Timeline for Using Schur-Weyl duality
Current License: CC BY-SA 4.0
20 events
| when toggle format | what | by | license | comment | |
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| Jan 5, 2023 at 20:12 | vote | accept | Trevor K | ||
| Jan 5, 2023 at 19:57 | comment | added | Sam Hopkins | @TrevorK: Indeed, although your inquiries seem more related to a different open problem about symmetric functions, namely "Thrall's problem" (see www-users.cse.umn.edu/~reiner/Talks/ThrallsProblem.pdf) | |
| Jan 5, 2023 at 18:16 | comment | added | Trevor K | As pointed out in a now-deleted comment, restricting from $GL(V)$ to $S_n$ is an open problem! realopacblog.wordpress.com/2019/11/17/the-restriction-problem | |
| Jan 5, 2023 at 7:48 | answer | added | Vladimir Dotsenko | timeline score: 2 | |
| Jan 5, 2023 at 5:20 | answer | added | Christopher Ryba | timeline score: 7 | |
| Jan 5, 2023 at 3:06 | comment | added | LSpice | @SamHopkins, re, I think your comment underrates the character: what you write down is the restriction to the diagonal torus of an honest, uniquely determined (subject to some restrictions) class function on $\operatorname{GL}_n$. Of course it takes some work to evaluate that character on non-semisimple elements … but a permutation matrix is semisimple (all in characteristic $0$), so there's no harm thinking about evaluating a character by looking at its eigenvalues. But maybe I misunderstood your point. | |
| Jan 5, 2023 at 1:43 | comment | added | Trevor K | I guess my last comment is a confusing way to ask: in finite group representation theory, the character values of the restriction of a subgroup are the original character values, you just forget some. In restricting from $GL(V)$ to $S_n$, does the same thing happen? | |
| Jan 5, 2023 at 1:36 | history | edited | Trevor K | CC BY-SA 4.0 |
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| Jan 5, 2023 at 1:36 | comment | added | Trevor K | @darijgrinberg I'm a little unsure about how to interpret applying the action of the diagonal matrix $\operatorname{diag}(x_1, x_2, \ldots, x_n)$. I get that the variables stand in for eigenvalues. In finite group representation theory we can fill in the character table. I see why no character table exists for $GL(V)$. Are we supposed to think of the symmetric polynomial output as a recipe for getting the characters of matrices? Does that mean that just restricting our attention to permutation matrices wont change the recipe so the character of the $S_n \leq GL(V)$ is the same polynomial? | |
| Jan 5, 2023 at 0:02 | comment | added | Trevor K | Thanks Darij and Sam. I read through the proof on Wikipedia which Sam linked. I wonder if I am missing something. Wikipedia seems to prove the decomposition of the tensor power of $V$ into $\mathbf{GL}(V)$ irreducibles, but not into $\mathbf{GL}(V)\times S_d$ irreducibles. Am I missing something there? | |
| Jan 3, 2023 at 17:18 | history | edited | Trevor K | CC BY-SA 4.0 |
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| Jan 2, 2023 at 18:01 | comment | added | Sam Hopkins | @darijgrinberg: At the beginning of Chapter 8, before the statement of Theorem 8.1, Reutenauer in fact considers both of these actions of the symmetric group on the tensor product space (he calls them the "place permutation action" and the "variable permutation action"). | |
| Jan 2, 2023 at 11:36 | comment | added | darij grinberg | Note that it may well be that the identity can be proved in a similar way to the Schur-Weyl duality character formula, viz., by computing the trace of some operator (probably a composition of the action of $\sigma$, the action of the diagonal matrix $\operatorname{diag}\left(x_1, x_2, \ldots, x_{\left|A\right|}\right)$, and perhaps the projection from $E_n$ onto $F \cap E_n$) in two different ways. But I don't find it obvious how. Fortunately, it seems that Theorem 8.1 is only used in the first proof of Theorem 8.3, while the second proof is self-contained (I hope; haven't checked). | |
| Jan 2, 2023 at 10:56 | comment | added | darij grinberg | The problem is that whatever Reutenauer uses doesn't seem to be standard Schur-Weyl duality. I got confused by his symmetric group too -- his symmetric group $S_n$ acts diagonally through $\operatorname{GL}\left(V\right)$, not by permuting the tensor factors! The latter action wouldn't preserve the free Lie subalgebra (viewed as a subspace of the tensor algebra), so this is not surprising in hindsight, but it makes me wonder if Schur-Weyl duality is applicable at all. | |
| Jan 2, 2023 at 2:29 | comment | added | Sam Hopkins | Also, permutation matrices do not have all eigenvalues $1$. But again maybe I am misunderstanding some of your assertions because I can't look at the text you're trying to understand. | |
| Jan 2, 2023 at 2:27 | comment | added | Sam Hopkins | I find several parts of your question confusing (but I think this may because you are discussing a proof that you don't link to so I can't read in detail). For example, in the usual set-up of Schur-Weyl duality $V$ is an $n$-dimensional complex vector space acted on by $GL(V) = GL_n$, and we are looking at the action of $GL_n \times \mathfrak{S}_k$ on $V^{\otimes k}$ where the first factor acts diagonally and the second by permuting the copies of $V$. In general $n$ and $k$ are completely unrelated and so there is no natural inclusion $\mathfrak{S}_k \subseteq GL_n$ like you suggest. | |
| Jan 1, 2023 at 23:41 | comment | added | Trevor K | That is helpful in understanding why Schur-Weyl duality applies. Thank you! But I am still confused about how this turns into a statement about symmetric functions. | |
| Jan 1, 2023 at 23:38 | comment | added | Qiaochu Yuan | The universal enveloping algebra of the free Lie algebra is the tensor algebra. I haven't looked at the rest of this but maybe that's already helpful. In any case, so that answering this question does not require having a copy of Reutenauer, it might be a good idea for you to include more of the text, e.g. the statement of the Theorem and a relevant section of the proof. | |
| S Jan 1, 2023 at 23:34 | review | First questions | |||
| Jan 2, 2023 at 1:25 | |||||
| S Jan 1, 2023 at 23:34 | history | asked | Trevor K | CC BY-SA 4.0 |