  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.
