# Distributive lattices with periodic Coxeter matrix

Let $$L$$ be a finite distributive lattice and $$U$$ its incidence matrix with entries $$u_{i,j}=1$$ iff $$i leq j$$ and $$u_{i,j}=0$$ else.
Then $$U^{-1}$$ is the Moebius matrix of $$L$$ and $$C_L:=- U^{-1} U^{T}$$ is the Coxeter matrix of $$L$$.

Question: For which $$L$$ is the Coxeter matrix periodic, that is $$C_L^n$$ is the identity matrix for some $$n$$?

Call such a lattice (or its incidence algebra) periodic. For example any divisor lattice (products of chains) is periodic.
For distributive lattices on $$r geq 2$$ points, the number of periodic distributive lattices starts with 1,1,2,3,5,7,11,8,15 for $$r leq 10$$ and oeis finds this : https://oeis.org/A053724 .

Question: Is this still true for $$r=11$$?

Algebras with periodic coxeter matrix are studied in representation theory of quivers, see for example https://www.sciencedirect.com/science/article/pii/S0024379505001709 and several other articles.

One might also ask what the period is for a given lattice. For example for the Boolean lattice of $$n$$ points it is 3 if n is odd and 6 if n is even.
Recall that the free distributive lattice is the lattice of order ideals of the Boolean lattice.

Question: For which $$n$$ is the free distributive lattice on $$n$$ points periodic and what is the period in case it is? It is periodic with period 4 for $$n=1$$, period 10 for $$n=2$$ and with period 6 for $$n=3$$ and with period 42 for $$n=4$$.

Some tests suggest that the lattice of order ideals of a divisor lattice is also periodic.

MathOverflow Asked on January 27, 2021

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