Rim Hook Lattices

A partial order is defined on partitions by the removal of rim hooks of a given length. This poset is isomorphic to a product of Young lattices, guaranteeing rim hook versions of Schensted correspondences. Analogous results are given for shifted shapes. 1. Main Results A shape (Young diagram) is a finite order ideal of the lattice P = {(k, l) : k, l ≥ 1}. Shapes form the so-called Young lattice Y (see, e.g., [St86]). An i’th diagonal of a shape λ is the set {(k, l) : l − k = i }. We use the so-called “English notation” for realizing shapes in the 4th quadrant. We denote by #λ the number of boxes in a shape λ. A rim hook is a set of elements (“boxes”) of P which forms a contiguous strip and has at most one box on each diagonal. Throughout the paper a positive integer r is fixed; all of the rim hooks contain exactly r boxes. (Exception: Definition 3.4(3).) *Partially supported by the Mittag-Leffler Institute. **Partially supported by the Mittag-Leffler Institute and by NSF grant #DMS90-01195. 1991 Mathematics Subject Classification. 05A.