On the Combinatorial Structure and Hodge Theory of Bott–Samelson Rings for Arbitrary Coxeter Groups

Core Concepts

This paper presents a combinatorial construction of Bott–Samelson rings associated with arbitrary Coxeter groups, proves their Koszulness, and establishes their Hodge-theoretic properties.

Abstract

Gui, T., Sun, L., Wang, S., & Zhu, H. (2024). On Bott–Samelson rings for Coxeter groups. arXiv preprint arXiv:2408.10155v2.

This paper aims to study the ring structure of Bott–Samelson modules for arbitrary Coxeter groups, going beyond their traditional association with crystallographic Coxeter systems. The authors seek to provide a combinatorial description of these rings and explore their Hodge-theoretic properties.

Key Insights Distilled From

On Bott--Samelson rings for Coxeter groups

by Tao Gui, Lin... at arxiv.org 11-06-2024

https://arxiv.org/pdf/2408.10155.pdf

Deeper Inquiries

How can the combinatorial framework developed in this paper be utilized to study other algebraic invariants associated with Coxeter groups or to establish further connections with matroid theory?

The combinatorial framework of Bott-Samelson rings, as presented in the paper, opens several avenues for studying Coxeter groups and their connections to matroids:
1. Kazhdan-Lusztig Polynomials and R-polynomials: The paper mentions that Soergel bimodules, which are closely related to Bott-Samelson rings, provide a combinatorial approach to Kazhdan-Lusztig polynomials. This connection could be further explored using the explicit presentations and Hodge-theoretic properties of Bott-Samelson rings. Similarly, the R-polynomials, another important invariant in Kazhdan-Lusztig theory, might be investigated within this framework.
2. Bruhat Order and Shellability: The Bott-Samelson rings are naturally indexed by words in the Coxeter group. This indexing suggests a potential connection to the Bruhat order on the group. Investigating how the structure of Bott-Samelson rings reflects properties of the Bruhat order, such as shellability, could be fruitful.
3. Matroid Analogues of Bott-Samelson Rings: The paper draws parallels between the combinatorial Hodge theory of matroids and the Hodge-theoretic properties of Bott-Samelson rings. This parallel suggests the possibility of defining and studying "matroid Bott-Samelson rings" associated with matroids. Such rings could provide a new perspective on matroid invariants and their relationship to Coxeter groups.
4. Generalized Demazure Operators and Invariant Theory: The Demazure operators play a crucial role in the construction and analysis of Bott-Samelson rings. Exploring generalizations of these operators to other representations of Coxeter groups or to more general settings could lead to new insights in invariant theory and representation theory.

Could there be alternative presentations of the Bott–Samelson ring that might offer different insights into its structure or lead to simpler proofs of its properties?

Yes, alternative presentations of the Bott-Samelson ring are certainly possible and could be advantageous:
1. Noncommutative Presentations: The current presentation is in terms of a commutative polynomial ring modulo relations. Exploring noncommutative presentations, perhaps using generators and relations inspired by the Coxeter group itself, might reveal deeper connections to the group structure and representation theory.
2. Graphical Presentations: The paper mentions Elias and Williamson's work on diagrammatic presentations of Bott-Samelson bimodules. Adapting these diagrammatic techniques to the ring setting could provide a more visual and intuitive understanding of the Bott-Samelson ring's structure.
3. Deformations and Specializations: Considering deformations or specializations of the Bott-Samelson ring, by introducing parameters into the relations for instance, could lead to simpler proofs of its properties by exploiting special cases or by using deformation-theoretic arguments.
4. Cellular or DG Algebra Structures: Investigating whether Bott-Samelson rings admit natural cellular algebra structures or differential graded algebra (DG algebra) structures could provide powerful tools for studying their homology and cohomology, potentially simplifying proofs of Hodge-theoretic properties.

What are the implications of the Hodge-theoretic properties of Bott–Samelson rings for the representation theory of Coxeter groups, particularly in the context of non-crystallographic groups?

The Hodge-theoretic properties of Bott-Samelson rings, particularly for non-crystallographic Coxeter groups, are significant for several reasons:
1. Geometric Intuition for Non-crystallographic Groups: Non-crystallographic Coxeter groups lack a direct connection to flag varieties and geometric representation theory. The Hodge theory of Bott-Samelson rings provides a "shadow" of the geometric structures that are absent in the non-crystallographic setting, offering a new perspective on their representation theory.
2. Graded Characters and Decomposition Numbers: The Hodge decomposition of Bott-Samelson rings could potentially be used to study graded characters of representations of Coxeter groups. This could lead to new insights into decomposition numbers and other representation-theoretic invariants.
3. Kazhdan-Lusztig Theory for Non-crystallographic Groups: As mentioned earlier, Bott-Samelson rings are closely related to Soergel bimodules, which are central to Kazhdan-Lusztig theory. The Hodge theory of Bott-Samelson rings might provide new tools for extending Kazhdan-Lusztig theory to non-crystallographic groups, where traditional geometric methods are not available.
4. Connections to Hecke Algebras: The Hodge structure on Bott-Samelson rings might shed light on the structure of Hecke algebras associated with Coxeter groups. This could be particularly interesting in the non-crystallographic case, where the representation theory of Hecke algebras is less well-understood.