Talk:Symmetric polynomial

Fundamental types of symmetric functions

Replace " There are a few" by "There are six" ... , namely the elementary symmetric functions, the power sums, the complete homogeneous functions, the monomial sums, the Doubilet functions, and the Schur functions.

John McKay 130.54.16.90 (talk) 09:15, 29 January 2008 (UTC)

We should definitely add the monomial sums (no article yet), and Schur polynomials to the list. I don't know anything about the Doubilet functions, but presumably they are discussed in MacDonald's book. Monomial sums are fairly natural. They are simply the sum of all monomials of a certain "shape". For instance the monomial sum for the shape of xxy in the variables x,y is xxy+xyy, and so the complete symmetric polynomial is the sum of the monomial sums of shape xxy and xxx, and each elementary symmetric polynomial is the monomial sum for the shape of a square-free monomial.

Should monomial sums get their own article, or just a short description here?

Power sums, elementary, and complete have a nice mutual context of generating the ring of symmetric polynomials.

Should we include the partition version of these polynomials, and present the basis proposition (prop 1) in ch 6.1 of Fulton's Young Tableaux? This would give natural mutual context for elementary, complete, monomial, and Schur polynomials.

What is the natural mutual context for including Doubilet functions? JackSchmidt (talk) 15:22, 29 January 2008 (UTC)

My feeling is that there should be a separate article on each type of symmetric function. That makes it easier to refer to the one type you happen to need when, say, reading math. Proofs should be in that article. The main facts should be summarized in the collective article (this one) to make the broad picture. This is what I tried to do in reorganizing the material last year.

As for which kinds to have article on, there's no reason to omit any, even less important ones. If each has a separate article, and the main article explains which ones are most basic or important and for what purpose, there shouldn't be any confusion for the readers. I'm sorry that I don't know enough to prepare articles on all the important types!

JackSchmidt, I think you mean "common context", not "mutual context". As for your idea, I think a short section on how to present a common formulation of different kinds is not out of place, but it is an advanced extra, so should be later in the article, since the basic need is to know the various types. Zaslav (talk) 17:04, 29 January 2008 (UTC)

I'll probably try to add the (imho badly named) monomial symmetric polynomials to this article soon. As for Doubilet functions, I've never seen those mentioned even though working in the area for years (I'm pretty sure they are neither mentioned by Macdonald not Stanley), so I think we should drop them. Macdonald does mention "forgotten" symmetric functions, the counterpart of the monomial ones in the duality that interchanges elementary and homogeneous symmetric functions (the power sums and Schur function are more or less self-dual). Maybe this is called Doubilet functions by some. I'm not sure about the idea of writing separate articles for each special class of symmetric polynomials/functions. There is for instance not much to say about monomial symmetric polynomials in isolation, their interest comes from the role they play in relation to other symmetric functions (see for instance Newton identities#As a telescopic sum of symmetric function identities). Similarly interest in complete homogeneous symmetric functions comes from their duality with elementary ones, and relation with monomial ones (dual bases for the natural scalar product). Probably Symmetric function is a good place for explaining such relations. Marc van Leeuwen (talk) 09:16, 8 April 2008 (UTC)

Anonymous request, moved from the article page

It is my understanding that the theorem above is attributable to Waring. A proof of this theorem would be a valuable addition to the page.

Moved by Oleg Alexandrov 00:53, 11 May 2005 (UTC)

Main theorem

I moved the statement and proof of the representation theorem for symmetric polynomials in terms of elementary symmetric polynomials to the article Elementary symmetric polynomial. This is to collect the main facts about the latter in one place. I hope similar proofs will be provided for power sum symmetric polynomials and complete homogeneous symmetric polynomials, and that this scheme (of partitioning specialized parts of the subject) meets with approval. Zaslav 00:52, 26 March 2007 (UTC)

Thanks, and approved – fits with “summary style” (specialized content on specialized pages)!

—Nils von Barth (nbarth) (talk) 17:49, 21 April 2009 (UTC)

Relation with the roots of a monic univariate polynomial

This section had a duplicate subsection with less information that I just removed tonight. — Preceding unsigned comment added by 64.134.64.115 (talk) 03:26, 4 November 2011 (UTC)

Error in the Galois theory subsection

The "only if" direction of the following claim is incorrect: "Moreover the fundamental theorem of symmetric polynomials implies that a polynomial function f of the n roots can be expressed as (another) polynomial function of the coefficients of the polynomial determined by the roots if and only if f is given by a symmetric polynomial."

For example, if "the polynomial" is $P(x)=x^{2}-3$ with roots $\pm {\sqrt {3}}$, and $f(x_{1},x_{2})=P(x_{1})$, then $f({\sqrt {3}},-{\sqrt {3}})=P({\sqrt {3}})=0$ yet $f$ is clearly not symmetric.

Certainly there are interesting true facts along these lines, e.g. a cubic polynomial has a non-square discriminant (or equivalently, the alternating product $(\alpha _{1}-\alpha _{2})(\alpha _{2}-\alpha _{3})(\alpha _{3}-\alpha _{1})$ of roots is not in $\mathbb {Q}$) if and only if it defines a $S_{3}$ Galois extension over the rationals. And we have a less interesting but still true result for quadratic polynomials: non-square discriminant is equivalent to the difference $\alpha _{1}-\alpha _{2}$ of roots not being in $\mathbb {Q}$, which is equivalent to the roots generating a nontrivial quadratic Galois extension over the rationals.

(I read somewhere that errors should be brought up on the talk page. Is this so? In this case, changing "if and only if" to "if" would suffice. Alternatively, a stronger statement that is useful in setting up Galois theory is that f can be expressed ... if and only if f can be expressed as (another) polynomial function symmetric in the n roots.) Vywang (talk) 23:24, 11 April 2018 (UTC)

I don't understand what you think is wrong with the given statement. A counter-example should consist of the following things:

an m-variate polynomial f and another m-variate polynomial g having the properties that whenever P is a monic single-variable polynomial with roots $r_{1},\ldots ,r_{m}$ and coefficients $a_{1},\ldots ,a_{m}$, one has $f(r_{1},\ldots ,r_{m})=g(a_{1},\ldots ,a_{m})$, and yet f is not symmetric. --JBL (talk) 01:32, 12 April 2018 (UTC)

My apologies. I missed a previous line, which had said that we view the roots as independent variables. (So I guess I was parsing "polynomial function" as "polynomial expression evaluated at...".) Big picture wise, I was confused by the placement of the statement, because I don't remember the statement actually being directly relevant in proving (say) the fundamental theorem of Galois theory. Vywang (talk) 01:56, 12 April 2018 (UTC)

@Vywang: No apologies necessary, to be sure. If you can think of a way to make it clearer in context, that would be great. --JBL (talk) 14:05, 15 April 2018 (UTC)