Easy way to understand theta basis for X-cluster algebras of finite type?

Question

For $\mathcal A$-cluster algebras of finite type, it is very easy to describe the theta-basis: it consists of the cluster monomials. Is there any similarly easy way to describe the theta-basis for $\mathcal X$-cluster algebras?

For $\mathcal A$-cluster algebras not of finite type, the cluster monomials always form a subset of the theta-basis, and if the description I asked for above extends to the non-finite type case in a similar way, I would like to know that too, but I would still be very happy with an answer which doesn't say anything about non finite type.

Answer 1

Probably the shortest answer, especially in finite-type:

X-type theta functions are monomials times F-polynomials

Making this precise is a bit fiddly, because there are multiple conventions and (more importantly) two different notions of `the' dual cluster algebra to an A-type cluster algebra:

• The "Fock-Goncharov dual" (sometimes called the cluster dual or the Langlands dual), which can be connected to the original algebra via an augmentation map
• The "GHKK dual" (sometimes called the mirror dual), which corresponds to the mirror dual directly constructed by the Gross-Siebert program.

They are closely related; if $B$ is the exchange matrix used to define one, the other is defined by $B^\top$. As a consequence, whenever $B$ is skew-symmetric and the full Fock-Goncharov conjecture holds, there is a twist isomorphism between them which sends theta functions to theta functions. However, even in this case, the set of theta functions have different parametrizations by the dual lattice $N$, and theta reciprocity only holds for the GHKK dual (without a twist of the parametrization).

With that caveat out of the way, I can tell you how to describe the X-type theta functions in the GHKK dual. If $\mathfrak{s}$ is the coefficient-free $A$-type seed defined by a skew-symmetrizable matrix $B$, then the GHKK dual theta function associated to $n\in N$ is

$$\vartheta_{\mathfrak{s}^\vee}[n] = y^n F_{B^\top}[B^\top n] $$

where $F_{B^\top}[B^\top n]$ is the F-polynomial obtained by taking the A-type seed of $B^\top$, adjoining principal coefficients (denoted by $y$s), computing the A-type theta function $\vartheta_{B^\top}[B^\top n]$, and specializing the initial $x$-variables to $1$.

In finite-type, the theta functions correspond to cluster monomials and the F-polynomials here correspond to the F-polynomials of Fomin-Zelevinsky. The latter may be computed by any number of enumerative problems. For example, given a simply-laced Dynkin quiver, the F-polynomial will be a generating function of Euler characteristics of quiver Grassmannians.

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An example of this construction

Let $B=\begin{bmatrix} 0 & -1 \\ 2 & 0 \end{bmatrix}$. With the convention I'm using, the six A-type cluster variables are \begin{align*} \vartheta_\mathfrak{s}[1,0] &= x^{(1,0)} \\ \vartheta_\mathfrak{s}[0,1] &= x^{(0,1)} \\ \vartheta_\mathfrak{s}[-1,0] &= x^{(-1,0)}+x^{(-1,2)} \\ \vartheta_\mathfrak{s}[0,-1] &= x^{(0,-1)}+x^{(-1,-1)}+x^{(-1,1)} \\ \vartheta_{\mathfrak{s}}[1,-2] &= x^{(1,-2)} + 2x^{(0,-2)} + x^{(-1,-2)}+x^{(-1,0)} \\ \vartheta_{\mathfrak{s}}[1,-1] &= x^{(1,-1)} + x^{(0,-1)} \end{align*} All theta functions are cluster monomials (products of adjacent pairs above). The scattering diagram and the Newton polytope of several theta functions (with a dot at $m$) are pictured below.

To cook up the F polynomials, we compute the six A-type cluster variables of $\mathsf{B}^\top=\begin{bmatrix} 0 & 2 \\ -1 & 0 \end{bmatrix}$ with principal coefficients. \begin{align*} \vartheta_{\mathfrak{s}^\top}[1,0] &= x^{(1,0)} \\ \vartheta_{\mathfrak{s}^\top}[0,1] &= x^{(0,1)} \\ \vartheta_{\mathfrak{s}^\top}[-1,1] &= x^{(-1,1)}+y^{(1,0)}x^{(-1,0)} \\ \vartheta_{\mathfrak{s}^\top}[-2,1] &= x^{(-2,1)} + 2y^{(1,0)}x^{(-2,0)} + y^{(2,0)} x^{(-2,-1)}+y^{(2,1)}x^{(0,-1)} \\ \vartheta_{\mathfrak{s}^\top}[-1,0] &= x^{(-1,0)}+y^{(1,0)}x^{(-1,-1)}+y^{(1,1)}x^{(1,-1)} \\ \vartheta_{\mathfrak{s}^\top}[0,-1] &= x^{(0,-1)} + y^{(0,1)}x^{(2,-1)} \end{align*} and specialize the $x$-variables to $1$: \begin{align*} F_{B^\top}[1,0] &= 1 \\ F_{B^\top}[0,1] &= 1 \\ F_{B^\top}[-1,1] &= 1+y^{(1,0)} \\ F_{B^\top}[-2,1] &= 1 + 2y^{(1,0)} + y^{(2,0)} + y^{(2,1)} \\ F_{B^\top}[-1,0] &= 1+y^{(1,0)}+y^{(1,1)} \\ F_{B^\top}[0,-1] &= 1 + y^{(0,1)} \end{align*} All other F-polynomials in this case can be computed as products of adjacent F-polynomials above; e.g. $$ F_{B^\top}[-2,-1] = F_{B^\top}[-1,0]^2F_{B^\top}[0,-1] =(1+y^{(1,0)}+y^{(1,1)})^2(1 + y^{(0,1)}) $$

