Following the methods used by Derksen–Weyman in [] and Chindris in [], we use quiver theory to represent the generalized Littlewood–Richardson coefficients for the branching rule for the diagonal embedding of as the dimension of a weight space of semi-invariants. Using this, we prove their saturation and investigate when they are nonzero. We also show that for certain partitions the associated stretched polynomials satisfy the same conjectures as single Littlewood–Richardson coefficients. We then provide a polytopal description of this multiplicity and show that its positivity may be computed in strongly polynomial time. Finally, we remark that similar results hold for certain other generalized Littlewood–Richardson coefficients.
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Keywords: Quiver representations, semi-invariants, Littlewood–Richardson coefficients, Horn’s Conjecture, branching rule, hive model.
CC-BY 4.0
@article{ALCO_2020__3_6_1365_0,
author = {Collins, Brett},
title = {Generalized {Littlewood{\textendash}Richardson} coefficients for branching rules of {GL}$(n)$ and extremal weight crystals},
journal = {Algebraic Combinatorics},
pages = {1365--1400},
publisher = {MathOA foundation},
volume = {3},
number = {6},
year = {2020},
doi = {10.5802/alco.143},
language = {en},
url = {https://alco.centre-mersenne.org/articles/10.5802/alco.143/}
}
TY - JOUR AU - Collins, Brett TI - Generalized Littlewood–Richardson coefficients for branching rules of GL$(n)$ and extremal weight crystals JO - Algebraic Combinatorics PY - 2020 SP - 1365 EP - 1400 VL - 3 IS - 6 PB - MathOA foundation UR - https://alco.centre-mersenne.org/articles/10.5802/alco.143/ DO - 10.5802/alco.143 LA - en ID - ALCO_2020__3_6_1365_0 ER -
%0 Journal Article %A Collins, Brett %T Generalized Littlewood–Richardson coefficients for branching rules of GL$(n)$ and extremal weight crystals %J Algebraic Combinatorics %D 2020 %P 1365-1400 %V 3 %N 6 %I MathOA foundation %U https://alco.centre-mersenne.org/articles/10.5802/alco.143/ %R 10.5802/alco.143 %G en %F ALCO_2020__3_6_1365_0
Collins, Brett. Generalized Littlewood–Richardson coefficients for branching rules of GL$(n)$ and extremal weight crystals. Algebraic Combinatorics, Volume 3 (2020) no. 6, pp. 1365-1400. doi : 10.5802/alco.143. https://alco.centre-mersenne.org/articles/10.5802/alco.143/
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