Irreducible characters and bitrace for the $q$-rook monoid
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866910025864183808 |
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| author | Jing, Naihuan Wu, Yu Liu, Ning |
| author_facet | Jing, Naihuan Wu, Yu Liu, Ning |
| contents | This paper studies irreducible characters of the $q$-rook monoid algebra $R_n(q)$ using the vertex algebraic method. Based on the Frobenius formula for $R_n(q)$, a new iterative character formula is derived with the help of the vertex operator realization of the Schur symmetric function. The same idea also leads to a simple proof of the Murnaghan-Nakayama rule for $R_n(q)$. We also introduce the bitrace for the $q$-rook monoid and derive its combinatorial formula as a generalization of the bitrace formula for the Iwahori-Hecke algebra. The character table of $R_n(q)$ with $|μ|=5$ is listed in the appendix. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2312_13681 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Irreducible characters and bitrace for the $q$-rook monoid Jing, Naihuan Wu, Yu Liu, Ning Combinatorics Rings and Algebras Representation Theory Primary: 20C08, Secondary: 17B69, 05E10 This paper studies irreducible characters of the $q$-rook monoid algebra $R_n(q)$ using the vertex algebraic method. Based on the Frobenius formula for $R_n(q)$, a new iterative character formula is derived with the help of the vertex operator realization of the Schur symmetric function. The same idea also leads to a simple proof of the Murnaghan-Nakayama rule for $R_n(q)$. We also introduce the bitrace for the $q$-rook monoid and derive its combinatorial formula as a generalization of the bitrace formula for the Iwahori-Hecke algebra. The character table of $R_n(q)$ with $|μ|=5$ is listed in the appendix. |
| title | Irreducible characters and bitrace for the $q$-rook monoid |
| topic | Combinatorics Rings and Algebras Representation Theory Primary: 20C08, Secondary: 17B69, 05E10 |
| url | https://arxiv.org/abs/2312.13681 |