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研究生: 周孟謙
Chou, Meng-Chien
論文名稱: On Schur expansions of Petrie Symmetric functions multiplied by a power symmetric function
On Schur expansions of Petrie Symmetric functions multiplied by a power symmetric function
指導教授: 游森棚
Eu, Sen-Peng
口試委員: 游森棚
Eu, Sen-Peng
郭君逸
Guo, Jun-Yi
丁建太
Ting, Chien-Tai
口試日期: 2022/06/08
學位類別: 碩士
Master
系所名稱: 數學系
Department of Mathematics
論文出版年: 2022
畢業學年度: 110
語文別: 中文
論文頁數: 27
英文關鍵詞: symmetric functions, Schur functions, Petrie symmetric function
DOI URL: http://doi.org/10.6345/NTNU202200720
論文種類: 學術論文
相關次數: 點閱:90下載:23
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  • 在 [2] 中,Darij Grinberg 證明了 Petrie symmetric function 是 signed multiplicity-free ,並使用代數的方法證明 Liu–Polo conjecture。本文章的主要結果是我們找到了組合的方法計算 k-Petrie number,因此我們可以知道 Petrie symmetric function 可以寫成那些 Schur functions 的線性組合。由這個組合解釋,Liu–Polo conjecture 有了簡單的證明;我們也用這個組合解釋證明並推廣了 Alexandersson conjecture。

    1 Introduction and Preliminary 1 1.1 Introduction 1 1.2 Notation 1 1.3 Symmetric function 4 1.4 Petrie symmetric function and k-Petrie number 5 2 Main Result 10 2.1 The combinatorial method of computing petk(λ) 10 2.2 The Proof of combinatorial method 10 2.3 Combinatorial proof of the Liu-Polo conjecture 16 3 Generalization of Alexandersson Conjecture 17 3.1 Proof of the Alexandersson conjecture 17 3.2 The case of k|n 21 4 Conclusion and Open problems 25 Bibliography 27

    [1] Gordon James and Adalbert Kerber. The representation theory of the symmetric group. Cambridge University Press, 1984.
    [2] Darij Grinberg, Petrie symmetric functions, arXiv:2004.11194.
    [3] R. P. Stanley, Enumerative Combinatorics (Volume 2), second edition (2011)
    [4] Stephen Doty, Grant Walker, Modular symmetric functions and irreducible modular representations of general linear groups, J. Pure Appl. Algebra 82 (1992), no. 1, pp. 1–26.
    [5] Grant Walker, Modular Schur functions, Trans. Amer. Math. Soc. 346 (1994), pp. 569–604.
    [6] Houshan Fu, Zhousheng Mei, Truncated Homogeneous Symmetric Functions, Linear and Multilinear Algebra, 70:3, 438-448, DOI: 10.1080/03081087.2020.1733460
    [7] Manfred Gordon, E. Martin Wilkinson, Determinants of Petrie matrices, Pacific Journal of Mathematics 51, no. 2, 1974.
    [8] Bergeron, François. Algebraic combinatorics and coinvariant spaces. CRC Press, 2009.

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