Fans and polytopes in tilting theory

Toshitaka Aoki, Akihiro Higashitani, Osamu Iyama, Ryoichi Kase, Yuya Mizuno

Published 2022-03-29Version 1

For a finite dimensional algebra $A$ over a field $k$, the 2-term silting complexes of $A$ gives a simplicial complex $\Delta(A)$ called the $g$-simplicial complex. We give tilting theoretic interpretations of the $h$-vectors and Dehn-Sommerville equations of $\Delta(A)$. Using $g$-vectors of 2-term silting complexes, $\Delta(A)$ gives a nonsingular fan $\Sigma(A)$ in the real Grothendieck group $K_0(\mathrm{proj } A)_\mathbb{R}$ called the $g$-fan. For example, the fan of $g$-vectors of a cluster algebra is given by the $g$-fan of a Jacobian algebra of a non-degenerate quiver with potential. We give several properties of $\Sigma(A)$ including idempotent reductions, sign-coherence, Jasso reductions and a connection with Newton polytopes of $A$-modules. Moreover, $\Sigma(A)$ gives a (possibly infinite and non-convex) polytope $P(A)$ in $K_0(\mathrm{proj } A)_\mathbb{R}$ called the $g$-polytope of $A$. We call $A$ $g$-convex if $P(A)$ is convex. In this case, we show that it is a reflexive polytope, and that the dual polytope is given by the 2-term simple minded collections of $A$. We give an explicit classification of $g$-convex algebras of rank $2$. We classify algebras whose $g$-polytopes are smooth Fano. We classify classical and generalized preprojective algebras which are $g$-convex, and also describe their $g$-polytope as the dual polytopes of short root polytopes of type $A$ and $B$. We also classify Brauer graph algebras which are $g$-convex, and describe their $g$-polytopes as root polytopes of type $A$ and $C$.