Timeline for Making the branching rule for the symmetric group concrete
Current License: CC BY-SA 2.5
5 events
| when toggle format | what | by | license | comment | |
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| Sep 10, 2014 at 15:41 | comment | added | darij grinberg | I never ended up writing systematic notes about it, but there is an exercise about induction and coinduction in Etingof's representation-theory notes (§4.10 in arXiv:0901.0827v5) and one (with solution) in my notes with Vic Reiner (Exercise 4.4 in web.mit.edu/~darij/www/algebra/HopfComb-sols.pdf ; the numbering is subject to change, but it directly follows the definition of restriction). Be warned that this is all about the notions of induction and coinduction as they are defined for finite groups. I think the definitions for compact Lie groups are different. | |
| Sep 10, 2014 at 13:09 | comment | added | Manuel Bärenz | @darijgrinberg, have you managed to write those notes by any chance? I reckon even for compact Lie groups induction and coinduction are not isomorphic? | |
| Jul 28, 2010 at 19:23 | comment | added | darij grinberg | While we're at it, two more remarks: (1) Induction are coinduction are isomorphic (as functors) if $\left|G:H\right|<\infty$, even when $G$ and $H$ may be infinite. (2) Restriction is similarly two-faced: it's both $\mathrm{Hom}_{A}\left(M_A,\ _B A_A\right)$ and $M_A \otimes_A \left(_A A_B\right)$ (this time the isomorphism actually works for any $A$ and $B$, methinks). This shows why Frobenius reciprocity is just the adjunction between Hom and $\otimes$, and also that there is a mirror version of Frobenius reciprocity, which is rarely discussed because everybody just talks about characters. | |
| Jul 28, 2010 at 19:15 | comment | added | darij grinberg | No, the iso between coinduction and induction doesn't depend on the choice. Actually I have promised to write some notes about this a week ago but probably they will have to wait yet a few weeks longer. | |
| Jul 28, 2010 at 19:12 | history | answered | Bruce Westbury | CC BY-SA 2.5 |