{\mathsf{mod}}\Lambda$ is called a brick if $\mathrm{End}_{\Lambda}(X)$ is a division ring. A set $\mathcal{X}$ of bricks is called a semibrick if $X\in Y^{\perp}\cap\prescript{\perp}{}{Y}$ for all distinct $X,Y\in\mathcal{X}$. We adopt the convention of using the term $\mathcal{X}$ to represent both a set of bricks and the module $\bigoplus_{X\in\mathcal{X}}X$. We denote by $\operatorname{\mathsf{brick}}\Lambda$ and $\operatorname{\mathsf{sbrick}}\Lambda$ the sets of (isomorphism classes of) bricks and semibricks in $\operatorname{\mathsf{mod}}\Lambda$, respectively. This leads to the following definition, which we state in this form in light of [BY13, Remark 4.11]. See Section 1.2 for historical notes and references.

The lattice of torsion classes is known to be a (completely) semidistributive lattice [DIR⁺23, GM19]. Therefore, each edge in the Hasse diagram of $\operatorname{\mathsf{tors}}\Lambda$ has a join-irreducible shard label as described at the end of Section 2. In this section, we recall from [Asa20] ($\tau$-tilting finite case) and [BCZ19, DIR⁺23] (general case) how this labeling can be formulated and interpreted representation-theoretically. We note that many of the results recalled in this section have been specialized to the $\tau$-tilting finite case.

The brick label of a cover relation $\mathcal{T}{\,\,<\!\!\!\!\cdot\,\,\,}\mathcal{T}^{\prime}$ in $\operatorname{\mathsf{tors}}\Lambda$ is the unique brick $X$ for which $j_{\mathcal{T},\mathcal{T}^{\prime}}=\mathsf{T}(X)$. For a fixed $\mathcal{T}\in\operatorname{\mathsf{tors}}\Lambda$, we (slightly abusing notation) denote by $\mathcal{D}(\mathcal{T})$ and $\mathcal{U}(\mathcal{T})$ the sets of bricks that label cover relations of the form $\mathcal{T}^{\prime}{\,\,<\!\!\!\!\cdot\,\,\,}\mathcal{T}$ and $\mathcal{T}{\,\,<\!\!\!\!\cdot\,\,\,}\mathcal{T}^{\prime}$, respectively.

Theorems 3.4 and 3.5 immediately yield the following characterization of the pop-stack operators.

We now recall how Theorem 3.4 relates to 2-term simple-minded collections. Once again, we note that the following theorem is simplified since we consider only the case where every torsion class is functorially finite. This result is essentially contained in [Asa20, Theorems 2.3 and 2.12], but we give a short argument utilizing [BTZ21, Corollary 5.1.8] and other results from [Asa20]. Note that it is implicitly included in Theorem 3.71 that the sets $\mathcal{D}(\mathcal{T})$ and $\mathcal{U}(\mathcal{T})$ are semibricks for any $\mathcal{T}\in\operatorname{\mathsf{tors}}\Lambda$.

In particular, Theorem 3.7 implies that any semibrick pair of the form $(\mathcal{X},\emptyset)$ or $(\emptyset,\mathcal{Y})$ is completable. For $(\mathcal{X},\mathcal{Y})\in\operatorname{2\textnormal{-}\mathsf{smc}}\Lambda$, we write $\mathsf{T}(\mathcal{X},\mathcal{Y})=\mathsf{T}(\mathcal{X})=\prescript{\perp}{}{\mathcal{Y}}$.

A subcategory $\mathcal{W}\subseteq\operatorname{\mathsf{mod}}\Lambda$ is called wide if it is closed under extensions, kernels, and cokernels. Alternatively, a subcategory is wide if it is an exact-embedded abelian subcategory. We denote by $\operatorname{\mathsf{wide}}\Lambda$ the set of wide subcategories of $\operatorname{\mathsf{mod}}\Lambda$.

It is a classical result of [Rin76] that there is a bijection $\operatorname{\mathsf{sbrick}}\Lambda\rightarrow\operatorname{\mathsf{wide}}\Lambda$ given by $\mathcal{X}\mapsto\operatorname{\mathsf{Filt}}(\mathcal{X})$. The inverse sends a wide subcategory $\mathcal{W}$ to the set of objects that are simple in $\mathcal{W}$ (i.e., those $X\in\mathcal{W}$ that admit no proper submodules in $\mathcal{W}$). Combining this with Theorem 3.7, we see that, under the $\tau$-tilting finite hypothesis, there are two bijections $\operatorname{\mathsf{tors}}\Lambda\rightarrow\operatorname{\mathsf{wide}}\Lambda$ given by $\mathcal{T}\mapsto\operatorname{\mathsf{Filt}}(\mathcal{D}(\mathcal{T}))$ and $\mathcal{T}\mapsto\operatorname{\mathsf{Filt}}(\mathcal{U}(\mathcal{T}))$. The inverses of these bijections are given by $\mathcal{W}\mapsto\mathsf{T}(\mathcal{W})$ and $\mathcal{W}\mapsto\prescript{\perp}{}{\mathcal{W}}$, respectively. Moreover, these bijections correspond to the so-called Ingalls–Thomas bijections of [IT09, MŠ17]; that is, we have the following. (See also [Eno23, Remark 4.20], which relates this result to the rowmotion operators.)

We conclude this section with a brief discussion of Serre subcategories. A subcategory ${\mathcal{W}\subseteq\operatorname{\mathsf{mod}}\Lambda}$ is called Serre if it is closed under extensions, submodules, and quotient modules. The following is well known.

Given a wide subcategory $\mathcal{W}$, we will also say that a wide subcategory $\mathcal{W}^{\prime}\subseteq\mathcal{W}$ is Serre in $\mathcal{W}$ if every object that is simple in $\mathcal{W}^{\prime}$ is also simple in $\mathcal{W}$.