Sign Types Corresponding to an Affine Weyl Group

The sign types corresponding to an affine Weyl group Wa were first studied in [3]. In the present paper, I generalize all the results of [3] on sign types to the case when Wa is an indecomposable affine Weyl group of an arbitrary type. As a result, I verify Carter's conjecture on the cardinality of sign types of type is an indecomposable root system of an arbitrary type. The main results are Theorems 2.1 and 8.1. We start with the definition of an admissible sign type in terms of a 0-tuple over Z. Then §§ 3-5 are reserved for the proof of Theorem 2.1. Theorem 2.1 asserts that the set 5^(0) of admissible sign types can be identified with the set of certain equivalence classes of Wa. We also deduce in §6 that £?( . Let 0 + , 0 ~ be the corresponding positive and negative root systems of 0 . Let E be the Euclidean space spanned by 0 with positive definite inner product such that |a| = = 1 for any short root a of 0 . For any oce0, a = 2a/ is called the coroot of a. The set 0 v = {a | aeO} of coroots is again a root system such that the set {a/, ...,aj} affords a choice of simple root system in it. Let — o be the highest short root of 0 . Then ( — o^) is the highest (co)root of 0 v . Let h be the Coxeter number of 0 . Then h is also the Coxeter number of 0 v . Let Wbt the Weyl group of 0 generated by the reflections^ on £"for a e 0 , where Received 14 February 1986. 1980 Mathematics Subject Classification 20H15. J. London Math. Soc. (2) 35 (1987) 56-74 SIGN TYPES CORRESPONDING TO AN AFFINE WEYL GROUP 57 sa sends x t o j c a . Let Q denote the root lattice 2 + over Z. The following results are well known. LEMMA 1.1 [4, Lemma 1.1]. Let Ax = n + ^t;fca witn kaeZ. Then Ak is an alcove of E if and only if for any cc,fie , the inequality