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More precisely, does there exist a sequence G1 < G2 < ... of finite groups such that the irreducible representations of Gn are parameterized by the plane partitions of total size n?

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  • $\begingroup$ You'd probably like to have the analogue of the branching rule hold as well, I imagine. $\endgroup$ – Reid Barton Oct 16 '09 at 18:35
  • $\begingroup$ Yes, that would be ideal. $\endgroup$ – Qiaochu Yuan Oct 16 '09 at 18:46
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    $\begingroup$ 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 Flath May 3 '15 at 12:45
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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.

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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.)

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