A symmetry of silting quivers

TL;DR

This paper explores symmetries in silting quivers of algebras induced by anti-automorphisms, revealing conditions under which the 2-silting quiver exhibits a bisection and has an even number of elements.

Contribution

It establishes a connection between anti-automorphisms fixing primitive idempotents and the symmetry properties of the 2-silting quiver, including its bisection and cardinality.

Findings

2-silting quiver has a bisection under certain conditions

Cardinality of finite 2-silting quivers is even

Symmetry is induced by algebra anti-automorphisms

Abstract

We investigate symmetry of the silting quiver of a given algebra which is induced by an anti-automorphism of the algebra. In particular, one shows that if there is a primitive idempotent fixed by the anti-automorphism, then the 2-silting quiver ($=$ the support $\tau$-tilting quiver) has a bisection. Consequently, in that case, we obtain that the cardinality of the 2-silting quiver is an even number (if it is finite).