Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The recording tableaux of the quantum Littlewood-Richardson map and the orthogonal transpose symmetry map

Olga Azenhas

Abstract

Recently Watanabe has given an algorithm to compute a bijection, that he calls (quantum) Littlewood-Richardson (LR) map, between semi-standard Young tableaux of shape a partition with at most $2n$ parts and pairs of tableaux consisting of a symplectic tableau with shape a partition with at most $n$ parts, and a recording tableau of skew-shape given by the two previous shapes. The recording tableaux in that algorithm are shown to be equinumerous to Littlewood-Richardson-Sundaram tableaux whose injectivity is shown combinatorially while the surjectivity is concluded via representation theory of a quantum symmetric pair of type AII. Henceforth, the algorithm to compute the quantum LR map provides a new branching model for the branching multiplicities from $GL_{2n}(C)$ to $Sp_{2n}(C)$. Here, as morally suggested by Watanabe, one provides a combinatorial proof of the surjectivity of the quantum LR map which in turn exhibits the restriction of the LR orthogonal transpose symmetry map to LR-Sundaram tableaux.

The recording tableaux of the quantum Littlewood-Richardson map and the orthogonal transpose symmetry map

Abstract

Recently Watanabe has given an algorithm to compute a bijection, that he calls (quantum) Littlewood-Richardson (LR) map, between semi-standard Young tableaux of shape a partition with at most parts and pairs of tableaux consisting of a symplectic tableau with shape a partition with at most parts, and a recording tableau of skew-shape given by the two previous shapes. The recording tableaux in that algorithm are shown to be equinumerous to Littlewood-Richardson-Sundaram tableaux whose injectivity is shown combinatorially while the surjectivity is concluded via representation theory of a quantum symmetric pair of type AII. Henceforth, the algorithm to compute the quantum LR map provides a new branching model for the branching multiplicities from to . Here, as morally suggested by Watanabe, one provides a combinatorial proof of the surjectivity of the quantum LR map which in turn exhibits the restriction of the LR orthogonal transpose symmetry map to LR-Sundaram tableaux.
Paper Structure (4 sections, 2 theorems, 25 equations)

This paper contains 4 sections, 2 theorems, 25 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Schensted column insertion and reverse
  4. The linear time bijection between the sets $\widetilde{R}ec_{2n}(\lambda/\mu)$ and LR-Sundaram tableaux $LRS_{2n}(\lambda/\mu)$

Key Result

Proposition 1

Given $Q\in \widetilde{R}ec_{2n}(\lambda/\mu)$ and $k>0$, the following conditions hold

Theorems & Definitions (8)

  • Remark 1
  • Definition 1
  • Remark 2
  • Definition 2
  • Remark 3
  • Proposition 1
  • proof
  • Corollary 1