臺灣博碩士論文加值系統

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.