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A Type B Analogue to Ribbon Tableaux

Ezgi Kantarcı Oğuz

TL;DR

This work extends the theory of ribbon tableaux to shifted diagrams by introducing shifted $k$-ribbon tableaux and their $k$-quotients, providing a bijection between shifted $k$-ribbon fillings and fillings of a $\,\lfloor k/2\rfloor$-tuple quotient. It develops the $k$-abacus machinery and folded tableaux to model these quotients, establishing positive expansions of shifted ribbon $Q$-functions into Schur $Q$-functions and connecting to Peak and LLT-type structures. A $q$-analogue is defined to obtain Schur $Q$-positivity in the shifted setting, yielding a Type B analogue of LLT polynomials and suggesting broader representation-theoretic and combinatorial applications. The results illuminate the interplay between shifted tableaux, ribbon tilings, and symmetric function bases, with potential impact on spin characters and shifted-Laplace-type polynomials in algebraic combinatorics.

Abstract

We introduce a shifted analogue of the ribbon tableaux defined by James and Kerber. For any positive integer $k$, we give a bijection between the $k$-ribbon fillings of a shifted shape and regular fillings of a $\lfloor k/2\rfloor$-tuple of shapes called its $k$-quotient. We also define the corresponding generating functions, and prove that they are symmetric, Schur positive and Schur $Q$-positive.

A Type B Analogue to Ribbon Tableaux

TL;DR

This work extends the theory of ribbon tableaux to shifted diagrams by introducing shifted -ribbon tableaux and their -quotients, providing a bijection between shifted -ribbon fillings and fillings of a -tuple quotient. It develops the -abacus machinery and folded tableaux to model these quotients, establishing positive expansions of shifted ribbon -functions into Schur -functions and connecting to Peak and LLT-type structures. A -analogue is defined to obtain Schur -positivity in the shifted setting, yielding a Type B analogue of LLT polynomials and suggesting broader representation-theoretic and combinatorial applications. The results illuminate the interplay between shifted tableaux, ribbon tilings, and symmetric function bases, with potential impact on spin characters and shifted-Laplace-type polynomials in algebraic combinatorics.

Abstract

We introduce a shifted analogue of the ribbon tableaux defined by James and Kerber. For any positive integer , we give a bijection between the -ribbon fillings of a shifted shape and regular fillings of a -tuple of shapes called its -quotient. We also define the corresponding generating functions, and prove that they are symmetric, Schur positive and Schur -positive.

Paper Structure

This paper contains 13 sections, 26 theorems, 36 equations, 22 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Schur Functions
  4. Ribbon Tableaux
  5. Shifted Tableaux
  6. Shifted Ribbon Tableaux
  7. Ribbons on Shifted Diagrams
  8. The $k$-Abacus Correspondence
  9. Shifted Ribbon Tableaux
  10. Folded Tableaux
  11. Quotients of Ribbon Tableaux
  12. Peak Functions of Ribbon Tableaux
  13. Type B LLT polynomials

Key Result

Lemma 2.1

If $i \in Des(T)$, then $i\in Des(T')$ if and only if $i$ is unmarked in $T'$. If $i \notin Des(T)$, then $i \in Des(T')$ if and only if $i+1$ is marked in $T$.

Figures (22)

  • Figure 1: The diagram $\mu=(4,2,2,1)$ with corresponding semi-standard and standard fillings.
  • Figure 2: The standard tableaux of shape $(3,2)$ and their reading words
  • Figure 3: A 3-ribbon tableau of shape $\mu=(6,3,3,2)$
  • Figure 4: The shifted diagram for $\lambda=(4,3,1)$ with diagonals labelled with diagonal values.
  • Figure 5: Shifted tableaux examples of shape $\lambda=(4,3,1)$
  • ...and 17 more figures

Theorems & Definitions (63)

  • Lemma 2.1
  • proof
  • Theorem 2.2
  • Definition 3.1
  • Definition 3.2
  • Definition 3.3
  • Proposition 3.4
  • proof
  • Proposition 3.5
  • proof
  • ...and 53 more