Abstract
R. Sulzgruber's rim hook insertion and the Hillman–Grassl correspondence are two distinct bijections between the reverse plane partitions of a fixed partition shape and multisets of rim-hooks of the same partition shape. It is known that Hillman–Grassl may be equivalently defined using the Robinson–Schensted–Knuth correspondence, and we show the analogous result for Sulzgruber's insertion. We refer to our description of Sulzgruber's insertion as diagonal RSK. As a consequence of this equivalence, we show that Sulzgruber's map from multisets of rim hooks to reverse plane partitions can be expressed in terms of Greene–Kleitman invariants.
Highlights
Reverse plane partitions are prominent combinatorial objects with connections to areas like symmetric functions and representation theory
The authors give a bijection between nonnegative integer arrays of shape λ— representing multisets of rim hooks of λ—and reverse plane partitions of λ, which is known as the Hillman–Grassl correspondence
Each diagonal of the reverse plane partition associated to a multiset of rim hooks encodes the Greene–Kleitman partition associated to a certain poset on a subset of the rim hooks–namely, the rim hooks whose support intersects the given diagonal
Summary
Reverse plane partitions are prominent combinatorial objects with connections to areas like symmetric functions and representation theory (see for example [9]). The authors give a bijection between nonnegative integer arrays of shape λ— representing multisets of rim hooks of λ—and reverse plane partitions of λ, which is known as the Hillman–Grassl correspondence. This correspondence has since been well studied, for example by Gansner in [2] and by Morales, Pak, and Panova in [7]. Each diagonal of the reverse plane partition associated to a multiset of rim hooks encodes the Greene–Kleitman partition associated to a certain poset on a subset of the rim hooks–namely, the rim hooks whose support intersects the given diagonal This idea is relevant to future work of the authors with Hugh Thomas that relates reverse plane partitions to the theory of quiver representations [3].
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
