Bijections of silting complexes and derived Picard groups

Abstract

We introduce a method that produces a bijection between the posets silt − A $\textrm {{{\bf silt}}-}{A}$ and silt − B $\textrm {{{\bf silt}}-}{B}$ formed by the isomorphism classes of basic silting complexes over finite-dimensional k $k$ -algebras A $A$ and B $B$ , by lifting A $A$ and B $B$ to two k [ [ X ] ] $k[\![X]\!]$ -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 TrPicent ${\operatorname{\bf TrPicent}}$ . As an application to the modular representation theory of finite groups, we show that if B $B$ and C $C$ are blocks with | IBr ( B ) | = | IBr ( C ) | $|\operatorname{IBr}(B)|=|\operatorname{IBr}(C)|$ whose defect groups are either both cyclic, both dihedral or both quaternion, then the posets tilt − B $\textrm {{{\bf tilt}}-}{B}$ and tilt − C $\textrm {{{\bf tilt}}-}{C}$ are isomorphic (except, possibly, in the quaternion case with | IBr ( B ) | = 2 $|\operatorname{IBr}(B)|=2$ ) and TrPicent ( B ) ≅ TrPicent ( C ) ${\operatorname{\bf TrPicent}}(B)\cong {\operatorname{\bf TrPicent}}(C)$ (except, possibly, in the quaternion and dihedral cases with | IBr ( B ) | = 2 $|\operatorname{IBr}(B)|=2$ ).