翻訳と辞書 Words near each other ・ Ring of polynomial functions ・ Ring of quotients ・ Ring of Red ・ Ring of Saturn ・ Ring of Scorpio ・ Ring of sets ・ Ring of Silvianus ・ Ring of Spies ・ Ring of steel ・ Ring of Steel (film) ・ Ring of Steel (Kabul) ・ Ring of steel (London) ・ Ring of Stone ・ Ring of Stones ・ Ring of Sweden ・ Ring of symmetric functions ・ Ring of Terror ・ Ring of the Fisherman ・ Ring of the Nibelungs ・ Ring of Time ・ Ring of Truth (band) ・ Ring Off (song) ・ Ring oiler ・ Ring on Her Finger, Time on Her Hands ・ Ring oscillator ・ Ring Out Bow Bells! ・ Ring Out Joy ・ Ring Out Your Dead ・ Ring Out, Wild Bells ・ Ring ouzel Dictionary Lists mini英和辞書 mini和英辞書 Webster 1913 Latin-English FOLDOC Wikipedia English ウィキペディア

翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Ring of symmetric functions ： ウィキペディア英語版 Ring of symmetric functions In algebra and in particular in algebraic combinatorics, the ring of symmetric functions is a specific limit of the rings of symmetric polynomials in ''n'' indeterminates, as ''n'' goes to infinity. This ring serves as universal structure in which relations between symmetric polynomials can be expressed in a way independent of the number ''n'' of indeterminates (but its elements are neither polynomials nor functions). Among other things, this ring plays an important role in the representation theory of the symmetric group. The ring of symmetric functions can be given a coproduct and a bilinear form making it into a positive selfadjoint graded Hopf algebra that is both commutative and cocommutative. == Symmetric polynomials == The study of symmetric functions is based on that of symmetric polynomials. In a polynomial ring in some finite set of indeterminates, a polynomial is called ''symmetric'' if it stays the same whenever the indeterminates are permuted in any way. More formally, there is an action by ring automorphisms of the symmetric group ''Sn'' on the polynomial ring in ''n'' indeterminates, where a permutation acts on a polynomial by simultaneously substituting each of the indeterminates for another according to the permutation used. The invariants for this action form the subring of symmetric polynomials. If the indeterminates are ''X''1,…,''X''''n'', then examples of such symmetric polynomials are : $X\_1+X\_2+\backslash cdots+X\_n,\; \backslash ,$ : $X\_1^3+X\_2^3+\backslash cdots+X\_n^3,\; \backslash ,$ and :$X\_1X\_2\backslash cdots\; X\_n.\; \backslash ,$ A somewhat more complicated example is ''X''13''X''2''X''3 +''X''1''X''23''X''3 +''X''1''X''2''X''33 +''X''13''X''2''X''4 +''X''1''X''23''X''4 +''X''1''X''2''X''43 +… where the summation goes on to include all products of the third power of some variable and two other variables. There are many specific kinds of symmetric polynomials, such as elementary symmetric polynomials, power sum symmetric polynomials, monomial symmetric polynomials, complete homogeneous symmetric polynomials, and Schur polynomials. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Ring of symmetric functions」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.