The X-type theta function in the GHKK dual with $n=(1,-1)$ is then \begin{align*} \vartheta_{\mathfrak{s}^\vee}[1,-1] &= y^{(1,-1)}F_{B^\top}[-2,-1] \\ % &= y^{(1,-1)}F_{B^\top}[-1,0]^2F_{B^\top}[0,-1] \\ &= y^{(1,-1)}(1+y^{(1,0)}+y^{(1,1)})^2(1 + y^{(0,1)}) \end{align*} We can check directly that theta reciprocity holds between $\vartheta_{\mathfrak{s}^\vee}[1,-1]$ and each of the cluster variables on the $A$-type side. $$\begin{array}{|c|ccccccc|} \hline m & (1,0) & (0,1) & (-1,0) & (0,-1) & (1,-2) & (1,-1) \\ \hline \mathrm{val}_{(1,-1)}(\vartheta_{\mathfrak{s}}[m]) & 1 & -1 & -3 & -2 & -1 & 1 \\ \hline \mathrm{val}_{m}(\vartheta_{\mathfrak{s}^\vee}[1,-1]) & 1 & -1 & -3 & -2 & -1 & 1 \\ \hline \end{array}$$ Note that the latter row is given by the tropicalization of $\vartheta_{\mathfrak{s}^\vee}[1,-1]$: $$\mathrm{val}_{m}(\vartheta_{\mathfrak{s}^\vee}[1,-1]) = (m_1-m_2) + 2\min(0,m_1,m_1+m_2)+ \min(0,m_2) $$

Answer 2

There's another answer which doesn't work in full generality, but is often easier to work with, especially if you know the A-type theta functions really well.

When the exchange matrix has full rank, X-type theta functions are certain A-type theta functions after a change of variables

Specifically, let $B$ be an $d\times r$ extended exchange matrix whose rank is equal to its width $r$. Choose a skew-symmetrizable matrix $\widehat{B}$ whose first $r$-many columns are $B$, and let $B'$ denote the first $r$-many columns of $\widehat{B}^\top$. (Note that $B'$ does not depend on the choice of $\widehat{B}$, and $B'=B^\top$ if $B$ is square)

Let $\rho_{\widehat{B}^\top}:\mathbb{Z}[y^{\mathbb{Z}^d}]\rightarrow \mathbb{Z}[x^{\mathbb{Z}^d}]$ be the ring homomorphism which sends $y^n$ to $x^{\widehat{B}^\top n}$. This is an augmentation map in the sense of Fock-Goncharov (or at least, it is when restricted to the cluster algebras/varieties). Then $$\rho_{\widehat{B}^\top}(\vartheta_{\mathfrak{s}^\vee}[n]) = \vartheta_{\mathfrak{s}'}[\widehat{B}^\top n] $$ where the latter theta function is an A-type theta function with exchange matrix $B'$.

In general, this doesn't determine the dual theta function. However, when $B$ has full rank (rank equal to its width), then this uniquely determines $\vartheta_{\mathfrak{s}^\vee}[n]$, which can be computed by replacing each $x^{\widehat{B}^\top n}$ by $y^n$.

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As an example, let $$B =\widehat{B} = \begin{bmatrix} 0 & -1 \\ 2 & 0 \end{bmatrix}$$ Then $\rho_{\widehat{B}^\top}(y^{(n_1,n_2)}) = x^{(2n_2,-n_1)}$, and $$ \rho_{\widehat{B}^\top}(\vartheta_{\mathfrak{s}^\vee}[1,-1]) = \vartheta_{\mathfrak{s}^\top}[-2,-1] $$ where the latter theta function is in the A-type cluster algebra of $B'=B^\top= \begin{bmatrix} 0 & 2 \\ -1 & 0 \end{bmatrix}$. One may compute that this is a cluster monomial; specifically, \begin{align*} \vartheta_{\mathfrak{s}^\top}[-2,-1] &= \vartheta_{\mathfrak{s}^\top}[-1,0]^2\vartheta_{\mathfrak{s}^\top}[0,-1]\\ &=(x^{(-1,0)}+x^{(-1,-1)}+x^{(1,-1)})^2( x^{(0,-1)} + x^{(2,-1)})\\ &= x^{(-2,-1)}(1+x^{(0,-1)}+x^{(2,-1)})^2( 1 + x^{(2,0)}) \end{align*} The last expression has been factored so the exponent of each monomial is in the image of $\widehat{B}^\top$. Replacing each $x^{\widehat{B}^\top n}$ with $y^n$ yields the corresponding X-type theta function. $$\vartheta_{\mathfrak{s}^\vee}[1,-1] = y^{(1,-1)}(1+y^{(1,0)}+y^{(1,1)})^2(1+y^{(0,1)}) $$