Timeline for G-modules and ideals of secant varieties
Current License: CC BY-SA 3.0
15 events
| when toggle format | what | by | license | comment | |
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| Sep 2, 2018 at 22:26 | history | notice removed | user117617 | ||
| Sep 1, 2018 at 21:31 | vote | accept | CommunityBot | ||
| Sep 1, 2018 at 21:30 | answer | added | Puzzled | timeline score: 0 | |
| Sep 1, 2018 at 21:18 | history | notice added | user117617 | Canonical answer required | |
| Nov 25, 2017 at 20:02 | comment | added | Zach Teitler | @JasonStarr Minor correction, the ideal of the $k$th secant variety is not defined by the degree $k+1$ part of the ideal of the Veronese, it is defined by the $k+1$ minors (which span a proper subspace of the degree $k+1$ part of the ideal of the Veronese, except if $k=1$). | |
| Nov 24, 2017 at 20:02 | comment | added | user117617 | You are right. I corrected it. | |
| Nov 24, 2017 at 20:02 | comment | added | Jason Starr | @AbdelmalekAbdesselam. Presumably the OP wants the generators $(f_1,\dots,f_r)$ to be a minimal set of generators, which forces the elements to be homogeneous. As Zach Teitler states, the issue is whether or not the graded ideal in degree $k+1$ is an irreducible representation. A related question is Exercise 15.45, p. 230 of Fulton-Harris, "Representation Theory, A First Course." | |
| Nov 24, 2017 at 20:02 | history | edited | user117617 | CC BY-SA 3.0 |
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| Nov 24, 2017 at 19:44 | comment | added | Abdelmalek Abdesselam | @J_Cole: I think the question as it stands is not well formulated. I would expect $H$ to be finite dimensional (e.g. of dimension 1 if $f_1$ is a classical invariant). However, the ideal will be an infinite-dimensional vector space, without restriction on the degree. | |
| Nov 24, 2017 at 19:18 | comment | added | Zach Teitler | @J_Cole I agree that with $G \times G$ acting on general matrices, it would be irreducible. | |
| Nov 24, 2017 at 18:41 | comment | added | Jason Starr | Why do you write, "But there is a subrepresentation spanned by the principal minors (same row and column indices)"? When $k$ equals $1$, this span is not a subrepresentation. Note, in this case, the question is simply whether the degree $2$ graded piece of the homogeneous ideal of the Veronese variety is an irreducible representation, which it is. | |
| Nov 24, 2017 at 18:38 | comment | added | user117617 | Thanks. Do you have a reference? If instead of considering symmetric matrices you consider general matrices it seems to me that the representation of $G\times G$ should be irreducible. | |
| Nov 24, 2017 at 17:53 | comment | added | Zach Teitler | Hmm, I don't think so. Here $\mathbb{P}^N$ is the space of $(n+1) \times (n+1)$ symmetric matrices ($N = \binom{n+2}{2}$). The Veronese is the variety of rank one symmetric matrices. Its $k$th secant variety is the variety of rank $k$ symmetric matrices. It is defined by the vanishing of the $(k+1) \times (k+1)$ minors. If I understand correctly, you are asking whether the $G$-representation on the vector space of size $k+1$ minors of a symmetric matrix is an irreducible representation. But there is a subrepresentation spanned by the principal minors (same row and column indices). | |
| Nov 24, 2017 at 14:21 | review | First posts | |||
| Nov 24, 2017 at 14:31 | |||||
| Nov 24, 2017 at 14:18 | history | asked | user117617 | CC BY-SA 3.0 |