Bangle functions are the generic basis for cluster algebras from punctured surfaces with boundary

Fomin and Zelevinsky introduced around 2001 cluster algebras [17, 18, 19] as a device to study dual canonical bases in quantum groups and the closely related theory of total positivity in complex reductive groups. They expected that in this context all cluster monomials should belong to the dual canonical basis. This question was only recently settled by Kang–Kashiwara–Kim–Oh [30] for the case of generalized symmetric Cartan matrices and by Qin [48] for generalized symmetrizable Cartan matrices. Previously, the first named author together with Leclerc and Schröer [25] had identified the dual of Lusztig’s semicanonical basis with the generic Caldero-Chapoton functions, which also contain all cluster monomials.

Since the set of cluster monomials is always linearly independent [4] this led more generally to the search of bases for (upper) cluster algebras which contain the cluster monomials, and which ideally expand positively with respect to each cluster. Moreover, such ”good” bases are typically parametrized by the by tropical points of the corresponding cluster Poisson variety. By the landmark work of Gross–Hacking–Keel–Kontsevich [28] on theta functions at least one such basis exists under rather weak hypothesis. See the remarkable paper [49] for a thorough discussion.

Parallel to these developments, cluster algebras have found applications in a surprising variety of subjects, see for example the surveys [14], [31],[32], [37].

A prominent class of cluster algebras stems from surfaces with marked points, thanks to works of Fock–Goncharov [13], Fomin–Shapiro–Thurston [15], Fomin–Thurston [16], Gekhtman–Shapiro–Vainshtein [27] and Penner [46].

Let $\mathbf{\Sigma}=(\Sigma,\mathbb{M},\mathbb{P})$ be a (bordered) surface with marked points in the sense of [15], where $\mathbb{M}$ denotes the set of marked points on the boundary and $\mathbb{P}$ is the set of punctures. See Definition 2.5 for more details. To associate a (coefficient-free) cluster algebra to $\mathbf{\Sigma}$ one considers tagged triangulations of $\mathbf{\Sigma}$. Following [15, Def. 9.6] each tagged triangulation $T$ has an associated skew-symmetric matrix $B(T)$, see also Definition 2.9 for details. The matrix $B(T)$ changes under any flip of an arc in $T$ according to the Fomin–Zelevinsky mutation rule. This allows us already to define a cluster algebra $\mathcal{A}(\mathbf{\Sigma})=\mathcal{A}(B(T),\mathbf{x})$ which is actually independent of the choice of the triangulation $T$. In fact, a fundamental result of Fomin–Shapiro–Thurston [16, Cor. 6.2] and [15, Thm. 7.11] states that provided $\mathbf{\Sigma}$ is not a closed surface with exactly one puncture, there is a bijection $\gamma\mapsto x_{\gamma}$ between the set of tagged arcs on $\mathbf{\Sigma}$ and the set of cluster variables of $\mathcal{A}(B(T_{0}),\mathbf{x})$, such that the induced assignment $T\mapsto(B(T),(x_{\gamma}))_{\gamma_{\in}T}$ is a bijection between the set of triangulations of $\mathbf{\Sigma}$ and the set of (unlabelled) seeds of $\mathcal{A}(B(T_{0}),\mathbf{x})$, such that whenever two tagged triangulations are related by the flip of a tagged arc, the seeds corresponding to them are related by the mutation of seeds corresponding to the arc being flipped.

Given a surface $\mathbf{\Sigma}=(\Sigma,\mathbb{M},\mathbb{P})$ and a tagged triangulation $T$ of $\mathbf{\Sigma}$, Musiker–Schiffler–Williams associated to each tagged arc (resp. simple closed curve) $\alpha$ on $\mathbf{\Sigma}$, a bipartite graph $G(T,\alpha)$, which they called snake graph (resp. band graph) [43, 44], with edges and tiles naturally labelled by the arcs in $T$ (and the boundary segments of $\mathbf{\Sigma}$). Using these labels, they assigned a weight monomial $w(P):=x(P)y(P)$ to each perfect (resp. good) matching $P$ of $G(T,\alpha)$, thus producing a generating function $\sum_{P}w(P)$ for the perfect matchings (resp. good matchings) of $G(T,\alpha)$. Dividing this generating function by the monomial $\operatorname{cross}(T,\alpha)$ that records the number of crossings of $\alpha$ with each arc in $T$, they define the bangle function

Furthermore, they define the bangle function associated to a lamination $L\in\operatorname{Lam}(\mathbf{\Sigma})$ (see Section 3.3 for details) to be

where $m_{\alpha}$ is the multiplicity of the single laminate $\alpha$ as a member of $L$. The set of bangle functions is

