Newton polytopes in cluster algebras and $τ$-tilting theory

We prove that the cluster monomials in non-initial cluster variables are uniquely determined by the Newton polytopes of their $F$-polynomials for skew-symmetrizable cluster algebras. Accordingly, we prove that the $τ$-rigid modules and the left finite multi-semibricks in $τ$-tilting theory are uniquely determined by the Newton polytopes of these modules. The key tools used in the proofs are the left Bongartz completion, $F$-invariant and partial $F$-invariant in the context of cluster algebras and $τ$-tilting theory.

💡 Research Summary

The paper establishes a striking parallel between cluster algebras and τ‑tilting theory: in both settings, the Newton polytope of the associated F‑polynomial uniquely determines the underlying combinatorial object. For a skew‑symmetrizable cluster algebra A with an initial seed (x₀,B₀), the authors consider cluster monomials u and v that involve only non‑initial cluster variables. Each such monomial can be expressed as x^{g_u}·F_u(ĥy), where g_u is the g‑vector and F_u is a polynomial with constant term 1. The Newton polytope P(F_u) is the convex hull of the exponent vectors of the monomials appearing in F_u. By introducing the F‑invariant—a function that evaluates the tropicalization of F_u at any integer vector—the authors show that the F‑invariant coincides with the support function of P(F_u). Using the cancellation property of Minkowski sums and the fact that the support function determines a polytope uniquely, they prove that if two non‑initial cluster monomials have identical Newton polytopes, then their F‑polynomials coincide, and consequently the monomials themselves are equal. The proof proceeds via a reduction argument: a left Bongartz completion of the partial seed generated by the support set of the product uv is constructed, and a sequence of left mutations is applied to shrink the support set while preserving the Newton polytope. Induction on the size of the support set forces equality.

In the τ‑tilting side, let A be a finite‑dimensional basic algebra over an algebraically closed field k. For a τ‑rigid module U, the F‑polynomial F_U (as defined by Fei) also has constant term 1, and Fei proved that the Newton polytope of F_U coincides with the Newton polytope of the module, defined as the convex hull of dimension vectors of all quotients of U. The authors define an F‑invariant for decorated modules (U,P) analogously to the cluster case. They then show that if two τ‑rigid modules U and V share the same Newton polytope, their F‑invariants agree. By taking the left Bongartz completion (M,P) of U⊕V—i.e., the basic τ‑tilting pair whose torsion class equals Fac(U⊕V)—they construct left mutations that produce new τ‑rigid modules U′∈add U and V′∈add V with the same Newton polytope but with strictly fewer indecomposable summands in U′⊕V′. An induction on the number of indecomposable summands yields that the minimal case forces U≅V. The argument relies on the cancellation law for Newton polytopes, the correspondence between functorially finite torsion classes and semistable torsion classes, and the brick‑τ‑rigid bijection of DIJ.

The paper further extends the result to left‑finite multi‑semibricks, which are direct sums of bricks with Hom‑orthogonal conditions and whose generated torsion class is functorially finite. Using the same reduction strategy and the fact that each brick’s Newton polytope determines the brick, the authors prove that two left‑finite multi‑semibricks with identical Newton polytopes must be isomorphic.

Overall, the work demonstrates that the Newton polytope of an F‑polynomial serves as a complete invariant for non‑initial cluster monomials, τ‑rigid modules, and left‑finite multi‑semibricks. The key tools—left Bongartz completion, F‑invariant, and partial F‑invariant—bridge the combinatorial geometry of polytopes with the representation‑theoretic structures of τ‑tilting theory, revealing a deep structural parallel between the two theories. Potential future directions include extending the results to initial objects, to non‑skew‑symmetrizable settings, and to broader classes of algebras.