Polynomial Assignments for Bott-Samelson manifolds

TL;DR

This paper studies polynomial assignment modules for torus actions on Bott-Samelson manifolds, providing a method to compute generators for these modules, which are algebraic structures capturing symmetry properties.

Contribution

It describes the assignment module for Bott-Samelson manifolds and introduces a computational method to find its generators, advancing understanding of equivariant algebraic structures.

Findings

Explicit description of the assignment module for Bott-Samelson manifolds

A new method for computing generators of the assignment module

Enhanced tools for studying torus actions on complex manifolds

Abstract

Polynomial assignments for a torus $T$-action on a smooth manifold $M$ were introduced by Ginzburg, Guillemin, and Karshon in 1999; they form a module over $\mathbb{S}(\mathfrak{t}^*)$, the algebra of polynomial functions on $\mathfrak{t}$, the Lie algebra of $T$. In this paper we describe the assignment module $\mathcal{A}_T(M)$ for a natural $T$-action on a Bott-Samelson manifold $M = BS^I$ and present a method for computing generators.