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This research introduces truncated homogeneous symmetric functions and explores their properties within the framework of symmetric function theory.

February 26, 2020

Truncated homogeneous symmetric functions

Key Points

This research introduces truncated homogeneous symmetric functions and explores their properties within the framework of symmetric function theory.
Defined truncated homogeneous symmetric functions hλ[d] for integer partitions λ.
Established the transition matrix M(h[d],p)=M′(p,m)z−1D[d] from hλ[d] to power sum symmetric functions pλ.
Showed that the generating function H[d](t) satisfies a specific involution property within the ring of symmetric functions.
Confirmed that {hλ[d]} forms a basis for the ring Λ of symmetric functions.
Demonstrated that the generating function H[d](t) behaves under the involution ω, satisfying ω(H[d](t))=H[d](−t)−1.
Showed that D[d] and z are nonsingular diagonal matrices, emphasizing their significance in the transition matrix.

Abstract

Extending the elementary and complete homogeneous symmetric functions, we introduce the truncated homogeneous symmetric function hλd in (1) for any integer partition λ, and show that the transition matrix from hλd to the power sum symmetric functions pλ is given by M (hd, p) =M′ (p, m) z−1Dd, where Dd and z are nonsingular diagonal matrices. Consequently, hλd forms a basis of the ring Λ of symmetric functions. In addition, we show that the generating function Hd (t) =∑n≥0hnd (x) tn satisfies ω (Hd (t) ) =Hd (−t) −1, where ω is the involution of Λ sending each elementary symmetric function eλ to the complete homogeneous symmetric function hλ.

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