Musiker–Schiffler–Williams show in [44] that if the underlying surface $(\Sigma,\mathbb{M},\mathbb{P})$ does not have punctures and has at least two marked points (i.e., $\mathbb{P}=\varnothing$ and $|\mathbb{M}|\geq 2$), then for the coefficient-free cluster algebra $\mathcal{A}(\mathbf{\Sigma})$ associated to $(\Sigma,\mathbb{M},\mathbb{P})$, and for any cluster algebra associated to $(\Sigma,\mathbb{M},\mathbb{P})$ under full-rank extended exchange matrices $\left[\begin{array}[]{c}B(T)\\ B^{\prime}\end{array}\right]$, that (after appropriate specialisations in the sense of Fomin-Zelevinsky’s separation of additions formula [Theorem 3.7, [19]]) both the set of bangle functions and the set of bracelet functions form bases of the associated cluster algebra, each containing the cluster monomials by [43]^†. They conjecture in [44] that $\mathcal{B}^{\circ}$

forms a basis also in the punctured case.

Let $Q$ be a 2-acyclic quiver with vertex set $Q_{0}=\{1,2,\ldots,n\}$, and $B(Q)=(b_{ij})\in\mathbb{Z}^{Q_{0}\times Q_{0}}$ the corresponding skew-symmetric matrix with with entries

as in [10, Equation (1.4)]. In this context we abbreviate $\mathcal{A}(Q):=\mathcal{A}(B(Q))$ for the corresponding cluster algebra with trivial coefficients.

Recall that for each basic $\mathbb{C}$-algebra $A=\mathbb{C}\langle\!\langle Q\rangle\!\rangle/I$ and a representation $M$ of $A$ we have the $F$-polynomial

where $\mathbf{y}^{\mathbf{e}}:=\displaystyle\prod_{i=1}^{n}y_{i}^{e_{i}}$ and $\chi(\operatorname{Gr}^{A}_{\mathbf{e}}(M))$ is the topological Euler characteristic of the quiver Grassmannian of subrepresentations of $M$ with dimension vector $\mathbf{e}$.

Building on this, one may define the Caldero–Chapoton function

where $\hat{y}_{j}:=\displaystyle\prod_{i=1}^{n}x_{i}^{b_{ij}}$ for each $j\in Q_{0}$ and $\mathbf{g}(M)\in\mathbb{Z}^{Q_{0}}$ is the injective g-vector of $M$.

Together with Leclerc and Schröer, the first named author explored in [25] the algebraic geometry of (affine) varieties of representations of Jacobi-finite non-degenerate quivers with potential $(Q,S)$ which appear in algebraic Lie theory. They isolated a class of irreducible components of these representation spaces with good geometric-homological properties and called them strongly reduced irreducible components. Later on, the adjective used was changed to generically $\tau^{-}$-reduced, or simply, $\tau^{-}$-reduced, cf. [23]. Roughly speaking, a component $Z\subseteq\operatorname{rep}(\mathcal{P}(Q,S),\mathbf{d})$ is $\tau^{-}$-reduced if the codimension in $Z$ of a top dimensional $\operatorname{GL}_{\mathbf{d}}(\mathbb{C})$-orbit $\mathcal{O}\subseteq Z$ is equal to the minimum value taken on $Z$ by Derksen-Weyman-Zelevinsky’s (injective) $E$-invariant. See [6, Sec. 3 and 5] for more details.

It is easy to see that each $\tau^{-}$-reduced irreducible component $Z$ has an open dense subset where the constructible function $M\mapsto CC_{\mathcal{P}(Q,S)}(M)$ takes a constant value, which is then denoted $CC_{\mathcal{P}(Q,S)}(Z)$. Resorting to decorated representations and spaces of decorated representations in order to be able to hit initial cluster variables through the Caldero-Chapoton function, the set of generic Caldero-Chapoton functions is

It is in fact quite easy to see with the help of [10] that for each Jacobi-finite non-degenerate quiver with potential $(Q,S)$ the set $\mathcal{B}_{\mathcal{P}(Q,S)}$ contains all cluster monomials of the cluster algebra $\mathcal{A}(Q)$, and that all elements of $\mathcal{B}_{\mathcal{P}(Q,S)}$ belong to the upper cluster algebra $\mathcal{U}(Q)$ corresponding to $Q$. If the set $\mathcal{B}_{\mathcal{P}(Q,S)}$ is a basis of the (coefficient-free) upper cluster algebra then it is called the generic basis of $\mathcal{U}(Q)$.

One of the main results of [25] is, that in the setting of unipotent cells for symmetric Kac-Moody groups, the generic Caldero–Chapoton functions can be identified with the dual of Lusztig’s semicanonical basis [38], and that this basis contains in particular all cluster monomials. This led the authors of [25] to conjecture that the set $\mathcal{B}_{\mathcal{P}(Q,S)}$ might be a basis of the upper cluster algebra in more situations beyond the algebraic Lie theoretic context.

