Quiver Grassmannians for the Bott-Samelson resolution of type A Schubert varieties

TL;DR

This paper establishes a geometric correspondence between Bott-Samelson resolutions of type A Schubert varieties and quiver Grassmannians, providing explicit descriptions and new structural insights.

Contribution

It introduces the concept of geometrically compatible decompositions and constructs a specific quiver to realize these resolutions as quiver Grassmannians.

Findings

Bott-Samelson resolutions are realized as quiver Grassmannians.

A new notion of geometrically compatible decomposition is introduced.

Explicit isomorphisms are constructed for smooth type A Schubert varieties.

Abstract

We realise the Bott-Samelson resolutions of type A Schubert varieties as quiver Grassmannians. In order to explicitly describe this isomorphism, we introduce the notion of a \textit{geometrically compatible} decomposition for any permutation in $S_n$. For smooth type A Schubert varieties, we identify a suitable dimension vector such that the corresponding quiver Grassmannian is isomorphic to the Schubert variety. To obtain these isomorphisms, we construct a special quiver with relations and investigate two classes of quiver Grassmannians for this quiver.