Timeline for proof of result from Ian Macdonald's paper "A New Class of Symmetric Functions"
Current License: CC BY-SA 4.0
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Mar 14, 2021 at 23:44 | comment | added | Jules Lamers | @dash1729 If you're happy with my answer you can accept it by clicking the check mark next to it, so that I won't be listed as an unanswered question any longer. | |
| Mar 10, 2021 at 1:10 | answer | added | Jules Lamers | timeline score: 2 | |
| Mar 2, 2021 at 2:24 | comment | added | dash1729 | Thanks so much for your answers @JulesLamers and PerAlexandersson. Very helpful. Again, sorry for the tone of the initial question--I was just frustrated to get halfway through a paper and find a key intermediate result was lacking a proof! | |
| Mar 1, 2021 at 8:20 | comment | added | Per Alexandersson | I think some of the work by Marshall is focused on this type of questions, arxiv.org/pdf/math/9812080.pdf | |
| Mar 1, 2021 at 4:15 | comment | added | Jules Lamers | Good. If you're happy with the alternative proof, use it to study what the involution does to $E$: it suffices to understand this on the eigenbasis of Macdonald polynomials. We know that $Q_\lambda(x;q,t) $ differs from $P_\lambda(x;q,t)$ only by a normalizing factor $b_\lambda = 1/\langle P_\lambda,P_\lambda \rangle_{q,t}$, so both are eigenfunctions of the same operator $E$. I would expect that this should be enough to find out what the involution does with the Macdonald operators, and thus with $E = t^{-n} D_n^1 - \sum_i t^{-i}$. | |
| Mar 1, 2021 at 4:00 | history | edited | David Roberts♦ | CC BY-SA 4.0 |
Added link
|
| Mar 1, 2021 at 3:59 | comment | added | dash1729 | If the paper appeared in 1988 but the proof only came later in 1995, then "somewhat seminal" seems a reasonable way to describe it to me. You misunderstood the emotional tone I intended. The intended emotional tone was not anger, but disappointment that Macdonald claimed a result that he didn't provide a proof for. The duality theorem in the paper is 3.5, not 3.1. It does look like the book provides an alternative proof of theorem 3.5, but not necessarily intermediate result 3.9, so I'm exploring that now. Yes, I have the Second Edition of Macdonald's book. Thanks for taking the time to reply! | |
| Mar 1, 2021 at 2:52 | comment | added | Jules Lamers | I find the phrase "somewhat seminal" a bit bizarre; either call it seminal or nothing at all. I do not understand the angry tone of the question either. Moreover, I think you could have made a little more effort to spell the name Macdonald correctly. With those comments out of the way -- in Macdonald's paper, equation (3.9) is used to prove the duality theorem 3.1. For another proof of that theorem see Section VI.5 of (at least the Second Edition, 1995) of Macdonald's "Symmetric Functions and Hall Polynomials". Depending on your goal that might be sufficient. | |
| Mar 1, 2021 at 1:10 | review | First posts | |||
| Mar 1, 2021 at 7:14 | |||||
| Mar 1, 2021 at 1:05 | history | asked | dash1729 | CC BY-SA 4.0 |