Bijections of silting complexes and derived Picard groups

TL;DR

This paper establishes a bijection between silting complexes of certain algebras via ring isomorphisms of their orders, revealing invariance properties and applications to modular representation theory of finite groups.

Contribution

It introduces a novel method to relate silting posets of algebras through ring isomorphisms of their orders, extending to derived Picard groups and modular group block classifications.

Findings

Silting posets are multiplicity-independent in many cases.

Large multiplicity-independent subgroups exist in derived Picard groups.

Posets of tilting modules are isomorphic for certain group blocks with similar defect groups.

Abstract

We introduce a method that produces a bijection between the posets ${\rm silt-}{A}$ and ${\rm silt-}{B}$ formed by the isomorphism classes of basic silting complexes over finite-dimensional $k$-algebras $A$ and $B$, by lifting $A$ and $B$ to two $k[[X]]$-orders which are isomorphic as rings. We apply this to a class of algebras generalising Brauer graph and weighted surface algebras, showing that their silting posets are multiplicity-independent in most cases. Under stronger hypotheses we also prove the existence of large multiplicity-independent subgroups in their derived Picard groups as well as multiplicity-invariance of $\rm TrPicent$. As an application to the modular representation theory of finite groups we show that if $B$ and $C$ are blocks with $|{\rm IBr}(B)|=|{\rm IBr}(C)|$ whose defect groups are either both cyclic, both dihedral or both quaternion, then the posets ${\rm tilt-}{B}$ and ${\rm tilt-}{C}$ are isomorphic (except, possibly, in the quaternion case with $|{\rm IBr}(B)|=2$) and ${\rm TrPicent}(B)\cong{\rm TrPicent}(C)$ (except, possibly, in the quaternion and dihedral cases with $|{\rm IBr}(B)|=2$).