Rowmotion on $m$-Tamari and BiCambrian Lattices

Abstract

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 $b\equiv 1\pmod a$; 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 $W$.

Figures (7)

• Figure 1: The lattice path $\mu=\text{NENENEEENE}$ in the $\nu$-Tamari lattice $\mathop{\mathrm{Tam}}\nolimits(\nu)$, where $\nu=\text{ENNEEEENNE}$. Each lattice point $p$ is labeled with its horizontal distance.
• Figure 2: The lattice path $\mu$ on the left is covered by the lattice path $\mu'$ on the right in $\mathop{\mathrm{Tam}}\nolimits(\nu)$.
• Figure 3: Rowmotion on $\mathop{\mathrm{Tam}}\nolimits_3(2)$. The solid pink and dotted orange arrows indicate the action of rowmotion. Notice that there is one orbit of size $9$ (solid pink) and one orbit of size $3$ (dotted orange).
• Figure 4: A noncrossing partition $\rho\in\mathop{\mathrm{NC}}\nolimits(12)$ (pink) together with $K'(\rho)$ (orange).
• Figure 5: Doubled root posets of types $A_5$, $B_3$, $H_3$, and $I_2(6)$ (from left to right).

Theorems & Definitions (78)

• Lemma 2.1
• Proposition 2.2: Semidistrim
• Conjecture 3.1: ThomasWilliams
• Definition 3.2
• Remark 3.3
• Remark 3.4
• Proposition 4.1: DefantMeeting
• Theorem 4.2
• Example 4.3
• proof : Proof of \ref{['thm:rowmotionbracket']}