Steinberg 通過考慮一些置換表示找出了GL_2(q)和GL_3(q)的所有不可約表示。在本文中，我們將介紹另一種由Green 和 Lusztig提出的方法以計算有限域上一般線性群的特征標。我們將以此方法計算出GL_2(q),GL_3(q) 的特征標。這種方法引用了對稱群表示的特征標和Green 函數。

Steinberg derived the irreducible representations of GL_2(q) and GL_3(q) by considering permutation representations. In this paper, we will present an alternative method from Green and Lusztig for calculating the characters of general linear groups over finite fields. Specifically, we will calculate the characters of GL_2(q) and GL_3(q) using this approach, which involves referencing the characters of symmetric groups and Green functions.

1 Preliminary 1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 Character theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.4 Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.5 Induced representations . . . . . . . . . . . . . . . . . . . . . . . . 7
2 Steinberg’s approach to find characters of GL2(q),GL3(q) 9
2.1 Notations of something parameterized by partitions . . . . . . . . . 9
2.2 Conjugacy class of GL_n(q) . . . . . . . . . . . . . . . . . . . . . . . 10
2.3 GL_2(q) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.4 GL_3(q) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3 Another approach to find characters of GL_n(q) 16
3.1 Character on torus defiend by a semisimple element . . . . . . . . . 16
3.2 Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.3 GL_2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.3.1 Conjugacy classes of GL_2 . . . . . . . . . . . . . . . . . . . 20
3.3.2 Calculate E (GL_2, s′), where (s′ = A′_1,A′_3,B′_1 ) . . . . . . . . . 20
3.3.3 Calculate R_{T,s′} on each class . . . . . . . . . . . . . . . . . . 22
3.3.4 Character table of GL_2 . . . . . . . . . . . . . . . . . . . . . 25
3.4 GL_3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.4.1 Conjugacy classes of GL_3 . . . . . . . . . . . . . . . . . . . . 26
3.4.2 Calculate E (GL_3, s′), where (s′ = A_1,A_4,A_6,B_1,C_1) . . . . . 27
3.4.3 Calculate R_{T,s′} on each class . . . . . . . . . . . . . . . . . . 29
3.4.4 Character table of GL_3 . . . . . . . . . . . . . . . . . . . . . 39
Bibliography 45

[Car93] Roger W. Carter, Finite groups of Lie type: conjugacy classes and complex characters, Wiley Classics Library, John Wiley & Sons Inc, 1993.

[DM20] Fran¸cois Digne and Jean Michel, Representations of finite groups of Lie type, 2 ed., London Mathematical Society Student Texts, vol. 95, Cambridge University Press, 2020.

[FH91] William Fulton and Joe Harris, Representation theory: A first course, Springer Science & Business Media, 1991.

[Gre55] James A Green, The characters of the finite general linear groups, Transactions of the American Mathematical Society 80 (1955), no. 2, 402–447.

[Ser77] Jean-Pierre Serre, Linear representations of finite groups, Graduate Texts in Mathematics, vol. 42, Springer, 1977.

[SIB+70] R Steinberg, N Iwahori, A Borel, TA Springer, CW Curtis, and R Carter, Seminar on algebraic groups and related finite groups, Lecture Notes in Mathematics, vol. 131, Springer, 1970.

[Ste51] Robert Steinberg, The representations of GL(3, q), GL(4, q), PGL(3, q), and PGL(4, q), Canadian Journal of Mathematics 3 (1951), 225–235.