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    Please use this identifier to cite or link to this item: http://ir.lib.ncu.edu.tw/handle/987654321/7808

    Title: Group Representations on GL(2,F_q) Group Representations on GL(2,F_q)
    Authors: 黃秀戀;Chiu-Lien Huang
    Contributors: 數學研究所
    Keywords: 群表現;Group Representation;GL(2;F_q)
    Date: 2002-06-26
    Issue Date: 2009-09-22 11:05:58 (UTC+8)
    Publisher: 國立中央大學圖書館
    Abstract: 這篇論文是整理GL(2,F_q)的不可分解的群表現有哪些。為了做這些事情,我們必須要先知道一些關於有限群的表現理論之基本知識。所以我們敘述了一些關於關於有限群的表現理論在第二章,關於這部分的定理,我們都不給予證明,但你可以參考 Serre’s book Complex Representations of Finite Groups。 接著,我們就開始去找GL(2,F_q)的不可分解的群表現,我們試著用一些比較自然的方法去找。我們用一維的射影空間來做,因為我們知道GL(2,F_q)作用在P(F_q)上有雙重可遞移性(doubly transitive)。更進一步的計算,我們知道排列的群表現對應到P(F_q)可以分解成一個一維跟一個q 維的不可分解的群表現。 接著,我們想要用 Frobenius method of induced representation 來造GL(2,F_q)的群表現, 所以我們開始去找有哪些重要 的子群,然後在知道這些子群的群表現,在這個過程 中我們會用到一個重要的方法叫做Method of Little Group。 最後,我們把不是這些方式做出來的群表現叫做 Cuspidal representation。在這裡, 我們需要知道有關於cuspidal representation 的性質。 This paper is a collation of all irreducible representations of GL(2, F_q). In order to do this, we need the basic knowledge about finite group representations. We arrange the basic understanding about finite group representations in §2. We state the basic results without proofs from Serre’s book on complex representations of finite groups [2]. For the proofs of all these results in §2, we refer to Serre’s book [2]. In §3, we start to find irreducible representations of GL(2, F_q). We use the projective line P(F_q) throught out the work. We can find q − 1 one-dimensional and q − 1 q-dimensional irreducible representations of GL(2, F_q). The part we refer to the paper[5] and Fulton’s book [7]. In §4, we use Frobenius method of induced representation which enables one to construct a representation of a group if a an irreducible representation of a subgroup is known. We use characters of Borel subgroup of GL(2, F_q) induces representations of GL(2, F_q). In §5, we bring in Cuspidal representation of GL(2, F_q). We can construct other irreducible representations of GL(2, F_q) by using cuspidal representation.
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