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| In algebra, an '''alternating polynomial''' is a [[polynomial]] <math>f(x_1,\dots,x_n)</math> such that if one switches any two of the variables, the polynomial changes sign:
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| :<math>f(x_1,\dots,x_j,\dots,x_i,\dots,x_n) = -f(x_1,\dots,x_i,\dots,x_j,\dots,x_n).</math>
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| Equivalently, if one [[permutes]] the variables, the polynomial changes in value by the [[sign of a permutation|sign of the permutation]]:
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| :<math>f\left(x_{\sigma(1)},\dots,x_{\sigma(n)}\right)= \mathrm{sgn}(\sigma) f(x_1,\dots,x_n).</math>
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| More generally, a polynomial <math>f(x_1,\dots,x_n,y_1,\dots,y_t)</math> is said to be ''alternating in'' <math>x_1,\dots,x_n</math> if it changes sign if one switches any two of the <math>x_i</math>, leaving the <math>y_j</math> fixed.<ref>''Polynomial Identities and Asymptotic Methods,'' [http://books.google.com/books?id=ZLW_Kz_zOP8C&pg=PA12&lpg=PA12&dq=%22alternating+polynomial%22 p. 12]</ref>
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| ==Relation to symmetric polynomials==
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| Products of [[symmetric polynomial|symmetric]] and alternating polynomials (in the same variables <math>x_1,\dots,x_n</math>) behave thus:
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| * the product of two symmetric polynomials is symmetric,
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| * the product of a symmetric polynomial and an alternating polynomial is alternating, and
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| * the product of two alternating polynomials is symmetric.
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| This is exactly the addition table for [[parity (mathematics)|parity]], with "symmetric" corresponding to "even" and "alternating" corresponding to "odd". Thus, the direct sum of the spaces of symmetric and alternating polynomials forms a [[superalgebra]] (a <math>\mathbf{Z}_2</math>-[[graded algebra]]), where the symmetric polynomials are the even part, and the alternating polynomials are the odd part.
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| This grading is unrelated to the grading of polynomials by [[degree of a polynomial|degree]]. | |
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| In particular, alternating polynomials form a [[module (mathematics)|module]] over the algebra of symmetric polynomials (the odd part of a superalgebra is a module over the even part); in fact it is a free module of rank 1, with as generator the [[Vandermonde polynomial]] in ''n'' variables.
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| If the [[Characteristic of a ring|characteristic]] of the coefficient [[ring (mathematics)|ring]] is 2, there is no difference between the two concepts: the alternating polynomials are precisely the symmetric polynomials.
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| ==Vandermonde polynomial==
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| {{main|Vandermonde polynomial}}
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| The basic alternating polynomial is the [[Vandermonde polynomial]]:
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| :<math>v_n = \prod_{1\le i<j\le n} (x_j-x_i).</math>
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| This is clearly alternating, as switching two variables changes the sign of one term and does not change the others.<ref>Rather, it only rearranges the other terms: for <math>n=3</math>, switching <math>x_1</math> and <math>x_2</math> changes <math>(x_2-x_1)</math> to <math>(x_1-x_2) = -(x_2-x_1)</math>, and exchanges <math>(x_3-x_1)</math> with <math>(x_3-x_2)</math>, but does not change their sign.</ref>
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| The alternating polynomials are exactly the Vandermonde polynomial times a symmetric polynomial: <math>a = v_n \cdot s</math> where <math>s</math> is symmetric. | |
| This is because:
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| * <math>v_n</math> is a factor of every alternating polynomial: <math>(x_j-x_i)</math> is a factor of every alternating polynomial, as if <math>x_i=x_j</math>, the polynomial is zero (since switching them does not change the polynomial, we get
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| :<math>f(x_1,\dots,x_i,\dots,x_j,\dots,x_n) = f(x_1,\dots,x_j,\dots,x_i,\dots,x_n) = -f(x_1,\dots,x_i,\dots,x_j,\dots,x_n),</math>
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| :so <math>(x_j-x_i)</math> is a factor), and thus <math>v_n</math> is a factor.
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| * an alternating polynomial times a symmetric polynomial is an alternating polynomial; thus all multiples of <math>v_n</math> are alternating polynomials
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| Conversely, the ratio of two alternating polynomials is a symmetric function, possibly rational (not necessarily a polynomial), though the ratio of an alternating polynomial over the Vandermonde polynomial is a polynomial.
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| [[Schur polynomial]]s are defined in this way, as an alternating polynomial divided by the Vandermonde polynomial.
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| ===Ring structure===
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| Thus, denoting the ring of symmetric polynomials by Λ<sub>''n''</sub>, the ring of symmetric and alternating polynomials is <math>\Lambda_n[v_n]</math>, or more precisely <math>\Lambda_n[v_n]/\langle v_n^2-\Delta\rangle</math>, where <math>\Delta=v_n^2</math> is a symmetric polynomial, the [[discriminant]].
