Triangulations of prisms and
preprojective algebras of type $A$

$\tau$-tilting theory was introduced in [AIR14] as a generalisation of cluster-tilting theory. A $\Lambda$-module $M$ is called $\tau$-rigid if $\operatorname{Hom}_{\Lambda}(M,$ $\tau M)=0$. A pair $(M,P)$ of $\Lambda$-modules where $P$ is projective is called $\tau$-rigid if $M$ is $\tau$-rigid and $\operatorname{Hom}(P,M)=0$. A $\tau$-rigid pair $(M,P)$ is called support $\tau$-tilting if $|M|+|P|=|\Lambda|$, where $|X|$ denotes the number of non-isomorphic indecomposable direct summands of $X$. Here $M$ is called a support $\tau$-tilting module. We call a $\tau$-rigid pair indecomposable if it is of the form $(M,0)$ with $M$ indecomposable, or of the form $(0,P)$ with $P$ indecomposable. We refer to $\tau$-rigid pairs of the form $(0,P)$ as shifted projectives. We write $\mathsf{s\tau\mbox{-}tilt}\,\Lambda$ for the set of (isomorphism classes of) basic support $\tau$-tilting pairs over $\Lambda$ and $\mathsf{ind\mbox{-}\tau\mbox{-}rigid\mbox{-}pair}\,\Lambda$ for the set of (isomorphism classes of) basic indecomposable $\tau$-rigid pairs over $\Lambda$.

Two-term silting complexes over $\Lambda$ are in bijection with support $\tau$-tilting pairs over $\Lambda$ and are in many ways nicer to work with [AIR14]. An object $T$ of $\mathcal{K}_{\Lambda}$ is presilting if

for all $i>0$. A presilting complex $T$ is silting if, additionally, $\operatorname{thick}T=\mathcal{K}_{\Lambda}$. Here $\operatorname{thick}T$ denotes the smallest full subcategory of $\mathcal{K}_{\Lambda}$ which contains $P$ and is closed under cones, $[\pm 1]$, direct summands, and isomorphisms. For a two-term complex $T$ to be presilting, it suffices that $\operatorname{Hom}_{\mathcal{K}_{\Lambda}}(T,T[1])=0$. Moreover, for a presilting two-term complex $T$ to be silting, it suffices that $|T|=|\Lambda|$ by [AIR14, Proposition 3.3(b)]. We write $\mathsf{2\mbox{-}silt}\,\Lambda$ for the set of (isomorphism classes of) two-term silting complexes over $\Lambda$ and $\mathsf{ind\mbox{-}2\mbox{-}psilt}\,\Lambda$ for the set of (isomorphism classes of) indecomposable two-term presilting complexes over $\Lambda$.

Given two-term silting complexes $T,T^{\prime}\in\mathsf{K}^{[-1,0]}(\mathsf{proj}\,\Lambda)$, we say that $T^{\prime}$ is a mutation of $T$ if and only if $T=E\oplus X,T^{\prime}=E\oplus Y$, where $X$ and $Y$ are indecomposable with $X\not\cong Y$.

Adachi, Iyama, and Reiten showed that there was a bijection between $\mathsf{ind\mbox{-}2\mbox{-}psilt}\,\Lambda$ and $\mathsf{ind\mbox{-}\tau\mbox{-}rigid\mbox{-}pair}\,\Lambda$ which induced a bijection between $\mathsf{2\mbox{-}silt}\,\Lambda$ and $\mathsf{s\tau\mbox{-}tilt}\,\Lambda$ [AIR14, Theorem 3.2]. Here, if $(M,P)$ is an indecomposable support $\tau$-rigid pair, then $P^{-1}\oplus P\to P^{0}$ is a two-term presilting complex, where $P^{-1}\to P^{0}$ is a minimal projective presentation of $M$.

