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研究生： 李劭勉
Lee, Shau-Mien
論文名稱： 楊氏圖及其應用
A Survey On Young Tableau and Its Application
指導教授： 林正洪
Lam, Ching-Hung
學位類別： 碩士
Master
系所名稱： 理學院 - 數學系應用數學碩博士班
Department of Mathematics
論文出版年： 2009
畢業學年度： 97
語文別： 英文
論文頁數： 34
外文關鍵詞： Specht module, group representation, Young Tableau
相關次數： 點閱：82  下載：2 
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    • In this thesis, we give a survey on Young Tableau and its
    application. We first introduce some definitions and properties about representations of finite groups and the construction of Specht modules in the beginning of the thesis. Then we study the irreducible representations of symmetric group constructed by Young symmetrizer, which is a equivalent construction with Specht modules.

    1 Representations of finite groups......................3 1.1 Matrix representations................................3 1.2 Submodules and Maschke's theorem......................5 1.3 G-homomorphisms.......................................6 1.4 Structure of Hom(V,W).................................7 1.5 The number of irreducible submodules.................10 1.6 Characters of groups.................................11 1.7 The relation between characters and reducibility.....................................12 2 Representations of symmetric groups..................18 2.1 Young diagrams.......................................18 2.2 Definition of Young tabloids.........................19 2.3 The $S_n$-module grnerated by Young tabloids of a fixed shape.....................................20 2.4 Some elements in $mathbb{C}[S_n]$ and definition of Specht modules....................................21 2.5 $S^lambda$ is irreducible ..........................22 2.6 A basis for Specht module............................25 2.7 Representations constructed by Young ymmetrizers.....29 2.8 $V_lambda$ is irreducible...........................29

    [1] G.D. James, “The Representation Theory of the Symmetric Groups”, Lecture notes in Mathematics, Vol. 682, Springer-Verlag,New York, NY, 1978, 13∼14.
    [2] W. Fulton and J. Harris, “Representations Theory : A First Cours”, Graduate text in Mathematics, Springer-Verlag, 1991, 45∼46, 53∼54.
    [3] Bruce E. Sagan, “The Symmetric Group : Representations, Combinatorial algorithms, and Symmetric Functions”, Graduate text in mathematics, Springer Verlag, 2000, 1∼73.
    [4] Willard Miller, Jr. “Symmetric Groups And Their Applications”, Academic Press, 1972, 123∼125.
    [5] K.I. Beidar, “Advanced Linear Algebra”, Lecture Notes, 2002, 94.
    [6] T.Y. Lam, “Representations of Finite Groups : A Houndred Years, Part I”, Notice of the AMS, March 1998.
    [7] H. Boerner, “Representations of Groups with Special Consideration for the Needs of Merdern Physics”, North-Holland, New York, NY, 1970.
    [8] R. P, Stanley, “Some aspects of groups acting on finite posets”, J. Combin. Theory Ser. A 32(1982),132∼161.

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