Timeline for Products of Conjugacy Classes in S_n
Current License: CC BY-SA 3.0
19 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Dec 6, 2011 at 18:02 | answer | added | Jafar | timeline score: 0 | |
| Apr 19, 2011 at 13:40 | answer | added | David E Speyer | timeline score: 6 | |
| Apr 19, 2011 at 9:21 | comment | added | Paul Johnson | That was really vague -- I can fill it out a bit more if you're a bit more precise on what you're looking for/curious about. | |
| Apr 19, 2011 at 9:20 | comment | added | Paul Johnson | John -- what do you mean, "get"? My answer below is mostly about the geometry, and though I'm not sure how useful it is, it does set up a geometric meaning for arbitrary $\mu,\nu,\lambda$ -- change the genus, $g$. As for algorithmically computing them, I'm am not sure what the state of the art is. I'd want to use character theory to do this, as multiplication is semisimple in that basis. My gut feel for this is that things are nice when you force $\ell(\mu)$ to be close to zero (most characters vanish) or $|\mu|$ (the permutation is mostly the identity), but messier in between. | |
| Apr 18, 2011 at 23:24 | comment | added | john mangual | It sounds like there's no way to get connection numbers for arbitrary $K_\mu K_\nu$. | |
| Apr 18, 2011 at 16:06 | vote | accept | john mangual | ||
| Apr 18, 2011 at 14:34 | comment | added | David E Speyer | No, as Paul Johnson says above. There should be a distant relationship as follows: Connection numbers are the multiplication constants for the center of $k[S_d]$, written in the basis $K_{\mu}$ in Paul Johnson's answer. Another basis for the center of $k[S_d]$ is characters of $S_d$-irreps. Multiplication in that basis is given by the Kronecker coefficients. LR numbers are a very special case of Kronecker coefficients. | |
| Apr 18, 2011 at 13:42 | comment | added | john mangual | Are "connection numbers" also known as Littlewood-Richardson numbers? | |
| Apr 18, 2011 at 11:09 | comment | added | Paul Johnson | Amritanshu -- I address this a bit at the end of my answer, but he's talking in part about: M. Lehn and C. Soerger. Symmetric groups and the cup product on the cohomology of Hilbert schemes. | |
| Apr 18, 2011 at 11:07 | comment | added | Paul Johnson | Denis -- not immediately. In his question, we'll want $|\mu|=|\nu|=|\lambda|$ to have nonzero coefficients $C^\lambda_{\mu,\nu}$, and for LR we want them to add. LR is about exterior tensor product of representations -- take a rep of $S_n$, one of $S_m$, tensor to get one of $S_n\times S_m$, and induct. John's question as stated is all taking place in fixed $S_n$, and you could relate it to representation theory some in that multiplication in the group algebra is semisimple in the character basis. | |
| Apr 18, 2011 at 11:02 | answer | added | Paul Johnson | timeline score: 24 | |
| Apr 18, 2011 at 9:46 | comment | added | Denis Serre | Are these formulae related to partitions and Littlewood--Richardson coefficients ? | |
| Apr 18, 2011 at 9:45 | history | edited | Denis Serre | CC BY-SA 3.0 |
deleted 1 characters in body
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| Apr 18, 2011 at 6:58 | comment | added | Amritanshu Prasad | Mariano: can you give some references for your comments? | |
| Apr 18, 2011 at 6:50 | comment | added | Mariano Suárez-Álvarez | You get some of them (the top ones, in some sense) as the structure coefficients of the cohomology algebra of the Hilbert scheme of points in the plane. Indeed, there is a filtration of the group algebra by the "age" of group elements, such that that cohomology algebra is the associated graded algebra to the induced filtration on the center of the group algebra. | |
| Apr 18, 2011 at 6:45 | comment | added | Mariano Suárez-Álvarez | Those multiplicities are called connection numbers, and have been investigated by combinatorists. | |
| Apr 18, 2011 at 6:23 | comment | added | Michael Lugo | I'm curious -- where did you see this? | |
| Apr 18, 2011 at 5:36 | history | edited | john mangual | CC BY-SA 3.0 |
clarify
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| Apr 18, 2011 at 5:30 | history | asked | john mangual | CC BY-SA 3.0 |