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A sign-reversing involution for the antipode of Schur functions

Younggwang Cho, Byung-Hak Hwang, Hojoon Lee

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.

A sign-reversing involution for the antipode of Schur functions

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.
Paper Structure (2 sections, 4 theorems, 18 equations, 2 figures)

This paper contains 2 sections, 4 theorems, 18 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Proof of Theorem

Key Result

Theorem 1

The antipode in $\mathsf{Sym}$ is given by where $\lambda^t/\mu^t$ denotes the conjugate skew shape.

Figures (2)

  • Figure 1: Examples of the involution $\Phi$
  • Figure 2: A row-strict plane partition and a semistandard Young tableau

Theorems & Definitions (8)

  • Theorem : Hall1959
  • Definition
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof