Department of Mathematics

Seminar in Cluster algebras

  •  Bruce E. Sagan, MSU https://msu.zoom.us/j/93361845408
  •  On a rank-unimodality conjecture of Morier-Genoud and Ovsienko
  •  10/07/2020
  •  4:00 PM - 5:00 PM
  •  Online (virtual meeting)
  •  Michael Shapiro (mshapiro@msu.edu)

Let $\alpha=(a,b,\ldots)$ be a composition. Consider the associated poset $F(\alpha)$, called a fence, whose covering relations are $$ x_1\lhd x_2 \lhd \ldots\lhd x_{a+1}\rhd x_{a+2}\rhd \ldots\rhd x_{a+b+1}\lhd x_{a+b+2}\lhd \ldots\ . $$ We study the associated distributive lattice $L(\alpha)$ consisting of all lower order ideals of $F(\alpha)$. These lattices are important in the theory of cluster algebras and their rank generating functions can be used to define $q$-analogues of rational numbers. In particular, we make progress on a recent conjecture of Morier-Genoud and Ovsienko that $L(\alpha)$ is rank unimodal. We show that if one of the parts of $\alpha$ is greater than the sum of the others, then the conjecture is true. We conjecture that $L(\alpha)$ enjoys the stronger properties of having a nested chain decomposition and having a rank sequence which is either top or bottom interlacing, the latter being a recently defined property of sequences. We verify that these properties hold for compositions with at most three parts and for what we call $d$-divided posets, generalizing work of Claussen and simplifying a construction of Gansner.

 

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