Promotion and rowmotion

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 certain cases of recent work of D. Armstrong, C. Stump, and H. Thomas on noncrossing and nonnesting 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. Finally, we define two actions with related orders on alternating sign matrices and totally symmetric self-complementary plane partitions. ► We present an equivariant bijection between two actions on order ideals. ► We obtain equivariant bijections from order ideals to objects under rotation. ► We consider order ideals in products of chains and root posets, among other posets. ► We extend our ideas to ASMs and TSSCPPs.