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Some consequences of the standard polynomial


Author: Qing Chang
Journal: Proc. Amer. Math. Soc. 104 (1988), 707-710
MSC: Primary 16A38
DOI: https://doi.org/10.1090/S0002-9939-1988-0964846-8
MathSciNet review: 964846
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Abstract: The standard polynomial of degree $ m$ is the polynomial $ \sum {\{ {\text{sign(}}\rho {\text{)}}{x_{\rho (1)}}{x_{\rho (2)}} \cdots {x_{\rho (m)}}\vert\rho \in {S_m}} \} $, where $ {S_m}$ is the symmetric group on $ m$ letters. We show that the polynomial

$\displaystyle \sum {\{ {\text{sign(}}\rho \sigma {\text{)}}{x_{\rho (1)}}{y_{\s... ...gma (2)}} \cdots {x_{\rho (m)}}{y_{\sigma (m)}}\vert\rho ,\sigma \in {S_m}\} } $

is a consequence of the standard polynomial of degree $ m$. We also show that certain polynomials of the form $ \sum \{ {\text{sign(}}\rho {\text{)}}{x_{\rho (1)}}{x_{\rho (2)}} \cdots {x_{\rho (n)}}\vert\rho \in Q\} $, where $ n \geq m$ and $ Q$ is a suitable subset of $ {S_n}$, are consequences of the standard polynomial of degree $ m$.

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1988-0964846-8
Keywords: Polynomial identity, standard polynomial, Capelli polynomial
Article copyright: © Copyright 1988 American Mathematical Society

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