Homomorphisms between Bott-Samelson bimodules corresponding to sequences of reflections

TL;DR

This paper investigates bimodule homomorphisms related to Bott-Samelson bimodules, establishing reflexivity under certain conditions and providing counterexamples that reveal complex cohomological properties.

Contribution

It proves reflexivity of bimodule homomorphism modules under restrictions and demonstrates their non-freeness through counterexamples in symmetric groups.

Findings

Modules are reflexive under certain restrictions.

Counterexamples show modules are not necessarily free.

Dual modules have projective dimension n-3 in symmetric groups.

Abstract

We study the space of all bimodule homomorphisms $R_x\otimes_R R(\underline{t})\otimes_R R_y\to R_z\otimes_R R(\underline{t}')\otimes_R R_w$ as a one-sided module, where $R_x,R_y,R_z,R_w$ are standard twisted bimodules and $R(\underline{t})$ and $R(\underline{t}')$ are the Bott-Samelson bimodules corresponding to sequences of reflections $\underline{t}$ and $\underline{t}'$ respectively. We prove that this module is always reflexive under some reasonable restrictions on the representation of the underlying Coxeter group. However, unlike the case where $\underline{t}$ and $\underline{t}'$ contain only simple reflections, this module does not need any longer to be free. We provide a series of counterexamples already for the symmetric groups $S_n$, where $n\ge4$. The projective dimension of the modules dual to them is $n-3$ and thus serves to measure the deviation from the free modules. When placed within a geometric framework, these examples show how to find fibers of points fixed by the compact torus in the Bott-Samelson resolutions (as in the original definition by Raoul Bott and Hans Samelson) with non-vanishing odd cohomology.