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

Naoki Fujita, Eunjeong Lee and Dong Youp Suh

Vol. 309 (2020), No. 1, 145–194

DOI: 10.2140/pjm.2020.309.145

Abstract

We define and study flag Bott–Samelson varieties which generalize both Bott–Samelson varieties and flag varieties. Using a birational morphism from an appropriate Bott–Samelson variety to a flag Bott–Samelson variety, we compute the 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 degenerate into flag Bott manifolds with higher rank torus actions, and we describe the Duistermaat–Heckman measures of the moment map images of flag Bott–Samelson varieties with torus actions and invariant closed 2-forms.

Keywords

flag Bott–Samelson varieties, Bott–Samelson varieties, Newton–Okounkov bodies, flag Bott manifolds

Mathematical Subject Classification

Primary: 05E10

Secondary: 14M15, 57S25

Milestones

Received: 14 July 2019

Revised: 13 April 2020

Accepted: 26 August 2020

Published: 26 December 2020