Truncated Homogeneous Symmetric Functions

Extending the elementary and complete homogeneous symmetric functions, we introduce the truncated homogeneous symmetric function $h_λ^{\dd}$ in $(\ref{THSF})$ for any integer partition $λ$, and show that the transition matrix from $h_λ^{\dd}$ to the power sum symmetric functions $p_λ$ is given by \[M(h^{\dd},p)=M'(p,m)z^{-1}D^{\dd},\] where $D^{\dd}$ and $z$ are nonsingular diagonal matrices. Consequently, $\{h_λ^{\dd}\}$ forms a basis of the ring $Λ$ of symmetric functions. In addition, we show that the generating function $H^{\dd}(t)=\ssum_{n\ge 0}h_n^{\dd}(x)t^n$ satisfies \[ω(H^{\dd}(t))=\left(H^{\dd}(-t)\right)^{-1},\] where $ω$ is the involution of $Λ$ sending each elementary symmetric function $e_λ$ to the complete homogeneous symmetric function $h_λ$.

No reviews yet.

No comments yet.