Timeline for Identity involving Jack polynomials at $x^{-1}$
Current License: CC BY-SA 4.0
25 events
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| Aug 19, 2022 at 13:01 | comment | added | darij grinberg | @Marcel: Thus a question :) It does mention the rotated complement, at least. | |
| Aug 19, 2022 at 12:51 | comment | added | Marcel | @darijgrinberg but that paper doesn't mention computing a polynomial with argument $x^{-1}$. How does it relate? | |
| Aug 18, 2022 at 22:51 | comment | added | darij grinberg | Does Corollary 16 in Paolo Bravi and Jacopo Gandini, Some Combinatorial Properties of Skew Jack Symmetric Functions help in any way? | |
| Aug 18, 2022 at 18:00 | history | bumped | CommunityBot | This question has answers that may be good or bad; the system has marked it active so that they can be reviewed. | |
| Apr 20, 2022 at 17:04 | history | bumped | CommunityBot | This question has answers that may be good or bad; the system has marked it active so that they can be reviewed. | |
| S Mar 21, 2022 at 20:50 | history | bounty ended | Marcel | ||
| S Mar 21, 2022 at 20:50 | history | notice removed | Marcel | ||
| Mar 21, 2022 at 14:43 | answer | added | thedude | timeline score: 1 | |
| Mar 16, 2022 at 21:50 | history | edited | Marcel | CC BY-SA 4.0 |
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| Mar 16, 2022 at 15:06 | comment | added | ArB | The Macdonald $P$ polynomials are the unique orthogonal basis of $\mathbb{C}[P]^W$ for this scalar product with unitriangularity wrt monomial basis . So to show image of $P_{\lambda}$ under the transformation $x^{\alpha} \mapsto x^{-w_0\alpha}$ is $P_{-w_0\lambda}$ you may show that the image satisfies orthogonality and triangularity. | |
| Mar 16, 2022 at 15:01 | comment | added | ArB | I am using the scalar product $(f,g) = ct(f\bar{g}\Delta)$ where $\bar{g}(x_1,...,x_n)=g(x_1^{-1},...,x_n^{-1})$ and $\Delta$ is given in chapter VI sec 9 of Macdonald's Symmetric Functions and Hall Polynomials book. This is for the Macdonald $P$ polynomials, and you can in-fact prove your statement for the $P$ polynomials and then take the limit for Jack case. | |
| Mar 16, 2022 at 14:34 | comment | added | Marcel | @ArB Its relatively clear and quite helpful. I could use a bit more detail on the "use the constant term scalar product since the kernel is invariant" part. You lost me there. | |
| Mar 16, 2022 at 14:16 | comment | added | ArB | ... $w_0$ is the longest permutation that takes dominant weights to antidominant weights... in one line notation $[n , n-1,..., 1]$... $P = \mathbb{Z}^n$ is the weight lattice for $GL_n$ and $\mathbb{C}[P]$ is the group algebra spanned by $x^{\alpha} : \alpha \in P$... | |
| Mar 16, 2022 at 14:12 | comment | added | ArB | Here's a sketch: (Working in type $GL_n$): take the transformation $x^{\alpha} \mapsto x^{-w_0\alpha}$ from $\mathbb{C}[P] \to \mathbb{C}[P]$, and show that the Jack polynomial $J_{\lambda} \mapsto J_{-w_0\lambda}$ via this transformation. (You can use the constant term scalar product for showing this since the kernel $\Delta$(in Macdonald's notation) is $S_n$ invariant. Now use the fact that $J_{\lambda+(1^n)} = x_1...x_nJ_{\lambda}$. Does this help? (continued below for more...) | |
| Mar 16, 2022 at 13:04 | history | edited | Marcel | CC BY-SA 4.0 |
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| Mar 15, 2022 at 11:57 | history | edited | Marcel | CC BY-SA 4.0 |
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| Mar 14, 2022 at 14:40 | comment | added | Jules Lamers | You could try $N=2$, for which the Jack polynomials can be written down explicitly as well | |
| Mar 14, 2022 at 14:37 | comment | added | Jules Lamers | @Marcel ...which is an invertible $N\times N$ matrix whose eigenvalues are meant to be the $N$ arguments of the Jack polynomials? | |
| Mar 14, 2022 at 13:52 | comment | added | Marcel | @JulesLamers of matrix $x$ | |
| Mar 14, 2022 at 13:02 | comment | added | Jules Lamers | Of what matrix is the determinant taken? | |
| S Mar 14, 2022 at 10:26 | history | bounty started | Marcel | ||
| S Mar 14, 2022 at 10:26 | history | notice added | Marcel | Draw attention | |
| Mar 11, 2022 at 16:35 | comment | added | Marcel | I don't think that would help. But the book is Log-gases and Random Matrices, by Peter Forrester, and the paper is also by him. | |
| Mar 11, 2022 at 16:34 | comment | added | Sam Hopkins | Perhaps you could cite the paper/book. | |
| Mar 11, 2022 at 13:34 | history | asked | Marcel | CC BY-SA 4.0 |