Promotion and Rowmotion

Details Promotion and Rowmotion Promotion and Rowmotion

2011-08-04

arxiv.org/abs/1108.1172v2

Mathematics

Combinatorics

23 pages, 20 figures; Sections 4,5, and 6 merged; Conjecture 10.21 disproved; Conjecture 10.30 proved; added references for se

Scientific paper

We present an equivariant bijection between two actions--promotion and rowmotion--on order ideals in certain posets. This bijection simultaneously generalizes a result of R. Stanley concerning promotion on the linear extensions of two disjoint chains and recent work of D. Armstrong, C. Stump, and H. Thomas on root posets and noncrossing partitions. We apply this bijection to several classes of posets, obtaining equivariant bijections to various known objects under rotation. We extend the same idea to give an equivariant bijection between alternating sign matrices under rowmotion and under B. Wieland's gyration. Lastly, we define two actions with related orders on alternating sign matrices and totally symmetric self-complementary plane partitions.

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