The set $\mathcal{B}_{\mathcal{P}(Q,S)}$ does not change (up to cluster automorphism) under mutations as a subset of $\mathcal{U}(Q)$. This was shown by Plamondon when $(Q,S)$ is Jacobi-finite non-degenerate [47], and by Schröer and the first two authors when $(Q,S)$ is arbitrary non-degenerate [22].

Let $\mathbf{\Sigma}$ be a surface with marked points. To each tagged triangulation $T$ of $\mathbf{\Sigma}$ we associate a quiver $Q^{\prime}(T)$ such that $B(Q^{\prime}(T))=B(T)$. This convention allows us to state and prove our results without the roundabout with dual CC-functions as for example in [23]. Schröer and the first two named authors showed [22, Thm. 1.18], relying heavily on results of Mills [40], Muller [41, 42] and Qin [49], that when $(Q,S)=(Q^{\prime}(T),S^{\prime}(T))$ for some triangulation $T$ of a surface $\mathbf{\Sigma}=(\Sigma,\mathbb{M},\mathbb{P})$ with at least two marked points on the boundary, the set of generic CC-functions $\mathcal{B}_{\mathcal{P}(Q,S)}$ is indeed a basis for the (upper) cluster algebra $\mathcal{A}(\mathbf{\Sigma})=\mathcal{U}(\mathbf{\Sigma})$, where $S^{\prime}(T)$ is the potential defined by Cerulli Irelli and the second author in [5]. Let us write from now on $A(T)$ for the corresponding complex Jacobian algebra $\mathcal{P}_{\mathbb{C}}(Q^{\prime}(T),W^{\prime}(T))$. Note that due to our convention here, the algebra $A(T)$ is opposite to the one which was considered for example in [24]. We are now ready to state our main result, Theorem 6.3, in a slightly simplified version.

The paper is organized as follows. In Section 2 we provide the basics about mutations and cluster algebras from surfaces. In Section 3 we summarize the main constructions and results from [24] upon which we will rely in Section 6, the most relevant one being [24, Proposition 2.8] (stated as Proposition 3.8 below), which gives a precise recursive formula for the change that the generic projective $g$-vector of any $\tau$-reduced component undergoes under mutation of $\tau$-reduced components.

It should be stressed that, since elements of $\mathcal{B}^{\circ}$ were defined directly from the associated snake or band graphs, then, a priori, this viewpoint says nothing about whether $\mathcal{B}^{\circ}$ is canonical in a cluster-theoetrical sense. That is, whether or not $\mathcal{B}^{\circ}$ is independent of the choice of triangulation, up to a cluster isomorphism.

Throughout this paper, our approach works by flipping the perspective of [43] — we fix a curve on the surface and alter the underlying triangulations through the process of cluster mutation. Since we wish to work with principal coefficients, extra care is needed here. Indeed, we consequently require a framework that describes how cluster variables transform from one triangulation to the next, with respect to principal coefficients at those triangulations. This is achieved through the following lemma:

As an immediate corollary, we obtain that the set of bangle functions $\mathcal{B}^{\circ}(\mathbf{\Sigma})=\mathcal{B}^{\circ}(\mathbf{\Sigma},T)$ is, up to cluster automorphism, independent of the choice of $T$. Note that a version of this lemma first appeared in the work of Derksen-Weyman-Zelevinsky [[10], Lemma 5.2] which was restricted the setting of cluster variables — our result shows this lemma holds on all elements of the basis $\mathcal{B}^{\circ}$.

In Section 4 we show that for simple closed curve $\alpha$, there exists a tagged $T_{\alpha}$ for which one can explicitly compute a (decorated) representation $\mathcal{M}(T_{\alpha},\alpha)$ such that

• •

  the projective $g$-vector $\mathbf{g}_{A(T_{\alpha})}(\mathcal{M}(T_{\alpha},\alpha))$ is equal to the vector of dual shear coordinates $\operatorname{Sh}_{T_{\alpha}}(\alpha)$ and to the snake $g$-vector $\mathbf{g}_{G(T_{\alpha},\alpha)}$ coming from the snake or band graph $G(T_{\alpha},\alpha)$;

• •

  the representation-theoretic (dual) $F$-polynomial $F_{M(T_{\alpha},\alpha)}$ is equal to the snake $F$-polynomial $F_{G(T_{\alpha},\alpha)}$ coming from the snake or band graph $G(T_{\alpha},\alpha)$;

• •

  the orbit closure of $\mathcal{M}(T_{\alpha},\alpha)$ is a $\tau$-reduced component $Z_{T_{\alpha},\alpha}$ and any decorated representation mutation-equivalent to $\mathcal{M}(T_{\alpha},\alpha)$ is generic, for the values of the projective $g$-vector, the representation-theoretic (dual) $F$-polynomial and Derksen-Weyman-Zelevinsky’s $E$-invariant, in the corresponding $\tau$-reduced component.