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| That is, the ring of symmetric and alternating polynomials is a [[quadratic extension]] of the ring of symmetric polynomials, where one has adjoined a square root of the discriminant.
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| Alternatively, it is:
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| :<math>R[e_1,\dots,e_n,v_n]/\langle v_n^2-\Delta\rangle.</math>
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| If 2 is not invertible, the situation is somewhat different, and one must use a different polynomial <math>W_n</math>, and obtains a different relation; see Romagny.
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| ==Representation theory==
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| {{details|Representation theory of the symmetric group}}
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| From the perspective of [[representation theory]], the symmetric and alternating polynomials are subrepresentations of [[Representation theory of the symmetric group|the action of the symmetric group]] on ''n'' letters on the polynomial ring in ''n'' variables. (Formally, the symmetric group acts on ''n'' letters, and thus acts on derived objects, particularly [[free object]]s on ''n'' letters, such as the ring of polynomials.)
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| The symmetric group has two 1-dimensional representations: the trivial representation and the sign representation. The symmetric polynomials are the trivial representation, and the alternating polynomials are the sign representation. Formally, the scalar span of any symmetric (resp., alternating) polynomial is a trivial (resp., sign) representation of the symmetric group, and multiplying the polynomials tensors the representations.
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| In characteristic 2, these are not distinct representations, and the analysis is more complicated.
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| If <math>n>2</math>, there are also other subrepresentations of the action of the symmetric group on the ring of polynomials, as discussed in [[representation theory of the symmetric group]].
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| ==Unstable==
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| Alternating polynomials are an unstable phenomenon (in the language of [[stable homotopy theory]]): the ring of symmetric polynomials in ''n'' variables can be obtained from the ring of symmetric polynomials in arbitrarily many variables by evaluating all variables above <math>x_n</math> to zero: symmetric polynomials are thus ''stable'' or ''compatibly defined.'' However, this is not the case for alternating polynomials, in particular the [[Vandermonde polynomial]].
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| ==Characteristic classes==
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| In [[characteristic classes]], the [[Vandermonde polynomial]] corresponds to the [[Euler class]], and its square (the discriminant) corresponds to the top [[Pontryagin class]]. This is formalized in the [[splitting principle]], which connects characteristic classes to polynomials.
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| From the point of view of [[stable homotopy theory]], the fact that the [[Euler class]] is an unstable class corresponds to the fact that alternating polynomials (and the Vandermonde polynomial in particular) are unstable.
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| ==See also==
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| * [[Symmetric polynomial]]
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| * [[Euler class]]
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| ==Notes==
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| <references/>
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| ==References==
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| * A. Giambruno, Mikhail Zaicev, ''Polynomial Identities and Asymptotic Methods,'' AMS Bookstore, 2005 ISBN 978-0-8218-3829-7, pp. 352
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| * [http://perso.univ-rennes1.fr/matthieu.romagny/notes/FTAF.pdf The fundamental theorem of alternating functions], by Matthieu Romagny, September 15, 2005
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| [[Category:Polynomials]]
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| [[Category:Symmetric functions]]
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If you present photography effectively, it helps you look much more properly at the globe around you. Also, you may want to opt for a more professioanl theme if you are planning on showing your site off to a high volume of potential customers each day. Should you go with simple HTML or use a platform like Wordpress. Keep reading for some great Word - Press ideas you can start using today. The number of options offered here is overwhelming, but once I took the time to begin to review the video training, I was amazed at how easy it was to create a squeeze page and a membership site.
Luckily, for Word - Press users, WP Touch plugin transforms your site into an IPhone style theme. If you loved this report and you would like to acquire a lot more facts regarding wordpress backup plugin kindly pay a visit to our own web page. If you wish to sell your services or products via internet using your website, you have to put together on the website the facility for trouble-free payment transfer between customers and the company. You are able to set them within your theme options and so they aid the search engine to get a suitable title and description for the pages that get indexed by Google. This is identical to doing a research as in depth above, nevertheless you can see various statistical details like the number of downloads and when the template was not long ago updated. W3C compliant HTML and a good open source powered by Word - Press CMS site is regarded as the prime minister.
ve labored so hard to publish and put up on their website. The following piece of content is meant to make your choice easier and reassure you that the decision to go ahead with this conversion is requited with rich benefits:. Possibly the most downloaded Word - Press plugin, the Google XML Sitemaps plugin but not only automatically creates a site map linking to everyone your pages and posts, it also notifies Google, Bing, Yahoo, and Ask. Our skilled expertise, skillfulness and excellence have been well known all across the world. Websites using this content based strategy are always given top scores by Google.
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