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I would like to merge Symmetric tensor into this article. Rationale: symmetric tensors are elements of the symmetric algebra. The content of two articles overlap. Justpasha ( talk) 10:18, 10 October 2010 (UTC)
Oppose: I think there are many cases in which ones deals with symmetric tensors without ever needing to know about the tensor algebra (or the symmetric algebra) and that symmetric tensors certainly need not be thought of as elements of the tensor algebra. RobHar ( talk) 16:59, 13 October 2010 (UTC)
Oppose. Essentially, per RobHar. This level of abstraction simply isn't useful for a scientist who just wants to know what a symmetric tensor is, and what in physical terms that symmetry represents, ie what about the physical nature of the world leads to that symmetry. The mere fact that information about some structure A can be hidden away in some immense amount of unnecessary irrelevance about some other structure B does not mean that is where any and all useful information about A should be buried. Jheald ( talk) 14:17, 14 October 2010 (UTC)
Well, I am giving up and removing the merge templates. justpasha ( talk) 12:37, 15 October 2010 (UTC)
Te confusion comes from the similarity between the symmetric algebra, which is naturally defined as a quotient of then tensor algebra and the symmetric COALGEBRA which is naturally defined as a subspace of the tensor power vector space. The symmetrizer $sym: T(V) \to S(V) (v_1 \otimes ... \otimes v_n) \mapsto \frac{1}{n!}\sum_{\sigma \in \Sigma_n}(v_{\sigma(1)}\otimes ...\otimes v_{\sigma(n)})$, which projects to symmetric tensors, defines not an algebra structure but a coalgebra structure.
If someone is conffused about the realtion beteeen symmetric tensors and elements of te symmetric tensor algebra, I would advice him, to read about cofree coalgebras. — Preceding unsigned comment added by 92.78.51.191 ( talk) 10:01, 27 May 2013 (UTC)
This article shows a complete lack of comprehension of the distinction between an encyclopedia article and Bourbaki. Despite the article's careful definition of its subject, it doesn't even give one example of a symmetric algebra, or even of a symmetric square. Another clarifying feature might be to explain why the dimension of a symmetric kth power of a vector space is what it is said to be. Daqu ( talk) 17:20, 3 December 2012 (UTC)
Completely missing from this article is the representation theory, a la Fulton and Harris, where the symmetric algebra is front row and center for the first 2-4 chapters! 67.198.37.16 ( talk) 23:55, 18 September 2016 (UTC)
This article is as poorly written as any math article I've ever seen in Wikipedia.
If someone doesn't already know what the symmetric algebra over a vector space is, the introductory section — and the next section — are not going to help very much at all. It would be a million times better the article avoided being so "correct" and first gave an example of S2(V) before going further. The statement that it "corresponds to polynomials" shows that whoever wrote that has no idea how to communicate mathematics to someone who doesn't already know what they are being told. 50.205.142.35 ( talk) 01:16, 16 January 2020 (UTC)
![]() | This article is rated C-class on Wikipedia's
content assessment scale. It is of interest to the following WikiProjects: | ||||||||||
|
I would like to merge Symmetric tensor into this article. Rationale: symmetric tensors are elements of the symmetric algebra. The content of two articles overlap. Justpasha ( talk) 10:18, 10 October 2010 (UTC)
Oppose: I think there are many cases in which ones deals with symmetric tensors without ever needing to know about the tensor algebra (or the symmetric algebra) and that symmetric tensors certainly need not be thought of as elements of the tensor algebra. RobHar ( talk) 16:59, 13 October 2010 (UTC)
Oppose. Essentially, per RobHar. This level of abstraction simply isn't useful for a scientist who just wants to know what a symmetric tensor is, and what in physical terms that symmetry represents, ie what about the physical nature of the world leads to that symmetry. The mere fact that information about some structure A can be hidden away in some immense amount of unnecessary irrelevance about some other structure B does not mean that is where any and all useful information about A should be buried. Jheald ( talk) 14:17, 14 October 2010 (UTC)
Well, I am giving up and removing the merge templates. justpasha ( talk) 12:37, 15 October 2010 (UTC)
Te confusion comes from the similarity between the symmetric algebra, which is naturally defined as a quotient of then tensor algebra and the symmetric COALGEBRA which is naturally defined as a subspace of the tensor power vector space. The symmetrizer $sym: T(V) \to S(V) (v_1 \otimes ... \otimes v_n) \mapsto \frac{1}{n!}\sum_{\sigma \in \Sigma_n}(v_{\sigma(1)}\otimes ...\otimes v_{\sigma(n)})$, which projects to symmetric tensors, defines not an algebra structure but a coalgebra structure.
If someone is conffused about the realtion beteeen symmetric tensors and elements of te symmetric tensor algebra, I would advice him, to read about cofree coalgebras. — Preceding unsigned comment added by 92.78.51.191 ( talk) 10:01, 27 May 2013 (UTC)
This article shows a complete lack of comprehension of the distinction between an encyclopedia article and Bourbaki. Despite the article's careful definition of its subject, it doesn't even give one example of a symmetric algebra, or even of a symmetric square. Another clarifying feature might be to explain why the dimension of a symmetric kth power of a vector space is what it is said to be. Daqu ( talk) 17:20, 3 December 2012 (UTC)
Completely missing from this article is the representation theory, a la Fulton and Harris, where the symmetric algebra is front row and center for the first 2-4 chapters! 67.198.37.16 ( talk) 23:55, 18 September 2016 (UTC)
This article is as poorly written as any math article I've ever seen in Wikipedia.
If someone doesn't already know what the symmetric algebra over a vector space is, the introductory section — and the next section — are not going to help very much at all. It would be a million times better the article avoided being so "correct" and first gave an example of S2(V) before going further. The statement that it "corresponds to polynomials" shows that whoever wrote that has no idea how to communicate mathematics to someone who doesn't already know what they are being told. 50.205.142.35 ( talk) 01:16, 16 January 2020 (UTC)