Littlewood-Richardson coefficients via mirror symmetry for cluster varieties

Abstract

I prove that the full Fock-Goncharov conjecture holds for ${\mathrm{Conf}}_3^{\times }(\widetilde{\mathcal{F}\ell })$-- the configuration space of triples of decorated flags in generic position. As a key ingredient of this proof, I exhibit a maximal green sequence for the quiver of the initial seed. I compute the Landau-Ginzburg potential $W$ on ${\mathrm{Conf}}_3^{\times }(\widetilde{\mathcal{F}\ell }{)}^{\vee }$ associated to the partial minimal model ${\mathrm{Conf}}_3^{\times }(\widetilde{\mathcal{F}\ell })\subset {\mathrm{Conf}}_3(\widetilde{\mathcal{F}\ell })$. The integral points of the associated "cone" $\Xi :=\left\{W^T\ge 0\right\}\subset {\mathrm{Conf}}_3^{\times }(\widetilde{\mathcal{F}\ell }{)}^{\vee }\left({\mathbb{R}}^T\right)$ parametrize a basis for $\mathcal{O}\left({\mathrm{Conf}}_3(\widetilde{\mathcal{F}\ell })\right)=⨁{\left(V_{\alpha }\otimes V_{\beta }\otimes V_{\gamma }\right)}^G$ and encode the Littlewood-Richardson coefficients $c_{\alpha \beta }^{\gamma }$. In the initial seed, the inequalities defining $\Xi$ are exactly Zelevinsky's tail positivity conditions. I exhibit a unimodular $p^{\ast }$ map that identifies $W$ with the potential of Goncharov-Shen on ${\mathrm{Conf}}_3^{\times }(\widetilde{\mathcal{F}\ell })$ and $\Xi$ with the Knutson-Tao hive cone.

51 pages, Many of the proofs are lifted from [Mag15]. See Remark 3. To appear in P. Lond. Math. Soc

References in corpus (4)

• Cluster Structures on Higher Teichmuller Spaces for Classical Groups
• Quiver mutation and combinatorial DT-invariants
• Fock-Goncharov conjecture and polyhedral cones for $U\subset S{L}_n$ and base affine space $S{L}_n/U$
• Lecture notes on Geometric Crystals and their combinatorial analogues

Cited by in corpus (5)

• The canonical wall structure and intrinsic mirror symmetry
• Toric degenerations of cluster varieties and cluster duality
• Strong positivity for quantum theta bases of quantum cluster algebras
• Tropical Fock-Goncharov coordinates for ${\mathrm{SL}}_3$-webs on surfaces I: construction
• Newton--Okounkov bodies and minimal models for cluster varieties