Character factorizations for representations of GL(n,C)
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2022
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| _version_ | 1866910103535353856 |
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| author | Karmakar, Chayan |
| author_facet | Karmakar, Chayan |
| contents | We give another proof of a theorem of D. Prasad (Theorem 2, \textit{Israel J. Math.} 2016), which is also a classical result of Littlewood--Richardson (Theorem VI, \textit{Q. J. Math.} 1934). For integers $m,n \ge 2$, this result calculates the character of an irreducible representation of $\GL(mn,\C)$ at diagonal elements with eigenvalues $ω^{j-1}_nt_i$ for $1 \le i \le m$, $1 \le j \le n$, where $ω_n=e^{2π\imath/n}$, expressing it as a product of certain characters for $\GL(m,\C)$ evaluated at $\underline{t}^n={\rm diag}(t_1^{n},t_{2}^{n},\dots,t_{m}^{n})$. Unlike previous approaches that rely on determinantal identities, our proof utilizes a direct combinatorial cancellation argument within the Weyl group. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2210_03544 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Character factorizations for representations of GL(n,C) Karmakar, Chayan Representation Theory Group Theory We give another proof of a theorem of D. Prasad (Theorem 2, \textit{Israel J. Math.} 2016), which is also a classical result of Littlewood--Richardson (Theorem VI, \textit{Q. J. Math.} 1934). For integers $m,n \ge 2$, this result calculates the character of an irreducible representation of $\GL(mn,\C)$ at diagonal elements with eigenvalues $ω^{j-1}_nt_i$ for $1 \le i \le m$, $1 \le j \le n$, where $ω_n=e^{2π\imath/n}$, expressing it as a product of certain characters for $\GL(m,\C)$ evaluated at $\underline{t}^n={\rm diag}(t_1^{n},t_{2}^{n},\dots,t_{m}^{n})$. Unlike previous approaches that rely on determinantal identities, our proof utilizes a direct combinatorial cancellation argument within the Weyl group. |
| title | Character factorizations for representations of GL(n,C) |
| topic | Representation Theory Group Theory |
| url | https://arxiv.org/abs/2210.03544 |