Combinatorics of extended affine root systems (type A 1)

Abstract

We establish extensions of some important features of affine theory to affine reflection systems (extended affine root systems) of type A1. We present a positivity theory which decomposes in a natural way the nonisotropic roots into positive and negative roots, then using that, we give an extended version of the well-known exchange condition for the corresponding Weyl group, and finally give an extended version of the Bruhat ordering and the Z-Lemma. Furthermore, a new presentation of the Weyl group in terms of the parity permutations is given, this in turn leads to a parity theorem which gives a characterization of the reduced words in the Weyl group. All root systems involved in this work appear as the root systems of certain well-studied Lie algebras. © 2019 World Scientific Publishing Company.