Homomorphisms between Bott-Samelson bimodules corresponding to sequences of reflections
Vladimir Shchigolev
TL;DR
This work extends the theory of Bott-Samelson bimodules beyond sequences of simple reflections by studying the space of bimodule homomorphisms for reflection expressions and showing that these spaces are reflexive yet not necessarily free. The authors develop a localization framework, define and analyze subexpression combinatorics, and construct bases for image spaces via even-subset and tree-based methods, yielding concrete counterexamples in symmetric groups where dual Hom-spaces have positive projective dimension. They connect these algebraic phenomena to geometric Bott-Samelson resolutions, demonstrating fixed-point fibres with nonvanishing odd cohomology, thus revealing a sharp contrast with the classical simple-reflection case where fibres have affine pavings. The paper integrates Demazure operators, Gröbner bases, and intricate combinatorial structures to provide a robust toolkit for understanding nonfree Hom spaces and their duals, with both algorithmic (Algorithm 1 and 2) and theoretical insights. This work advances the understanding of reflection-expressions in the Soergel framework and highlights unexpected geometric behavior arising from non-simple reflections, with potential implications for positivity phenomena and representation-theoretic conjectures in a broader setting.
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.