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Towards equivalent thickened ribbon Schur functions

Emma Yu Jin, Shu Xiao Li

Abstract

Two skew diagrams are defined to be equivalent if their corresponding skew Schur functions are equal. The equivalence classes for ribbons have been classified by Billera, Thomas and van Willigenburg in 2006. In this paper, we provide a complete characterization of equivalence classes for connected skew diagrams with exactly one $2\times m$ or $m\times 2$ block of boxes for all $m\ge 2$. In particular, possible sizes of equivalence classes are one, two or four, confirming special cases of the elusive conjecture on equivalent skew connected diagrams proposed by McNamara and van Willigenburg in 2009.

Towards equivalent thickened ribbon Schur functions

Abstract

Two skew diagrams are defined to be equivalent if their corresponding skew Schur functions are equal. The equivalence classes for ribbons have been classified by Billera, Thomas and van Willigenburg in 2006. In this paper, we provide a complete characterization of equivalence classes for connected skew diagrams with exactly one or block of boxes for all . In particular, possible sizes of equivalence classes are one, two or four, confirming special cases of the elusive conjecture on equivalent skew connected diagrams proposed by McNamara and van Willigenburg in 2009.
Paper Structure (13 sections, 29 theorems, 29 equations, 2 figures)

This paper contains 13 sections, 29 theorems, 29 equations, 2 figures.

Table of Contents

  1. Introduction and main results
  2. Background and preliminaries
  3. A composition of thickened ribbons
  4. A classification of positive integers
  5. When two ribbons are of the same size
  6. A classification of equivalent classes
  7. Some auxiliary lemmas
  8. Proof of Theorem \ref{['T:1']} for the case $|\alpha|=|\beta|$
  9. When two ribbons are of different sizes
  10. A classification of equivalent classes
  11. Some auxiliary lemmas
  12. Proof of Theorem \ref{['T:1']} for the case $|\alpha|\ne |\beta|$
  13. Acknowledgement

Key Result

Theorem 2

Two thickened ribbons $\CMcal{D}$ and $\CMcal{E}$ with exactly one $2\times m$ or $m\times 2$ block for a fixed $m\ge 2$ satisfy $\CMcal{D}\sim \CMcal{E}$ if and only if $\CMcal{D}=\CMcal{E}_1\circ_{\CMcal{W}}\CMcal{E}_2$ and $\CMcal{E}=\CMcal{E}'_1\circ_{\CMcal{W}}\CMcal{E}'_2$ such that The equivalence class of $\CMcal{D}$ will contain $2^{\kappa}$ elements, where $\kappa$ counts the occurrence

Figures (2)

  • Figure 2.1: A ribbon $31231$ and a thickened ribbon $3\boxdot 3\boxdot 3 \boxdot 31$
  • Figure 3.1: A composition of two ribbons: $21\circ_{1}312=31412\boxdot 312$ where $w$ and $o$ represent a box of $\CMcal{W}$ and $\CMcal{O}$ respectively.

Theorems & Definitions (77)

  • Definition 1: BTvWMvWRSvW
  • Conjecture 1
  • Theorem 2
  • Definition 2
  • Definition 3
  • Proposition 3
  • proof
  • Example 1
  • Example 2
  • Corollary 4
  • ...and 67 more