Timeline for Is the appearance of Schur functions a coincidence?
Current License: CC BY-SA 4.0
16 events
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| Oct 11, 2023 at 16:42 | vote | accept | matha | ||
| Oct 8, 2023 at 16:38 | answer | added | Antoine Labelle | timeline score: 8 | |
| Oct 4, 2023 at 11:42 | comment | added | Joel Kamnitzer | As luck would have it, my undergraduate student, Antoine Labelle, just posted a paper which partially answers this question, arxiv.org/abs/2310.00855. | |
| Sep 24, 2023 at 18:09 | history | edited | matha | CC BY-SA 4.0 |
Addressed the corrections from the comments and clarified my question
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| Sep 24, 2023 at 18:06 | comment | added | matha | I want to understand the rigidity of the Schur functions in these examples when modified. For instance, the equivariant cohomology of the Grassmannian yields the factorial Schur functions. Are there analogous concepts in the representation theory of the general linear group? The representation theory of the super Lie algebra for the general linear group gives the supersymmetric Schur functions. Is there an analogue in the cohomology of the Grassmannian? Or is the appearance of Schur functions in these contexts coincidental, without any generalizations having a corresponding analogue? | |
| Sep 24, 2023 at 13:04 | comment | added | Timothy Chow | I'm not sure I understand the question. How is the question "is it a coincidence?" different from the question "is it false in other types?"? Are you asking if there is a way to generalize Schur functions other than by considering other types? One can add parameters and get Macdonald polynomials, for example; does that count? | |
| Sep 23, 2023 at 19:41 | comment | added | lambda | I added too many hypotheses in my last comment. What I should have said is that they are the only monomial-positive orthonormal basis over $\mathbb Z$, and also the only one over $\mathbb Q$ that has nonnegative integer structure coefficients. | |
| Sep 23, 2023 at 18:42 | comment | added | lambda | Over $\mathbb Z$, the Schur functions are the only monomial-positive homogeneous orthonormal basis with nonnegative structure coefficients. | |
| Sep 23, 2023 at 16:43 | comment | added | Richard Stanley | Apparently (is this right?) the fact that the symmetric group is the Weyl group of the general linear group and that it appears in Schur-Weyl duality is just a coincidence. I found one such statement on pages 5-6 of math.columbia.edu/~samdehority/files/spring_2021_seminar_1/…. | |
| Sep 23, 2023 at 16:33 | comment | added | Sam Hopkins | @MarkWildon: but $p_{\lambda}/z_{\lambda}$ is only defined over $\mathbb{Q}$; $s_{\lambda}$ being defined over $\mathbb{Z}$ is indeed a special feature. | |
| Sep 23, 2023 at 16:25 | comment | added | Mark Wildon | I'd add also, in reference to 5, that there are, of course, infinitely many orthonormal bases of the ring of symmetric functions. Okay the Schur function basis is particularly nice/important, but for instance the scaled power sum functions $p_\lambda / z_\lambda$ are also orthonormal. In fact it's this basis that's used to define the characteristic isometry. | |
| Sep 23, 2023 at 16:23 | comment | added | Mark Wildon | The irreducible characters of the symmetric groups are not themselves Schur functions. Instead the Schur functions are their images under the characteristic isometry, and there is a close connection between the symmetric group characters and symmetric functions via the Murnaghan--Nakayama rule in the form $s_\lambda p_k = \sum_\mu \epsilon(\mu/\lambda) s_\mu$, where the sum is over skew-partitions of size $k$. This makes connection 1 a little less direct than your question might suggest. | |
| Sep 23, 2023 at 15:44 | comment | added | Sam Hopkins | While it is true that in other types the close connection between these various guises of Schur functions (irreducible representations of algebraic groups, Schubert classes, etc.) breaks down, I would assert that the appearance of Schur functions in all these places is not just "a coincidence." Possibly Allen Knutson will chime in explaining how the fundamental idea is the category $\mathrm{Rep}(\mathbf{Vect})$ or something like that... | |
| Sep 23, 2023 at 15:34 | history | edited | matha | CC BY-SA 4.0 |
fixed the title
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| S Sep 23, 2023 at 15:32 | review | First questions | |||
| Sep 23, 2023 at 20:17 | |||||
| S Sep 23, 2023 at 15:32 | history | asked | matha | CC BY-SA 4.0 |