Cyclic symmetries of A_n-quiver representations

TL;DR

This paper presents a combinatorial approach to symmetries in A_n-quiver representations, connecting symplectic geometry, algebraic geometry, and K-theory, with applications to quantizations of Lagrangian skeleta.

Contribution

It introduces a novel combinatorial construction of symmetries in A_n-quiver representations relevant to multiple mathematical fields.

Findings

Provides a combinatorial construction of symmetries in A_n-quivers.

Connects symmetries to quantizations of Lagrangian skeleta.

Offers an explicit solution in the one-dimensional case of ribbon graphs.

Abstract

This short note contains a combinatorial construction of symmetries arising in symplectic geometry (partially wrapped or infinitesimal Fukaya categories), algebraic geometry (derived categories of singularities), and K-theory (Waldhausen's S-construction). Our specific motivation (in the spirit of expectations of Kontsevich, and to be taken up in general elsewhere) is a combinatorial construction of quantizations of Lagrangian skeleta (equivalent to microlocal sheaves in their many guises). We explain here the one dimensional case of ribbon graphs where the main result of this paper gives an immediate solution.