Counterexamples to the forking path conjeture

Abstract

In 2011, N. Libedinsky studied in [1] morphisms induced by paths in the reduced expression graph of extra-large Coxeter systems. He showed that mor phisms induced by complete paths are idempotents that act as projectors. In 2016, B. Elias provides in [2] an extension of his work with M. Khovanov [3], where they gave a diagrammatic presentation of the category of Bott-Samelson bimodules BSBim. Here, morphisms can be translated into linear combinations of planar graphs, and stacking planar graphs can be interpreted as composing morphisms. Elias uses his diagrammatic calculus to construct an idempotent in the reduced expressions graph for the longest element w0 in the symmetric group Sn, Rex(w0,n). This idempotent can also be described by complete paths. These observations, plus a considerable number of computer comprobations, motivate Libedinsky to formulate the Forking Path Conjecture [4, Section 6.3]. Conjecture 1 (Forking Path Conjecture) Let x ∈ Sn, let p, q be two complete paths with the same starting points and the same ending points in the reduced expression graph of x. The morphisms induced by these paths are equal. While trying to prove Libedinsky’s conjecture, a counterexample was found. In this document we prove the conjecture for all but one element in S4. The outstanding element is the one that sends 1 to 4, 2 to 2, 3 to 3, and 4 to 1