The Quantum Bruhat Graph for $\widehat{SL}_2$ and Double Affine Demazure Products

TL;DR

This paper extends the concept of Demazure products to double affine settings for type f7SL_2, utilizing the quantum Bruhat graph to establish a well-defined associative product and verifying its consistency with known examples.

Contribution

It introduces a new approach to defining the double affine Demazure product for f7SL_2 using quantum Bruhat graphs, generalizing previous affine cases.

Findings

Quantum Bruhat graph for f7SL_2 has key properties enabling Demazure product construction.

Constructed a well-defined associative Demazure product for the double affine Weyl semigroup.

Validated the new product against existing examples from Kac-Moody affine Hecke algebra.

Abstract

We investigate the Demazure product in a double affine setting. Work by Muthiah and Pusk\'as gives a conjectural way to define this in terms of the $q=0$ specialisation of these Hecke algebras. We instead take a different approach generalising work by Felix Schremmer, who gave an equivalent formula for the (single) affine Demazure product in terms of the quantum Bruhat graph. We focus on type $\widehat{SL}_2$, where we prove that the quantum Bruhat graph of this type satisfies some nice properties, which allows us to construct a well-defined associative Demazure product for the double affine Weyl semigroup $W_{\mathcal{T}}$ (for level greater than one). We give results regarding the Demazure product and Muthiah and Orr's length function for $W_{\mathcal{T}}$, and we verify that our proposal matches specific examples computed by Muthiah and Pusk\'as using the Kac-Moody affine Hecke algebra