Rowmotion on \(m\)-Tamari and biCambrian lattices. (English) Zbl 07923222
Summary: Thomas and Williams conjectured that rowmotion acting on the rational \((a, b)\)-Tamari lattice has order \(a + b - 1\). We construct an equivariant bijection that proves this conjecture when \(a \equiv 1(\operatorname{mod} b)\); in fact, we determine the entire orbit structure of rowmotion in this case, showing that it exhibits the cyclic sieving phenomenon. We additionally show that the down-degree statistic is homomesic for this action. In a different vein, we consider the action of rowmotion on Barnard and Reading’s biCambrian lattices. Settling a different conjecture of Thomas and Williams, we prove that if \(c\) is a bipartite Coxeter element of a coincidental-type Coxeter group \(W\), then the orbit structure of rowmotion on the \(c\)-biCambrian lattice is the same as the orbit structure of rowmotion on the lattice of order ideals of the doubled root poset of type \(W\).
MSC:
| 05E18 | Group actions on combinatorial structures |
| 06B10 | Lattice ideals, congruence relations |
| 06D75 | Other generalizations of distributive lattices |
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