More precisely, does there exist a sequence $G_1 < G_2 < \cdots$ of finite groups such that the irreducible representations of $G_n$ are parameterized by the plane partitions of total size $n$?
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asked Oct 16, 2009 at 16:39
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$\begingroup$ I take it this is supposed to be by analogy with the representation theory of the symmetric group? (But it may not be too useful to point this out; anybody who doesn't recognize that isn't going to be able to help.) $\endgroup$– Michael LugoCommented Oct 16, 2009 at 18:18
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1$\begingroup$ You'd probably like to have the analogue of the branching rule hold as well, I imagine. $\endgroup$– Reid BartonCommented Oct 16, 2009 at 18:35
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$\begingroup$ Yes, that would be ideal. $\endgroup$– Qiaochu YuanCommented Oct 16, 2009 at 18:46
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2$\begingroup$ More generally, what are some objects naturally parameterized by plane partitions? I ask this question, because ordinary partitions seems to be a ubiquitous index set. $\endgroup$– Dan FlathCommented May 3, 2015 at 12:45
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2$\begingroup$ A relevant paper is arxiv.org/pdf/1110.5310.pdf. $\endgroup$– Richard StanleyCommented Sep 14, 2020 at 3:19
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1 Answer
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Not if you want the direct analogue of the branching rule to hold: namely, if V is the representation of Gn corresponding to a plane partition A of n, then the restriction of V to Gn-1 is the direct sum of one copy of the representation corresponding to each plane partition of n-1 contained in A. That would allow you to compute the dimension of the representation corresponding to A as the number of paths in the containment poset of plane partitions from the empty partition to A. Some computation then shows that the order of G3 would be 1+4+4+1+4+1=15, but there's only one group of order 15, the abelian one, which doesn't work.
You could imagine some variations of the branching rule, though, such as "if B is obtained from A by replacing k by k-1 then the irrep corresponding to A contains k copies of the irrep corresponding to B", and maybe something like that would work.
answered Oct 16, 2009 at 18:59