Talk:Symmetric algebra

Proposed merger

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)

Except this rationale isn't true. Symmetric tensors are elements of the tensor algebra, whereas the symmetric algebra is a quotient of the tensor algebra. There is a natural quotient mapping that takes the space of symmetric tensors into the symmetric tensor algebra. However this is not an isomorphism in general (e.g., over fields of prime characteristic). Sławomir Biały (talk) 13:41, 10 October 2010 (UTC)

As I see the things: characteristic zero case is general enough. In characteristic zero, symmetric algebra can be viewed both as a quotient and as a subalgebra of the full tensor algebra, and such double view is often fruitful. Most of material, including all applications in natural sciences mentioned in the Symmetric tensor article, require characteristic zero. Granted, in characteristic p things are getting more complicated, but this is not the reason to have two separate articles. I was probably too cryptic in the initial description of my rationale, sorry for that. Justpasha (talk) 21:30, 11 October 2010 (UTC)

Except I have to point out that the symmetric algebra is not a subalgebra of the tensor algebra either, even over fields of characteristic zero. It is true that in that case it can be identified with a linear subspace of the tensor algebra, since the quotient mapping onto the symmetric algebra has a natural complement (which is also only a linear subspace). Anyway, since there are many applications of the symmetric algebra that do not assume characteristic zero, I don't think a merger is appropriate. Sławomir Biały (talk) 22:11, 11 October 2010 (UTC)

Again, I agree with all your substantial claims, and I was wrong claiming that symmetric algebra is isomorphic to a subalgtebra of the full tensor algebra. Still, these are highly intertwined and related constructions which, in my opinion, should be featured in one article. The most general (characteristic-free!) definition of Schur functors, which are generalizations of the symmetric and exterior products, involves interplay between subspaces and quotients of tensor algebras. Bourbaki, which is a major reference for both articles, speaks about symmetric tensors inter alia in the chapter devoted to symmetric algebras.justpasha (talk) 16:29, 12 October 2010 (UTC)

A case in point: the Schur functors generalize the graded parts of the symmetric algebra, and the Young symmetrizers generalize the symmetric tensors. We have separate articles for these things as well, although it is true that they are related. Sławomir Biały (talk) 04:59, 14 October 2010 (UTC)

Well, that's bad that you have separate articles for these things. justpasha (talk) 14:01, 14 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)

First - sorry, I am not sure I understand what do you mean. Can you give an example of such a case? Second, as you probably know better than me, mathematical structures often have many facets, and a mere fact that some structure A can be thought, in some context, without any relation to some other structure B does not automatically imply that A and B deserve separate articles. justpasha (talk) 14:01, 14 October 2010 (UTC)

For example, in quantum physics, when one is dealing with a system of N identical particles one deals simply with rank N symmetric tensors, i.e. just some finite-dimensional vector space (at least if the original particles were represented by vectors in some finite-dimensional space as in the case of just considering spin). One need not consider the infinite-dimensional space of all symmetric tensors of all ranks and its structure as an algebra. Another example is in the case of the theory of representations of say SL(2). Here one has the standard two-dimensional representation, V, and all other irreducible representations are SymⁿV for some n, and every finite-dimensional representation breaks up into a direct sum of these. Nowhere does one introduce the tensor algebra (later on one might wish to study the universal enveloping algebra, but until then it's overkill). I think I would turn around what you said and say that just because A can be thought of as B doesn't mean that A should be merged into the article on B. The treatment of topics in Bourbaki is generally meant to be pushing the limits of dealing with everything at once, and that's great, but that's not the point of wikipedia. I think that in the case at hand, it is overkill to define symmetric tensors as elements of the tensor algebra. RobHar (talk) 01:14, 15 October 2010 (UTC)

Ok, thank you for your thorough explanation. justpasha (talk) 12:37, 15 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)

How about getting this article right first?

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)

Missing topic: representation theory

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)

Poorly written

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)