On decomposition of Bott-Samelson sheaves

TL;DR

This paper presents an exact algorithm for computing the matrix describing the embedding of Bott-Samelson sheaves, enabling decomposition analysis and character calculation in previously unexplored cases.

Contribution

It introduces a novel algorithm for calculating the embedding matrix of Bott-Samelson sheaves under GKM restrictions, facilitating decomposition and character computations.

Findings

Calculated initial decomposition terms of Bott-Samelson modules.

Determined characters of Braden--MacPherson sheaves in new cases.

Provided an explicit method for embedding matrix computation.

Abstract

We give an exact algorithm to calculate (under some GKM-restriction) the matrix describing the embedding $\B(\s)_x\subset\B(\s)^x$, where the first module is the costalk and the second one is the stalk at $x$ of a Bott-Samelson module (sheaf) $\B(\s)$. This allows us to calculate the first few terms of the decomposition of $\B(\s)$ into a sum of indecomposable modules (sheaves) and to calculate the characters of Braden--MacPherson sheaves in some previously unknown cases.