Algebraic and geometric properties of flag Bott-Samelson varieties and applications to representations

TL;DR

This paper introduces flag Bott-Samelson varieties, generalizes key geometric objects, and applies these to compute Newton-Okounkov bodies, decompose tensor products, and analyze degenerations and measures in representation theory.

Contribution

It defines flag Bott-Samelson varieties, computes their Newton-Okounkov bodies, and explores their degenerations and symplectic measures, extending the understanding of geometric representation theory.

Findings

Newton-Okounkov bodies as generalized string polytopes

Polyhedral formulas for tensor product decompositions

Degeneration into flag Bott manifolds with torus actions

Abstract

We introduce the notion of flag Bott-Samelson variety as a generalization of Bott-Samelson variety and flag variety. Using a birational morphism from an appropriate Bott-Samelson variety to a flag Bott-Samelson variety, we compute Newton-Okounkov bodies of flag Bott-Samelson varieties as generalized string polytopes, which are applied to give polyhedral expressions for irreducible decompositions of tensor products of $G$-modules. Furthermore, we show that flag Bott-Samelson varieties are degenerated into flag Bott manifolds with higher rank torus actions, and find the Duistermaat-Heckman measures of the moment map images of flag Bott-Samelson varieties with the torus action together with invariant closed $2$-forms.