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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Place-permutation action ： ウィキペディア英語版 Place-permutation action In mathematics, there are two natural interpretations of the place-permutation action of symmetric groups, in which the group elements act on positions or ''places''. Each may be regarded as either a left or a right action, depending on the order in which one chooses to compose permutations. There are just two interpretations of the meaning of "acting by a permutation $\backslash sigma$" but these lead to four variations, depending whether maps are written on the left or right of their arguments. The presence of so many variations often leads to confusion. When regarding the group algebra of a symmetric group as a ''diagram algebra''〔For a readable overview of various diagram algebras generalizing group algebras of symmetric groups, see Halverson and Ram 2005.〕 it is natural to write maps on the right so as to compute compositions of diagrams from left to right. == Maps written on the left == First we assume that maps are written on the left of their arguments, so that compositions take place from right to left. Let $\backslash mathfrak\_n$ be the symmetric group〔See James 1978 for the representation theory of symmetric groups. Weyl 1939, Chapter IV treats the important topic now known as Schur–Weyl duality, which is an important application of the place-permutation action.〕 on $n$ letters, with compositions computed from right to left. Imagine a situation in which elements of $\backslash mathfrak\_n$ act〔Hungerford 1974, Chapter II, Section 4〕 on the “places” (i.e., positions) of something. The places could be vertices of a regular polygon of $n$ sides, the tensor positions of a simple tensor, or even the inputs of a polynomial of $n$ variables. So we have $n$ places, numbered in order from 1 to $n$, occupied by $n$ objects that we can number $x\_1,\; \backslash dots,\; x\_n$. In short, we can regard our items as a ''word'' $x\; =\; x\_1\; \backslash cdots\; x\_n$ of length $n$ in which the position of each element is significant. Now what does it mean to act by “place-permutation” on $x$? There are two possible answers: # an element $\backslash sigma\; \backslash in\; \backslash mathfrak\_n$ can move the item in the $j$th place to the $\backslash sigma(j)$th place, or # it can do the opposite, moving an item from the $\backslash sigma(j)$th place to the $j$th place. Each of these interpretations of the meaning of an “action” by $\backslash sigma$ (on the places) is equally natural, and both are widely used by mathematicians. Thus, when encountering an instance of a "place-permutation" action one must take care to determine from the context which interpretation is intended, if the author does not give specific formulas. Consider the first interpretation. The following descriptions are all equivalent ways to describe the rule for the first interpretation of the action: * For each $j$, move the item in the $j$th place to the $\backslash sigma(j)$th place. * For each $j$, move the item in the $\backslash sigma^(j)$th place to the $j$th place. * For each $j$, replace the item in the $j$th position by the one that was in the $\backslash sigma^(j)$th place. This action may be written as the rule $x\_1\; \backslash cdots\; x\_n\; \backslash overset\; x\_\; \backslash cdots\; x\_$. Now if we act on this by another permutation $\backslash tau$ then we need to first relabel the items by writing $y\_1\; \backslash cdots$ y_n = x_\cdots x_. Then $\backslash tau$ takes this to $y\_\; \backslash cdots\; y\_\; =\; x\_(1)\}\; \backslash cdots\; x\_(n)\}\; =$ x_ \cdots x_. This proves that the action is a left action: $\backslash tau\; \backslash cdot\; (\backslash sigma$ \cdot x) = (\tau \sigma) \cdot x. Now we consider the second interpretation of the action of $\backslash sigma$, which is the opposite of the first. The following descriptions of the second interpretation are all equivalent: * For each $j$, move the item in the $j$th place to the $\backslash sigma^(j)$th place. * For each $j$, move the item in the $\backslash sigma(j)$th place to the $j$th place. * For each $j$, replace the item in the $j$th position by the one that was in the $\backslash sigma(j)$th place. This action may be written as the rule $x\_1\; \backslash cdots\; x\_n\; \backslash overset\; x\_\; \backslash cdots$ x_. In order to act on this by another permutation $\backslash tau$, again we first relabel the items by writing $y\_1\; \backslash cdots$ y_n = x_ \cdots x_. Then the action of $\backslash tau$ takes this to $y\_\; \backslash cdots\; y\_\; =\; x\_\; \backslash cdots\; x\_\; =\; x\_\; \backslash cdots\; x\_.$ This proves that our second interpretation of the action is a right action: $(x\; \backslash cdot\; \backslash sigma)\; \backslash cdot\; \backslash tau\; =\; x\; \backslash cdot\; (\backslash sigma\; \backslash tau)$. 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「Place-permutation action」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.