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Promotion and growth diagrams for fans of Dyck paths and vacillating tableaux

Joseph Pappe, Stephan Pfannerer, Anne Schilling, Mary Claire Simone

Abstract

We construct an injection from the set of $r$-fans of Dyck paths (resp. vacillating tableaux) of length $n$ into the set of chord diagrams on $[n]$ that intertwines promotion and rotation. This is done in two different ways, namely as fillings of promotion-evacuation diagrams and in terms of Fomin growth diagrams. Our analysis uses the fact that $r$-fans of Dyck paths and vacillating tableaux can be viewed as highest weight elements of weight zero in crystals of type $B_r$ and $C_r$, respectively, which in turn can be analyzed using virtual crystals. On the level of Fomin growth diagrams, the virtualization process corresponds to the Roby-Krattenthaler blow up construction. One of the motivations for finding rotation invariant diagrammatic bases such as chord diagrams is the cyclic sieving phenomenon. Indeed, we give a cyclic sieving phenomenon on $r$-fans of Dyck paths and vacillating tableaux using the promotion action.

Promotion and growth diagrams for fans of Dyck paths and vacillating tableaux

Abstract

We construct an injection from the set of -fans of Dyck paths (resp. vacillating tableaux) of length into the set of chord diagrams on that intertwines promotion and rotation. This is done in two different ways, namely as fillings of promotion-evacuation diagrams and in terms of Fomin growth diagrams. Our analysis uses the fact that -fans of Dyck paths and vacillating tableaux can be viewed as highest weight elements of weight zero in crystals of type and , respectively, which in turn can be analyzed using virtual crystals. On the level of Fomin growth diagrams, the virtualization process corresponds to the Roby-Krattenthaler blow up construction. One of the motivations for finding rotation invariant diagrammatic bases such as chord diagrams is the cyclic sieving phenomenon. Indeed, we give a cyclic sieving phenomenon on -fans of Dyck paths and vacillating tableaux using the promotion action.
Paper Structure (33 sections, 25 theorems, 85 equations, 13 figures)

This paper contains 33 sections, 25 theorems, 85 equations, 13 figures.

Table of Contents

  1. Introduction
  2. Crystal bases
  3. Background on crystals
  4. Virtual crystals
  5. Virtual crystal $B_r \hookrightarrow C_r$ spin to vector
  6. Virtual crystal $B_r \hookrightarrow C_r$ vector to vector
  7. Highest weights of weight zero
  8. Oscillating tableaux
  9. $r$-fans of Dyck paths
  10. Vacillating tableaux
  11. Promotion via crystal commutor
  12. Promotion via local rules
  13. Chord diagrams
  14. Promotion matrices
  15. Chord diagrams for oscillating tableaux
  16. ...and 18 more sections

Key Result

Lemma 2.5

$\mathcal{V}\sqcup \{\emptyset\}$ is closed under the operators $f_{i}$ and $e_{i}$ for $1\leqslant i \leqslant r$.

Figures (13)

  • Figure 1: Overview of strategy and results for $r$-fans of Dyck paths
  • Figure 2: Overview of strategy and results for vacillating tableaux
  • Figure 3: Left: One component of the crystal $\widehat{\mathcal{V}} = \mathcal{C}_\square^{\otimes 3}$ of type $C_3$. Middle: The virtual crystal $\mathcal{V}$ inside $\widehat{\mathcal{V}}$ of type $B_3$. Right: The spin crystal $\mathcal{B}_{\mathsf{spin}}$ of type $B_3$.
  • Figure 4: Left: The crystal $\mathcal{C}_\square$ of type $C_2$. Right: The crystal $\mathcal{B}_\square$ of type $B_2$.
  • Figure 5: Far Left: One connected component $\widehat{\mathcal{S}}$ of the crystal $\widehat{\mathcal{V}}^{\otimes 2} = (\mathcal{C}_\square^{\otimes 2})^{\otimes 2}$ of type $C_2$. Middle Left: The connected component $\mathcal{S}$ of the virtual crystal $\mathcal{V}^{\otimes 2}$ inside $\mathcal{S}$ induced by Definition \ref{['definition.V vector']}. Middle Right: The corresponding connected component $\mathcal{T}$ of the crystal $\mathcal{B}_\square^{\otimes 2}$ of type $B_2$ that corresponds to $\mathcal{S}$ under the embedding given in Definition \ref{['definition.Psi vector']}. Far Right: The connected component $\mathcal{U}$ of $(\mathcal{B}_{\mathsf{spin}}\otimes\mathcal{B}_{\mathsf{spin}})^{\otimes 2}$ of type $B_2$ corresponding to $\mathcal{T}$ under the isomorphism given in Figure \ref{['figure.B vector']}.
  • ...and 8 more figures

Theorems & Definitions (90)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Lemma 2.5
  • proof
  • Lemma 2.6
  • proof
  • Definition 2.7
  • Lemma 2.8
  • ...and 80 more