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$\DeclareMathOperator\Co{Co}$Let $P$ be the root poset associated to a simple Lie algebra.

Let $L=L(P)$ denote the distributive lattice of order ideals of $P$ and let $\Co_L$ denote the Coxeter matrix of $L(P)$ which is defined as $-C^{-1} C^T$ when $C$ is the matrix with entries $c_{i,j}=1$ when $i \leq j$ and $c_{i,j}=0$ else for $i,j \in L$.

Question: For which types is the matrix $\Co_L$ periodic? That is when do we have $\Co_L^t=\operatorname{id}$ for some $t \geq 1$? What is the period (the smallest such $t$) in case this is true for a given type?

The quesiton has a positive answer for all simple Lie algebras of rank at most 6. So maybe it is always periodic?

For $A_n$ it is $2n+4$ for $n=2,3,4,5,6$ but for $A_1$ it is 3. Is it $2n+4$ in general for $n \geq 2$?

For $D_4$ it is 14 and for $D_5$ it is 18 and for $D_6$ it is 22. Is it given by $4n-2$?

For $B_2$ it is 10 and for $B_3$ it is 7 and for $B_4$ it is 18 and for $B_5$ it is 11 and for $B_6$ it is 26. Is it $8n+2$ for $B_{2n}$? Is it $4n+3$ for $B_{2n+1}$?

For $G_2$ it is 14 and for $F_4$ it is 26 and for $E_6$ it is 26.

edit: I saw that in the introduction of https://arxiv.org/pdf/1710.10632.pdf there is a conjecture of Chapoton stated that says the the incidence algebras of those distributive lattices are fractionally Calabi-Yau. The truth of this conjecture would imply a positive answer to the question of this thread.

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  • $\begingroup$ Is the period always $2*(h+1)$ where $h$ is the Coxeter number? $\endgroup$
    – Sam Hopkins
    Commented Aug 29, 2020 at 23:54
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    $\begingroup$ @SamHopkins For $B_3$ it is 7. $\endgroup$
    – Mare
    Commented Aug 29, 2020 at 23:56
  • $\begingroup$ It seems that the period could be $h+1$ or $2(h+1)$. $\endgroup$
    – Sam Hopkins
    Commented Aug 30, 2020 at 0:17
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    $\begingroup$ It seems the first nonnegative entry in the row of $-C^{-1}C^T$ indexed by $i \in L$ is $\mathrm{Row}(i)$, the rowmotion of the order ideal (a.k.a., the Panyushev map). Since root posets are known to have good behavior of rowmotion (see arxiv.org/abs/0711.3353 and arxiv.org/abs/1101.1277), this may be key to your observation (although note that the order of rowmotion is $2h$ or $h$, slightly different than what you see here). Rowmotion appeared in another related Q of yours: mathoverflow.net/questions/343203/… $\endgroup$
    – Sam Hopkins
    Commented Aug 30, 2020 at 3:41
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    $\begingroup$ You can see Sage code here: cocalc.com/share/0efb266aca1eef099315e87e1750671ee5737323/… $\endgroup$
    – Sam Hopkins
    Commented Aug 30, 2020 at 14:02

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