Hive-type polytopes for quiver multiplicities and the membership problem for quiver moment cones
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| Format: | Preprint |
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2022
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| author | Chindris, Calin Collins, Brett Kline, Daniel |
| author_facet | Chindris, Calin Collins, Brett Kline, Daniel |
| contents | Let $Q$ be a bipartite quiver with vertex set $Q_0$ such that the number of arrows between any source vertex and any sink vertex is constant. Let $β=(β(x))_{x \in Q_0}$ be a dimension vector of $Q$ with positive integer coordinates.
Let $rep(Q, β)$ be the representation space of $β$-dimensional representations of $Q$ and $GL(β)$ the base change group acting on $rep(Q, β)$ be simultaneous conjugation. Let $K^β_{\underlineλ}$ be the multiplicity of the irreducible representation of $GL(β)$ of highest weight $\underlineλ$ in the ring of polynomial functions on $rep(Q, β)$.
We show that $K^β_{\underlineλ}$ 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 $(Q,β)$ in strongly polynomial time. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2211_01990 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Hive-type polytopes for quiver multiplicities and the membership problem for quiver moment cones Chindris, Calin Collins, Brett Kline, Daniel Combinatorics Computational Complexity Representation Theory 16G20 (Primary), 13A50, 05E10, 68Q17 (Secondary) Let $Q$ be a bipartite quiver with vertex set $Q_0$ such that the number of arrows between any source vertex and any sink vertex is constant. Let $β=(β(x))_{x \in Q_0}$ be a dimension vector of $Q$ with positive integer coordinates. Let $rep(Q, β)$ be the representation space of $β$-dimensional representations of $Q$ and $GL(β)$ the base change group acting on $rep(Q, β)$ be simultaneous conjugation. Let $K^β_{\underlineλ}$ be the multiplicity of the irreducible representation of $GL(β)$ of highest weight $\underlineλ$ in the ring of polynomial functions on $rep(Q, β)$. We show that $K^β_{\underlineλ}$ 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 $(Q,β)$ in strongly polynomial time. |
| title | Hive-type polytopes for quiver multiplicities and the membership problem for quiver moment cones |
| topic | Combinatorics Computational Complexity Representation Theory 16G20 (Primary), 13A50, 05E10, 68Q17 (Secondary) |
| url | https://arxiv.org/abs/2211.01990 |