The algebras we are interested in are the preprojective algebras of type $A$, which are defined as follows using quivers and relations. Preprojective algebras were originally defined by Gel\cprimefand and Ponomarev [GfP79]. They were subsequently generalised to non-simply-laced types in [GLS17]. The preprojective algebra of $A_{n}$, denoted $\Pi_{n}$, has quiver $Q_{n}$

with relations $\beta_{i}\alpha_{i}=\alpha_{i+1}\beta_{i+1}$ for $i\in\{1,2,\dots,n-1\}$, and $\alpha_{1}\beta_{1}=\beta_{n-1}\alpha_{n-1}=0$. We compose arrows using the convention $\xrightarrow{\gamma}\xrightarrow{\delta}=\gamma\delta$. We denote the idempotent at vertex $i$ by $e_{i}$ and the indecomposable projective $\Pi_{n}$-module at vertex $i$ by $P_{i}$. The module $P_{i}$ is also the indecomposable injective at vertex $n-i+1$.

In order to study the $\tau$-tilting theory of $\Pi_{n}$, it will be useful to introduce a particular quotient of it. We let

in $\Pi_{n}$. This is a central element of $\Pi_{n}$. We define

Then $\overline{\Pi}_{n}$ is the quotient of $\Pi_{n}$ by the ideal generated by all two-cycles. That is, $\overline{\Pi}_{n}=\Pi_{n}/I$, where $I=\langle\alpha_{i}\beta_{i},\beta_{i}\alpha_{i}\mid i\in\{1,2,\dots,n-1\}\rangle$. This algebra was considered in [EJR18, BCZ19, DIR⁺17]. Denote by $\overline{P}_{i}$ the projective $\overline{\Pi}_{n}$-module at vertex $i$. By [EJR18, Theorem 11], the functor $-\otimes_{\Pi_{n}}\overline{\Pi}_{n}\colon\mathcal{K}_{\Pi_{n}}\to\mathcal{K}_{\overline{\Pi}_{n}}$ induces a bijection $\mathsf{2\mbox{-}silt}\,\Pi_{n}\to\mathsf{2\mbox{-}silt}\,\overline{\Pi}_{n}$. Given a two-term silting complex $P^{\bullet}$ of $\Pi_{n}$, we write $\overline{P}^{\bullet}$ for the corresponding two-term silting complex over $\overline{\Pi}_{n}$.

Let $P^{\bullet}=P^{-1}\to P^{0}$ be a two-term complex over projectives over $\Pi_{n}$, where $P^{0}=\bigoplus_{i=1}^{n}P_{i}^{r_{i}}$ and $P^{-1}=\bigoplus_{i=1}^{n}P_{i}^{s_{i}}$. Then the $g$-vector of $P^{\bullet}$ is

Note that this is the class of $P^{\bullet}$ in the Grothendieck group $K_{0}(\mathcal{K}_{\Pi_{n}})=K_{0}(\mathsf{proj}\,\Pi_{n})$, identified with $\mathbb{Z}^{n}$. Given a $\tau$-rigid pair $(M,P)$, we define $g^{(M,P)}:=g^{P^{\bullet}}$ where $P^{\bullet}$ is the corresponding two-term presilting complex. It is known that two-term presilting complexes and support $\tau$-rigid pairs are uniquely determined by their $g$-vectors [AIR14, Theorem 5.5].

It was shown in [Miz14] (see also [BIRS09]) that support $\tau$-tilting modules over $\Pi_{n}$ were in bijection with permutations in $\mathfrak{S}_{n+1}$. This was generalised to non-simply-laced types in [FG19, Mur22]. The bijection is constructed as follows. Define $I_{i}:=\Pi_{n}(1-e_{i})\Pi_{n}$, the principal two-sided ideal of $\Pi_{n}$ generated by $(1-e_{i})$. Then, for $w\in\mathfrak{S}_{n+1}$ with reduced expression $w=s_{i_{1}}s_{i_{2}}\dots s_{i_{k}}$, define

The map $w\mapsto I_{w}$ then defines a bijection from $\mathfrak{S}_{n+1}$ to the support $\tau$-tilting modules over $\Pi_{n}$. In the case $w=e$, the identity permutation, the corresponding support $\tau$-tilting module is the regular module $\Pi_{n}$. We write $P_{w}$ for the projective module $P_{w}$ such that $(I_{w},P_{w})$ is a support $\tau$-tilting pair.

