Tensor Product Multiplicities via Upper Cluster Algebras

Abstract

For each valued quiver $Q$ of Dynkin type, we construct a valued ice quiver ${\Delta }_Q^2$. Let $G$ be a simple connected Lie group with Dynkin diagram the underlying valued graph of $Q$. The upper cluster algebra of ${\Delta }_Q^2$ is graded by the triple dominant weights $(\mu ,\nu ,\lambda )$ of $G$. We prove that when $G$ is simply-laced, the dimension of each graded component counts the tensor multiplicity $c_{\mu ,\nu }^{\lambda }$. We conjecture that this is also true if $G$ is not simply-laced, and sketch a possible approach. Using this construction, we improve Berenstein-Zelevinsky's model, or in some sense generalize Knutson-Tao's hive model in type $A$.

47 pages. v3 various improvements thanks to referees' comments -- title changed; correct a minor mistake in proving Lemma 7.4 (Lemma 8.3 in v2); simplify proofs in Section 5.1; v4 correct a wrong dual in the definition of twisted cyclic shift, and modify the proof of Theorem B.14 accordingly

References in corpus (1)

• Cluster Algebras and Semi-invariant Rings I. Triple Flags

Cited by in corpus (3)

• A survey on maximal green sequences
• Extending Upper Cluster Algebras
• Newton-Okounkov polytopes of Schubert varieties arising from cluster structures