A sign-reversing involution for the antipode of Schur functions

TL;DR

This paper constructs a sign-reversing involution on Takeuchi's expansion to explicitly compute the antipode of Schur functions within the ring of symmetric functions, addressing a previously unresolved question.

Contribution

It introduces a novel sign-reversing involution that provides a combinatorial formula for the antipode of Schur functions.

Findings

Explicit involution for antipode computation

Resolution of Benedetti and Sagan's question

Enhanced understanding of symmetric function antipodes

Abstract

We resolve a question posed by Benedetti and Sagan by constructing a signreversing involution on Takeuchi's expansion that yields the antipode for the ring of symmetric functions in terms of the Schur basis.