Timeline for Young diagrams for the block matrices
Current License: CC BY-SA 4.0
12 events
| when toggle format | what | by | license | comment | |
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| Jan 25, 2023 at 22:00 | comment | added | Andrey Radul | Let us continue this discussion in chat. | |
| Jan 25, 2023 at 17:26 | comment | added | Andrey Radul | No, no @SamHopkins. Let S_n act on V_n, S_m on V_m. Then S_n x S_m act on V_n \otimes V_m, which has dimension n*m. I am not talking about the usual product of Schur functions. | |
| Jan 25, 2023 at 14:44 | comment | added | Sam Hopkins | I don't follow your last comment: $S_n \times S_m$ is naturally a subgroup of $S_{n+m}$, not $S_{nm}$. It's true that induction from $S_n \times S_m$ to $S_{n+m}$ corresponds to the usual (not plethystic) product of Schur functions. But I don't think that's what your asking about. I still think that your "block embedding" of $S_n$ into $S_{nm}$ should correspond to some kind of plethysm. | |
| Jan 25, 2023 at 14:12 | comment | added | Andrey Radul | It seems I do know the answer. Consider S_n x S_m and two irreducible representations \lambda and \mu. Induce a representation of S_mn from the subgroup. It is irreducible. Young diagram Y for the induced representation may be computed from the diagrams for \lambda and \mu. Multiplicity for \lambda equals dimention of \mu. | |
| Jan 25, 2023 at 10:31 | comment | added | Andrey Radul | Thank you, @SamHopkins for your thoughts! | |
| Jan 25, 2023 at 2:57 | comment | added | Andrey Radul | Dear Sam, may be you can figure it out better than I will. The "block permutations" are actually permutations of the tensor product of n-dimensional by m-dimensional vector spaces, and it is easy to decompose the respective tensor powers as S_n representation. My doubts are related to the "rest" of the Young diagram for S_mn. Is it "negligible" in any reasonable way? | |
| Jan 25, 2023 at 2:43 | comment | added | Andrey Radul | It is not what I am looking for. The symmetric group changes as well | |
| Jan 25, 2023 at 2:33 | comment | added | Andrey Radul | Thank you, Sam, checking it out | |
| Jan 25, 2023 at 2:10 | comment | added | Sam Hopkins | The keyword you're looking for is "plethysm." See, e.g., Chapter 7 Appendix 2 of Stanley's "Enumerative Combinatorics, Vol. 2." But, for most plethysms, we do not have any kind of nice formula. So likely the answer is "it is not (at the moment) possible to give a formula for the decomposition." | |
| Jan 25, 2023 at 2:08 | history | edited | Andrey Radul | CC BY-SA 4.0 |
added 175 characters in body
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| S Jan 25, 2023 at 1:00 | review | First questions | |||
| Jan 25, 2023 at 7:16 | |||||
| S Jan 25, 2023 at 1:00 | history | asked | Andrey Radul | CC BY-SA 4.0 |