Motivated by the Saxl conjecture and the tensor square conjecture, which states that the tensor squares of certain irreducible representations of the symmetric group contain all irreducible representations, we study the tensor squares of irreducible representations associated with square Young diagrams. We give a formula for computing Kronecker coefficients, which are indexed by two square partitions and a three-row partition, specifically one with a short second row and the smallest part equal to 1. We also prove the positivity of square Kronecker coefficients for particular families of partitions, including three-row partitions and near-hooks.
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Keywords: Kronecker coefficients, symmetric group representations, Saxl conjecture
Zhao, Chenchen 1
CC-BY 4.0
@article{ALCO_2024__7_5_1575_0,
author = {Zhao, Chenchen},
title = {On the {Kronecker} product of {Schur} functions of square shapes},
journal = {Algebraic Combinatorics},
pages = {1575--1600},
publisher = {The Combinatorics Consortium},
volume = {7},
number = {5},
year = {2024},
doi = {10.5802/alco.381},
language = {en},
url = {https://alco.centre-mersenne.org/articles/10.5802/alco.381/}
}
TY - JOUR AU - Zhao, Chenchen TI - On the Kronecker product of Schur functions of square shapes JO - Algebraic Combinatorics PY - 2024 SP - 1575 EP - 1600 VL - 7 IS - 5 PB - The Combinatorics Consortium UR - https://alco.centre-mersenne.org/articles/10.5802/alco.381/ DO - 10.5802/alco.381 LA - en ID - ALCO_2024__7_5_1575_0 ER -
%0 Journal Article %A Zhao, Chenchen %T On the Kronecker product of Schur functions of square shapes %J Algebraic Combinatorics %D 2024 %P 1575-1600 %V 7 %N 5 %I The Combinatorics Consortium %U https://alco.centre-mersenne.org/articles/10.5802/alco.381/ %R 10.5802/alco.381 %G en %F ALCO_2024__7_5_1575_0
Zhao, Chenchen. On the Kronecker product of Schur functions of square shapes. Algebraic Combinatorics, Volume 7 (2024) no. 5, pp. 1575-1600. doi : 10.5802/alco.381. https://alco.centre-mersenne.org/articles/10.5802/alco.381/
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