We now explain the relevant background on prisms of simplices, following [DLRS10, Section 6.2.1]. The prism of the $n$-simplex $\Delta_{n}$ is the polytope $\Delta_{n}\times\Delta_{1}$. The particular geometric realisation of this polytope is not important for the results of this paper, but for the sake of clarity, we take $\Delta_{n}\times\Delta_{1}$ to be the convex hull of the points given by the column vectors of the $(n+3)\times(2n+2)$ matrix

We label the points given by the first $n+1$ columns of this matrix by $a_{0},a_{1},\dots,a_{n}$ and the points given by the last $n+1$ columns by $b_{0},b_{1},\dots,b_{n}$. We write $V=\{a_{0},a_{1},\dots,a_{n},b_{0},b_{1},\dots,b_{n}\}$ for the set of vertices of the prism $\Delta_{n}\times\Delta_{1}$. One can specify $k$-simplices in the prism $\Delta_{n}\times\Delta_{1}$ by giving subsets of $V$ of size $k+1$, provided the chosen vertices are affinely independent. An internal $k$-simplex is one which does not lie in the boundary of $\Delta_{n}\times\Delta_{1}$. We write $\mathsf{int\mbox{-}sim}_{n}(\Delta_{n}\times\Delta_{1})$ for the set of internal $n$-simplices of $\Delta_{n}\times\Delta_{1}$.

We again follow [DLRS10]. A polyhedral subdivision $\mathcal{S}$ of $\Delta_{n}\times\Delta_{1}$ is a set of $(n+1)$-dimensional convex polytopes $\{L_{1},\dots,L_{r}\}$ such that

• •

  $\bigcup_{i=1}^{r}L_{i}=\Delta_{n}\times\Delta_{1}$ and

• •

  $L_{i}\cap L_{j}$ is a face of both $L_{i}$ and $L_{j}$ for all $i,j$.

A polyhedral subdivision $\mathcal{T}$ is a triangulation if $L_{i}$ is a simplex for every $i$. We write $\mathsf{tri}(\Delta_{n}\times\Delta_{1})$ for the set of triangulations of $\Delta_{n}\times\Delta_{1}$.

One polyhedral subdivision $\mathcal{S}=\{L_{1},L_{2},\dots,L_{r}\}$ refines another polyhedral subdivision $\mathcal{S}^{\prime}=\{L^{\prime}_{1},L^{\prime}_{2},\dots,L^{\prime}_{r^{\prime}}\}$ if for every $L_{i}$ we have $L_{i}\subseteq L^{\prime}_{j}$ for some $j$, and $\mathcal{S}\neq\mathcal{S}^{\prime}$. A polyhedral subdivision is an almost triangulation if all of its refinements are triangulations. Two triangulations $\mathcal{T}$ and $\mathcal{T}^{\prime}$ are related by a bistellar flip if there is an almost triangulation $\mathcal{S}$ whose only two refinements are $\mathcal{T}$ and $\mathcal{T}^{\prime}$.

A circuit of $\Delta_{n}\times\Delta_{1}$ is a pair $(Z,Z^{\prime})$ of two disjoint vertex subsets $Z$ and $Z^{\prime}$ such that $\operatorname{conv}(Z)\cap\operatorname{conv}(Z^{\prime})\neq\varnothing$ and such that $Z$ and $Z^{\prime}$ are minimal with respect to this property. Here ‘$\operatorname{conv}$’ denotes the convex hull. The circuits of $\Delta_{n}\times\Delta_{1}$ are given by $(\{a_{i},b_{j}\},\{a_{j},b_{i}\})$ for $i\neq j$.

Triangulations of $\Delta_{n}\times\Delta_{1}$ are in bijection with permutations in the symmetric group $\mathfrak{S}_{n+1}$, as shown in [GKZ94, Chapter 7, Section 3C]. Indeed, for a permutation $w=i_{0}i_{1}\dots i_{n}\in\mathfrak{S}_{n+1}$, we have that

is the set of $(n+1)$-simplices of the corresponding triangulation, and all triangulations of $\Delta_{n}\times\Delta_{1}$ arise in this way. For a permutation $w\in\mathfrak{S}_{n+1}$, we write $\mathcal{T}_{w}$ for the corresponding triangulation of $\Delta_{n}\times\Delta_{1}$.