A bijection between the indecomposable summands of two multiplicity free tilting modules

TL;DR

The paper establishes a reflection-type bijection between indecomposable summands of two multiplicity free tilting modules, revealing structural correspondences and complement interchange properties.

Contribution

It introduces a novel bijection that relates summands of two tilting modules, preserving common parts and interchanging projective and injective summands.

Findings

Bijection fixes common summands

Interchanges projective and non-projective summands

Interchanges complements of almost complete 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.