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Zapiski Nauchnykh Seminarov POMI, 2020, Volume 496, Pages 43–60
(Mi znsl7013)
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Linear immanant converters on skew-symmetric matrices of order $4$
A. E. Gutermanab, M. A. Duffnerc, I. A. Spiridonovbde a Lomonosov Moscow State University
b Moscow Center for Fundamental and Applied Mathematics
c Universidade de Lisboa
d National Research University "Higher School of Economics", Moscow
e Moscow Center for Continuous Mathematical Education
Abstract:
Let $Q_n$ denote the space of all $n\times n$ skew-symmetric matrices over the complex field $\mathbb{C}$. It is proved that for $n = 4$, there are no linear maps $ T :Q_4\to Q_4$ satisfying the condition $ d_{\chi'} ( T (A) ) =d_{\chi} (A) $ for all matrices $ A\in Q_4$, where $\chi, \chi' \in \{1, \epsilon, [2,2]\}$ are two distinct irreducible characters of $S_4$. In the case $\chi=\chi'=1$, a complete characterization of the linear maps $T :Q_4\to Q_4$ preserving the permanent is obtained. This case is the only one corresponding to equal characters and remaining uninvestigated so far.
Key words and phrases:
immanants, skew-symmetric matrices, linear maps.
Received: 12.10.2020
Citation:
A. E. Guterman, M. A. Duffner, I. A. Spiridonov, “Linear immanant converters on skew-symmetric matrices of order $4$”, Computational methods and algorithms. Part XXXIII, Zap. Nauchn. Sem. POMI, 496, POMI, St. Petersburg, 2020, 43–60
Linking options:
https://www.mathnet.ru/eng/znsl7013 https://www.mathnet.ru/eng/znsl/v496/p43
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Abstract page: | 117 | Full-text PDF : | 33 | References: | 18 |
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