A Bijection Between the Indecomposable Summands of Two Multiplicity Free Tilting Modules

Abstract

We show that there is a reflection type bijection between the indecomposable summands of two multiplicity free tilting modules X and Y . This bijection fixes the common indecomposable summands of X and Y and sends indecomposable projective (resp., injective) summands of exactly one module to non-projective (resp., non-injective) summands of the other. Moreover, this bijection interchanges the two possible non-isomorphic complements of an almost complete tilting module.