Some consequences of the standard polynomial

Abstract

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