No abstract

Abstract. Affine Weyl groups and their parabolic quotients are used extensively as indexing sets for objects in combinatorics, representation theory, algebraic geometry, and number theory. Moreover, in the classical Lie types we can conveniently realize the elements of these quotients via intuitive geometric and combinatorial models such as abaci, alcoves, coroot lattice points, core partitions, and bounded partitions. In [1] Berg, Jones, and Vazirani described a bijection between n-cores with first part equal to k and (n − 1)-cores with first part less than or equal to k, and they interpret this bijection in terms of these other combinatorial models for the quotient of the affine symmetric group by the finite symmetric group. In this paper we discuss how to generalize the bijection of Berg-Jones-Vazirani to parabolic quotients of affine Weyl groups in type C. We develop techniques using the associated affine hyperplane arrangement to interpret this bijection geometrically as a projection of alcoves onto the hyperplane containing their coroot lattice points. We are thereby able to analyze this bijective projection in the language of various additional combinatorial models developed by Hanusa and Jones in [10], such as abaci, core partitions, and canonical reduced expressions in the Coxeter group.

No abstract

Motivated by the local theory of Banach spaces we introduce a notion of finite representability for metric spaces. This allows us to develop a new technique for comparing the generalized roundness of metric spaces. We illustrate this technique in two different ways by applying it to Banach spaces and metric trees. In the realm of Banach spaces we obtain results such as the following: (1) if U is any ultrafilter and X is any Banach space, then the second dual X * * and the ultrapower (X) U have the same generalized roundness as X, and (2) no Banach space of positive generalized roundness is uniformly homeomorphic to c 0 or ℓp, 2 < p < ∞. Our technique also leads to the identification of new classes of metric trees of generalized roundness one. In particular, we give the first examples of metric trees of generalized roundness one that have finite diameter. These results on metric trees provide a natural sequel to a paper of Caffarelli et al. [6]. In addition, we show that metric trees of generalized roundness one possess special Euclidean embedding properties that distinguish them from all other metric trees.2010 Mathematics Subject Classification. 46B07, 46B80, 05C05, 05C12.

No abstract