Spectral gap for random-to-random shuffling on linear extensions

In this paper, we propose a new Markov chain which generalizes random-to-random shuffling on permutations to random-to-random shuffling on linear extensions of a finite poset of size $n$. We conjecture that the second largest eigenvalue of the transition matrix is bounded above by $(1+1/n)(1-2/n)$ with equality when the poset is disconnected. This Markov chain provides a way to sample the linear extensions of the poset with a relaxation time bounded above by $n^2/(n+2)$ and a mixing time of $O(n^2 \log n)$. We conjecture that the mixing time is in fact $O(n \log n)$ as for the usual random-to-random shuffling.Comment: 16 pages, 10 figures; v2: typos fixed plus extra information in figures; v3: added explicit conjecture 2.2 + Section 3.6 on the diameter of the Markov Chain as evidence + misc minor improvements; v4: fixed bibliograph

On the wildness of cambrian lattices

In this note, we investigate the representation type of the cambrian lattices and some other related lattices. The result is expressed as a very simple trichotomy. When the rank of the underlined Coxeter group is at most 2, the lattices are of finite representation type. When the Coxeter group is a reducible group of type A 3 1 , the lattices are of tame representation type. In all the other cases they are of wild representation type