On character values of $GL_n(\mathbb F_q)$
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| Format: | Preprint |
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2024
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| _version_ | 1866912829589684224 |
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| author | Jing, Naihuan Wu, Yu |
| author_facet | Jing, Naihuan Wu, Yu |
| contents | In this paper, we use vertex operator techniques to compute character values on unipotent classes of $\GL_n(\mathbb F_q)$. By realizing the Grothendieck ring $R_G=\bigoplus_{n\geq0}^\infty R(\GL_n(\mathbb F_q))$ as Fock spaces, we formulate the Murnanghan-Nakayama rule of $\GL_n(\mathbb F_q)$ between Schur functions colored by an orbit $ϕ$ of linear characters of $\overline{\mathbb F}_q$ under the Frobenius automorphism on and modified Hall-Littlewood functions colored by $f_1=t-1$, which provides detailed information on the character table of $\GL_n(\mathbb F_q)$. As applications, we use vertex algebraic methods to determine the Steinberg characters of $\GL_n(\mathbb F_q)$, which were previously determined by Curtis-Lehrer-Tits via geometry of homology groups of spherical buildings and Springer-Zelevinsky utilizing Hopf algebras. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2412_18793 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On character values of $GL_n(\mathbb F_q)$ Jing, Naihuan Wu, Yu Representation Theory Group Theory Quantum Algebra Primary: 20C33, 17B69, Secondary: 05E10 In this paper, we use vertex operator techniques to compute character values on unipotent classes of $\GL_n(\mathbb F_q)$. By realizing the Grothendieck ring $R_G=\bigoplus_{n\geq0}^\infty R(\GL_n(\mathbb F_q))$ as Fock spaces, we formulate the Murnanghan-Nakayama rule of $\GL_n(\mathbb F_q)$ between Schur functions colored by an orbit $ϕ$ of linear characters of $\overline{\mathbb F}_q$ under the Frobenius automorphism on and modified Hall-Littlewood functions colored by $f_1=t-1$, which provides detailed information on the character table of $\GL_n(\mathbb F_q)$. As applications, we use vertex algebraic methods to determine the Steinberg characters of $\GL_n(\mathbb F_q)$, which were previously determined by Curtis-Lehrer-Tits via geometry of homology groups of spherical buildings and Springer-Zelevinsky utilizing Hopf algebras. |
| title | On character values of $GL_n(\mathbb F_q)$ |
| topic | Representation Theory Group Theory Quantum Algebra Primary: 20C33, 17B69, Secondary: 05E10 |
| url | https://arxiv.org/abs/2412.18793 |