# The formal affine Demazure algebra and real finite reflection groups

## TL;DR

This paper extends the formal affine Demazure algebra framework to all real finite reflection groups, providing new algebraic structures and explicit computations of structure coefficients for specific groups.

## Contribution

It generalizes the formal affine Demazure algebra and formal group ring to all real finite reflection groups, including explicit calculations for complex cases.

## Key findings

- Generalized formal group ring for all real finite reflection groups
- Defined and studied formal Demazure operators on these rings
- Computed structure coefficients for specific reflection groups

## Abstract

In this paper, we generalize the formal affine Demazure algebra of Hoffnung-Malag\'on-L\'opez-Savage-Zainoulline to all real finite reflection groups. We begin by generalizing the formal group ring of Calm\`es-Petrov-Zainoulline to all real finite reflection groups. We then define and study the formal Demazure operators that act on the formal group ring. Using these results and constructions, we define and study the formal affine Demazure algebra for all real finite reflection groups. Finally, we compute several structure coefficients that appear in a braid relation among the formal Demazure elements, and we conclude this paper by computing all structure coefficients for the reflection groups $I_2(5)$, $I_2(7)$, $H_3$, and $H_4$.