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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 Schur algebra ： ウィキペディア英語版 Schur algebra In mathematics, Schur algebras, named after Issai Schur, are certain finite-dimensional algebras closely associated with Schur–Weyl duality between general linear and symmetric groups. They are used to relate the representation theories of those two groups. Their use was promoted by the influential monograph of J. A. Green first published in 1980.〔J. A. Green, ''Polynomial Representations of GLn'', Springer Lecture Notes 830, Springer-Verlag 1980. , ISBN 978-3-540-46944-5, ISBN 3-540-46944-3〕 The name "Schur algebra" is due to Green. In the modular case (over infinite fields of positive characteristic) Schur algebras were used by Gordon James and Karin Erdmann to show that the (still open) problems of computing decomposition numbers for general linear groups and symmetric groups are actually equivalent.〔Karin Erdmann, Decomposition numbers for symmetric groups and composition factors of Weyl modules. ''Journal of Algebra'' 180 (1996), 316–320. 〕 Schur algebras were used by Friedlander and Suslin to prove finite generation of cohomology of finite group schemes.〔Eric Friedlander and Andrei Suslin, Cohomology of finite group schemes over a field. ''Inventiones Mathematicae'' 127 (1997), 209--270. 〕 == Construction == The Schur algebra $S\_k(n,\; r)$ can be defined for any commutative ring $k$ and integers $n,\; r\; \backslash geq\; 0$. Consider the algebra $k()$ of polynomials (with coefficients in $k$) in $n^2$ commuting variables $x\_$, 1 ≤ ''i'', ''j'' ≤ $n$. Denote by $A\_k(n,\; r)$ the homogeneous polynomials of degree $r$. Elements of $A\_k(n,\; r)$ are ''k''-linear combinations of monomials formed by multiplying together $r$ of the generators $x\_$ (allowing repetition). Thus : $k()\; =\; \backslash bigoplus\_\; A\_k(n,\; r).$ Now, $k()$ has a natural coalgebra structure with comultiplication $\backslash Delta$ and counit $\backslash varepsilon$ the algebra homomorphisms given on generators by : $\backslash Delta(x\_)\; =\; \backslash textstyle\backslash sum\_l\; x\_\; \backslash otimes\; x\_,\; \backslash quad\; \backslash varepsilon(x\_)\; =\; \backslash delta\_\backslash quad\backslash$    (Kronecker's delta). Since comultiplication is an algebra homomorphism, $k()$ is a bialgebra. One easily checks that $A\_k(n,\; r)$ is a subcoalgebra of the bialgebra $k()$, for every ''r'' ≥ 0. Definition. The Schur algebra (in degree $r$) is the algebra $S\_k\; (n,\; r)\; =\; \backslash mathrm\_k(\; A\_k\; (n,\; r),\; k)$. That is, $S\_k(n,r)$ is the linear dual of $A\_k(n,r)$. It is a general fact that the linear dual of a coalgebra $A$ is an algebra in a natural way, where the multiplication in the algebra is induced by dualizing the comultiplication in the coalgebra. To see this, let : $\backslash Delta(a)\; =\; \backslash textstyle\; \backslash sum\; a\_i\; \backslash otimes\; b\_i$ and, given linear functionals $f$, $g$ on $A$, define their product to be the linear functional given by : $\backslash textstyle\; a\; \backslash mapsto\; \backslash sum\; f(a\_i)\; g(b\_i).$ The identity element for this multiplication of functionals is the counit in $A$. 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「Schur algebra」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.