Coxeter groups preserving orthants

TL;DR

This paper proves that Coxeter groups preserving an orthant are products of symmetric groups and extends this to an affine setting, revealing that certain quantum geometric phenomena are specific to type A Coxeter groups.

Contribution

It establishes a characterization of Coxeter groups preserving orthants and their affine analogues, linking these to symmetric groups and type A phenomena in quantum geometry.

Findings

Coxeter groups preserving orthants are products of symmetric groups.

Affine analogues preserve orthants modulo invariant sublattices.

Quantum geometric Satake equivalence phenomena are specific to type A.

Abstract

In this short, elementary note we prove that if a faithful reflection representation of a Coxeter group preserves an orthant, then that Coxeter group is a product of symmetric groups acting on its natural permutation representation. We also prove an affine analogue of this statement, where an orthant is preserved modulo an invariant sublattice. As a consequence, the existence of two different versions of the quantum geometric Satake equivalence is a purely type A phenomenon.