In Section 5 we show that every time a flip of tagged triangulations is applied, the changes undergone by the snake $g$-vectors and snake $F$-polynomials of the band graphs constructed from $\alpha$, obey the same recursive formulas that govern the changes undergone by the projective $g$-vectors and representation-theoretic (dual) $F$-polynomials. In Section 6 we combine the results from Sections 4 and 5 to deduce our main result. In Section 7 we make some remarks about the extent to which the representations that yield the bangle functions have been really explicitly computed, and state a couple of open problems.

This section provides a brief review of (skew-symmetric) cluster algebras of geometric type. Let $n\leq m$ be positive integers. Furthermore, let $\mathcal{F}$ be the field of rational functions in $m$ independent variables. Fix a collection $X_{1},\ldots,X_{n},x_{n+1},\ldots,x_{m}$ of algebraically independent variables in $\mathcal{F}$. We define the coefficient ring to be $\mathbb{ZP}:=\mathbb{Z}[x^{\pm 1}_{n+1}\ldots x_{m}^{\pm 1}]$.

A quiver is a finite directed (multi) graph $Q=(Q_{0},Q_{1})$ where $Q_{0}$ is the set of vertices and $Q_{1}$ is the set of directed edges. It will often be convenient to encode (extended) skew-symmetric matrices as quivers. We describe this simple relationship in the definition below, which follows the conventions set out in the work of Derksen, Weyman and Zelevinsky [10].

In this subsection we recall the work of Fomin, Shapiro and Thurston [15], which establishes a cluster structure for triangulated orientable surfaces.

The simple-minded notion of partial Krull-Remak-Schmidt monoid was introduced in [24] as a convenient abstract framework to state the naturalness of the bijection between laminations and $\tau$-reduced components proved therein.

Let $A$ be a basic finite dimensional algebra over an algebraically closed field $\Bbbk$. We may identify the Grothendieck group $K_{0}(A)$ of $A$ with $\mathbb{Z}^{n}$ for some non-negative integer $n$. Consider the set $\operatorname{DecIrr}^{\tau^{-}}(A)$ of decorated, generically $\tau^{-}$-reduced, irreducible components of the representation varieties of $A$. See Section 1.3 for the terminology, and [6, Sec. 3 and 5] for more details. The set $\operatorname{DecIrr}^{\tau^{-}}(A)$ becomes a partial KRS-monoid under the function

that to each pair $(X,Y)\in\operatorname{DecIrr}^{\tau}(A)\times\operatorname{DecIrr}^{\tau}(A)$ associates the generic value $e_{A}(X,Y)$ that on the Zariski product $X\times Y$ takes the symmetrized (injective) $E$-invariant

and the partially-defined sum

given by the direct sum of $e_{A}$-orthogonal irreducible components, $X\oplus Y:=\overline{X\oplus Y}$.

The function $\mathbf{g}_{A}:\operatorname{DecIrr}^{\tau^{-}}(A)\rightarrow\mathbb{Z}^{n}$ that to each $X\in\operatorname{DecIrr}^{\tau}(A)$ associates the generic value taken on $X$ by the (injective) $g$-vector

is a framing for the partial KRS-monoid $(\operatorname{DecIrr}^{\tau^{-}}(A),e_{A},\oplus)$. More precisely:

Suppose now that the finite-dimensional $\Bbbk$-algebra $A$ is the Jacobian algebra of a non-degenerate quiver with potential $(Q,W)$. Following [9, Section 10], for each representation $M$ of $A=\mathcal{P}_{\Bbbk}(Q,W)$ and each vertex $k\in Q_{0}$ we consider the triangle of linear maps

In view of [10, Proposition 10.4 and Remark 10.8], we have

Here $\mathbf{g}_{A}^{\mathrm{DWZ}}(M)$ denotes the $g$-vector of a QP-representation in the sense of [10, (1.13)]. This can be easily extended to decorated QP-representations.

Now, write $A^{\prime}:=\mathcal{P}_{\Bbbk}(\mu_{k}(Q,W))$, where $\mu_{k}(Q,W)$ is the QP-mutation defined in [9], and consider the piecewise linear transformation of integer vectors

where $B(Q)=(b_{ij})$ is the matrix defined in (1.2). Note that, this is just another way of writing the transformation rule for g-vectors from [19, Conj. 7.12], which was proved in [10, Thm. 1.7]. Compare also with Equation (5.6).