In the nice paper "On the rank function of a differential poset" (2011) by Richard Stanley and Fabrizio Zanello a number of interesting questions was asked about such posets. I would like to know which of them are already answered, what are new questions appeared in the field and so on, what are now considered to be the most important and so on. Maybe, some new surveys appeared?
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asked Feb 27, 2019 at 19:08
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1$\begingroup$ By the way, if you are thinking differential posets are an interesting setting for "asymptotic algebraic combinatorics": I agree! Maybe a first thing to consider would be asymptotic behavior of large random oscillating tableaux. $\endgroup$– Sam HopkinsFeb 27, 2019 at 19:27
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$\begingroup$ @SamHopkins yes, I think so. This topic has many strong connections with what we (first of all Vershik, after him others, including me) think about in St. Petersburg. We need a good collection of examples of graded graphs for which a rich theory (like for Young graph) does exist. $\endgroup$– Fedor PetrovFeb 27, 2019 at 19:37
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2 Answers
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Check out this recent paper by Gaetz and Venkataramana: https://arxiv.org/abs/1806.03509
EDIT:
And actually, you'll also want to read this paper by Miller: https://link.springer.com/article/10.1007/s11083-012-9268-y
answered Feb 27, 2019 at 19:10
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1$\begingroup$ So: Question 14 was resolved by Miller (the ranks do strictly increase); but that was already mentioned in the paper of Stanley and Zanello in a note added in proof. As far as I am aware, the other questions remain unanswered (but the paper of Gaetz-Venkataramana will still be interesting to you). $\endgroup$ Feb 27, 2019 at 19:16
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I have a recent paper in which I prove that when $r=1$ or $r$ is prime, $Y^r$ is the only $r$-differential poset (or more generally $r$-dual graded graph) which comes from the branching rules for some sequence of groups $G_0 \subset G_1 \subset \cdots$. I also conjecture that this is true for general values of $r$; this may be interesting to people in Vershik's school.
answered Feb 28, 2019 at 14:37
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$\begingroup$ thank you, it certainly is interesting! $\endgroup$ Feb 28, 2019 at 14:40