Tropical F$F$‐polynomials and general presentations

Abstract

We introduce the tropical F$F$‐polynomial fM$f_M$ of a quiver representation M$M$. We study its interplay with the general presentation for any finite‐dimensional basic algebra. We give an interpretation of evaluating fM$f_M$ at a weight vector. As a consequence, we give a presentation of the Newton polytope sans-serifNfalse(Mfalse)${\sf N}(M)$ of M$M$. We study the dual fan and 1‐skeleton of sans-serifNfalse(Mfalse)${\sf N}(M)$. We propose an algorithm to determine the generic Newton polytopes, and show that it works for path algebras. As an application, we give a representation‐theoretic interpretation of Fock–Goncharov's duality pairing. We give an explicit construction of dual clusters, which consists of real Schur representations. We specialize the above general results to the cluster‐finite algebras and the preprojective algebras of Dynkin type.

Tensor Product Multiplicities via Upper Cluster Algebras

“…For a nondegernate quiver with potential, and any δ and ǫ we have that fδ pǫq " f ǫpδq " hompδ, ǫq. This is a stronger version of a conjecture in [17], where we only conjecture the reciprocity fδ pǫq " f ǫpδq. Lemma 4.1.…”

“…In Section 2 we first briefly review the theory of quivers with potentials following [11], then we introduce the extension construction and define the mutation of presentations. In Section 3 we briefly review the theory of general presentations and tropical F -polynomials following [13,17]. In Section 4 we review the raising and lowering operators r ǫ and l ǫ introduced in [18] in the special case when ǫ is rigid.…”

“…In Section 8 we briefly review upper cluster algebras and their generic bases following [5,50,17], and recall another tropical pairing. In Section 9 we show that the crystal structure can be algebraically lifted to the upper cluster algebras.…”

“…Tropical F -polynomials. Motivated by the F -polynomial of M defined in [12], we introduced its tropical version in [17]. We will review the F -polynomial of M in Section 8.2.…”

“…Note that what (2) requires is essentially Berenstein-Kazhdan's biperfect basis. The part (3) is already done in [17] (see Theorem 3.5). Although no group action is involved in the definition of the upper cluster algebra, we will see that each r i can always be lifted to a -derivation R i of Cp∆q (see Section 9.1 for the detail).…”