Let $\mathbf{\mathcal{Q}}$ be a bipartite quiver with vertex set $\mathbf{\mathcal{Q}}_0$ such that the number of arrows between any source vertex and any sink vertex is constant. Let $\beta =(\beta (x))_{x \in \mathbf{\mathcal{Q}}_0}$ be a dimension vector of $\mathbf{\mathcal{Q}}$ with positive integer coordinates.
Let $\mathrm{rep}(\mathbf{\mathcal{Q}}, \beta )$ be the representation space of $\beta $-dimensional representations of $\mathbf{\mathcal{Q}}$ and $\mathrm{GL}(\beta )$ the base change group acting on $\mathrm{rep}(Q, \beta )$ be simultaneous conjugation. Let $K^{\beta }_{\underline{\lambda }}$ be the multiplicity of the irreducible representation of $\mathrm{GL}(\beta )$ of highest weight $\underline{\lambda }$ in the ring of polynomial functions on $\mathrm{rep}(\mathbf{\mathcal{Q}}, \beta )$.
We show that $K^{\beta }_{\underline{\lambda }}$ can be expressed as the number of lattice points of a polytope obtained by gluing together two Knutson-Tao hive polytopes. Furthermore, this polytopal description together with Derksen-Weyman’s Saturation Theorem for quiver semi-invariants allows us to use Tardos’ algorithm to solve the membership problem for the moment cone associated to $(\mathbf{\mathcal{Q}},\beta )$ in strongly polynomial time.
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Keywords: Hive polytopes, Littlewood-Richardson coefficients, moment cones, strongly polynomial time, quiver exceptional sequences, quiver semi-invariants
Chindris, Calin 1; Collins, Brett 2; Kline, Daniel 3
CC-BY 4.0
@article{ALCO_2025__8_1_175_0,
author = {Chindris, Calin and Collins, Brett and Kline, Daniel},
title = {Hive-type polytopes for quiver multiplicities and the membership problem for quiver moment cones},
journal = {Algebraic Combinatorics},
pages = {175--199},
publisher = {The Combinatorics Consortium},
volume = {8},
number = {1},
year = {2025},
doi = {10.5802/alco.398},
language = {en},
url = {https://alco.centre-mersenne.org/articles/10.5802/alco.398/}
}
TY - JOUR AU - Chindris, Calin AU - Collins, Brett AU - Kline, Daniel TI - Hive-type polytopes for quiver multiplicities and the membership problem for quiver moment cones JO - Algebraic Combinatorics PY - 2025 SP - 175 EP - 199 VL - 8 IS - 1 PB - The Combinatorics Consortium UR - https://alco.centre-mersenne.org/articles/10.5802/alco.398/ DO - 10.5802/alco.398 LA - en ID - ALCO_2025__8_1_175_0 ER -
%0 Journal Article %A Chindris, Calin %A Collins, Brett %A Kline, Daniel %T Hive-type polytopes for quiver multiplicities and the membership problem for quiver moment cones %J Algebraic Combinatorics %D 2025 %P 175-199 %V 8 %N 1 %I The Combinatorics Consortium %U https://alco.centre-mersenne.org/articles/10.5802/alco.398/ %R 10.5802/alco.398 %G en %F ALCO_2025__8_1_175_0
Chindris, Calin; Collins, Brett; Kline, Daniel. Hive-type polytopes for quiver multiplicities and the membership problem for quiver moment cones. Algebraic Combinatorics, Volume 8 (2025) no. 1, pp. 175-199. doi : 10.5802/alco.398. https://alco.centre-mersenne.org/articles/10.5802/alco.398/
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