Some consequences of the standard polynomial

Abstract:

The standard polynomial of degree $m$ is the polynomial $\sum {\{ {\text {sign(}}\rho {\text {)}}{x_{\rho (1)}}{x_{\rho (2)}} \cdots {x_{\rho (m)}}|\rho \in {S_m}} \}$, where ${S_m}$ is the symmetric group on $m$ letters. We show that the polynomial \[ \sum {\{ {\text {sign(}}\rho \sigma {\text {)}}{x_{\rho (1)}}{y_{\sigma (1)}}{x_{\rho (2)}}{y_{\sigma (2)}} \cdots {x_{\rho (m)}}{y_{\sigma (m)}}|\rho ,\sigma \in {S_m}\} } \] is a consequence of the standard polynomial of degree $m$. We also show that certain polynomials of the form $\sum \{ {\text {sign(}}\rho {\text {)}}{x_{\rho (1)}}{x_{\rho (2)}} \cdots {x_{\rho (n)}}|\rho \in Q\}$, where $n \geq m$ and $Q$ is a suitable subset of ${S_n}$, are consequences of the standard polynomial of degree $m$.