Generalized Goulden-Yong duals and signed minimal factorizations

Abstract

In this paper, we give two combinatorial ways to study signed exceptional sequences. First, we show the equivalence between one-way reflections and relatively projective representations. Secondly, we construct generalized Goulden-Yong duals using reverse Garside element actions and folded chord diagrams. We then give two applications of the generalized Goulden-Yong duals: constructing generalized Prüfer codes and counting signed factorizations using the matrix-tree theorem.

arXiv comment34 pages, 15 figures. v2: updated type D Prüfer codes following a suggestion by Olivier Bernardi and made some minor changes. v3: minor changes