Stability of Stretched Root Systems, Root Posets and Shards

Inspired by the infinite families of finite and affine root systems, we define a "stretching" operation on general crystallographic root systems which, on the level of Coxeter diagrams, replaces a vertex with a path of unlabeled edges. We embed a root system into its stretched versions using a similar operation on individual roots. For a fixed root, we describe the long-term behavior of two associated structures as we lengthen the stretched path: the downset in the root poset and Reading's arrangement of shards. We show that both eventually admit a uniform description, and deduce enumerative consequences: the size of the downset is eventually a polynomial, and the number of shards grows exponentially.

Combinatorics of extended affine root systems (type A1)

We establish extensions of some important features of affine theory to affine reflection systems (extended affine root systems) of type [Formula: see text]. 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 [Formula: see text]-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.

A Length Function for Weyl Groups of Extended Affine Root Systems of Type A1

In this work, we study the concept of the length function and some of its combinatorial properties for the class of extended affine root systems of type A1. We introduce a notion of root basis for these root systems, and using a unique expression of the elements of the Weyl group with respect to a set of generators for the Weyl group, we calculate the length function with respect to a very specific root basis.

ROOT SYSTEMS ARISING FROM AUTOMORPHISMS

We study a combinatorial approach of producing new root systems from the old ones in the context of affine root systems and their new generalizations. The appearance of this approach in the literature goes back to the outstanding work of Kac in the realization of affine Kac–Moody Lie algebras. In recent years, this approach has been appeared in many other works, including the study of affinization of extended affine Lie algebras and invariant affine reflection algebras.

GENERALIZED mth ORDER JACOBI THETA FUNCTIONS AND THE MACDONALD IDENTITIES

We describe an mth order generalization of Jacobi's theta functions and use these functions to construct classes of theta function identities in multiple variables. These identities are equivalent to the Macdonald identities for the seven infinite families of irreducible affine root systems. They are also equivalent to some elliptic determinant evaluations proven recently by Rosengren and Schlosser.