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研究生： 宋狄謙
Di-Chian Sung
論文名稱： GLn(C)的不可約表現建構方式之討論
On Some Constructions of Irreducible Representations of GLn(C)
指導教授： 蔡宛育
Wan-Yu Tsai
口試委員:
學位類別： 碩士
Master
系所名稱： 理學院 - 數學系
Department of Mathematics
論文出版年： 2025
畢業學年度： 113
語文別： 英文
論文頁數： 48
中文關鍵詞： 不可約表現 、Weyl模 、最高權重
外文關鍵詞： Irreducible representations, Weyl modules, Highest weight
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    • 在這篇文章中，我們討論兩種GLn(C)有限維的不可約表現之建構方式: Weyl 模和Highest weight定理。結果我們發現，Weyl模並不涵蓋所有GLn(C)的有限維不可約表現，比如說，對偶表現無法透過Weyl模來建構。因此，我們將透過其他的建構方式，來明確地刻劃出所有GLn(C)的有限維不可約表現。

    There are two constructions of irreducible finite-dimensional representations of GLn(C):
    Weyl modules and Highest weight theory. It turns out that Weyl modules don’t give us all irreducible finite-dimensional representations of GLn(C). For example, dual representations are not included in Weyl modules. In this article, we explicitly describe all irreducible finite-dimensional representations of GLn(C) that don’t arise from Weyl modules.

    摘要 i Abstract ii Contents iii 1 Introduction 1 2 Preliminaries 3 2.1 Lie Algebras 3 2.2 Representations of Groups And Lie Algebras 6 3 Weyl Modules 11 3.1 Introduction to Weyl modules 11 3.1.1 Representations of Symmetric Group Sd 11 3.2 The Decomposition of V ⊗d 13 4 Representations of SLn(C) And sln(C) 17 4.1 Highest Weight Theory of sln(C) 17 4.1.1 Representations of sl2(C) 17 4.1.2 Representations of sln(C) 20 4.2 Irreducible Representations of SLn(V) 22 4.2.1 Weyl Module for SLn(V) 22 5 Relation Between Irr(GLn(C)) And Irr(SLn(C)) 32 5.1 Some Constructions of Irr(GLn(C)) 32 5.1.1 First Construction: Φ(a1,a2,...,an) 33 5.1.2 Second Construction: Ψ(λ1,λ2,...,λn) 35 5.2 Characterization of Irreducible Representations of GLn(C) 36 References 42

    [1] I. Schur, Über eine Klasse von Matrizen, die sich einer gegebenen Matrix zuordnen lassen. 1901.
    [2] H. Weyl, The classical groups: their invariants and representations. Princeton university press, 1946.
    [3] W. Fulton and J. Harris, Representation theory: a first course. Springer Science & Business Media, 2013, vol. 129.
    [4] B. C. Hall and B. C. Hall, Lie groups, Lie algebras, and representations. Springer, 2013.
    [5] B. E. Sagan, The symmetric group: representations, combinatorial algorithms, and symmetric functions. Springer Science & Business Media, 2013, vol. 203.
    [6] J. E. Humphreys, Introduction to Lie algebras and representation theory. Springer Science & Business Media, 2012, vol. 9.
    [7] A. A. Kirillov, An introduction to Lie groups and Lie algebras. Cambridge University Press, 2008, vol. 113.

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