[Paper Review] Rank 2 affine MV polytopes

Pierre Baumann, Thomas Dunlap+2 more|arXiv (Cornell University)|Feb 28, 2012

Advanced Combinatorial Mathematics9 references3 citations

TL;DR

This paper constructs a combinatorial realization of the crystal $B(-∞)$ for the affine Kac-Moody algebra $ˇ{\mathrm{sl}}_2$ using decorated polytopes, generalizing Mirkoviác-Vilonen polytopes to the affine setting. It establishes that these MV polytopes are uniquely determined by their left or right sides, transform under crystal operators via edge-length changes, and are closed under Kashiwara's involution. The construction is extended to $A_2^{(2)}$ via a lattice-scaling map, yielding a crystal isomorphism to $B^{A_2^{(2)}}(-∞)$.

ABSTRACT

We give a realization of the infinity crystal for affine sl(2) using decorated polygons. The construction and proof are combinatorial, making use of Kashiwara and Saito's characterization of the infinity crystal in terms of the * involution. The polygons we use have combinatorial properties suggesting they are the analogues in this case of the Mirkovic-Vilonen polytopes defined by Anderson and the third author in finite type. Using Kashiwara's similarity of crystals we also give MV polytopes for $A_2^{(2)}$, the only other rank two affine Kac-Moody algebra.

Motivation & Objective

• To provide a combinatorial realization of the crystal $B(-∞)$ for $ˇ{\mathrm{sl}}_2$ using decorated polytopes in the root lattice.
• To generalize the finite-type Mirkoviác-Vilonen polytope construction to the affine setting, preserving key structural properties.
• To extend the construction to the other rank two affine Kac-Moody algebra $A_2^{(2)}$ using a scaling map and crystal isomorphism.
• To verify that the resulting polytopes satisfy the defining axioms of crystals, including Kashiwara's involution and crystal operator actions.

Proposed method

• Define decorated polytopes in the $ˇ{\mathrm{sl}}_2$ root lattice using systems of non-intersecting diagonals, generalizing finite-type MV polytope constructions.
• Use Kashiwara's similarity of crystals to lift the $ˇ{\mathrm{sl}}_2$ construction to $A_2^{(2)}$ via a lattice-scaling map $γ$ that sends $α_0 \mapsto \tilde{\alpha}_0$, $\alpha_1 \mapsto \tilde{\alpha}_1/2$.
• Characterize integral $A_2^{(2)}$ MV polytopes as the image under $\gamma$ of integral $ˇ{\mathrm{sl}}_2$ MV polytopes, with integrality conditions on Lusztig data and edge labels.
• Prove that the crystal operators $e_0$, $e_1$, $f_0$, $f_1$ act by increasing/decreasing edge lengths on the left or right side, preserving the polytope structure.
• Verify that the crystal structure on the polytopes matches $B(-∞)$ via a strict morphism, using the fact that $B^{A_2^{(2)}}(-∞)$ is generated by its lowest weight element under $e_0$ and $e_1$.
• Confirm that Kashiwara's involution corresponds to negation on the polytopes, and that $\varphi_i$ and $\varphi_i^*$ are given by top and bottom edge lengths.

Experimental results

Research questions

• RQ1 Can Mirkoviác-Vilonen polytopes be generalized to the affine Kac-Moody algebra $ˇ{\mathrm{sl}}_2$ using decorated lattice polytopes?
• RQ2 Do the resulting polytopes satisfy the same combinatorial axioms as finite-type MV polytopes, such as side-determination and non-intersecting diagonal systems?
• RQ3 How can the crystal structure of $B(-∞)$ for $A_2^{(2)}$ be realized combinatorially via a transformation of $ˇ{\mathrm{sl}}_2$ MV polytopes?
• RQ4 Is there a crystal isomorphism between the image of $ˇ{\mathrm{sl}}_2$ MV polytopes under the scaling map $\gamma$ and $B^{A_2^{(2)}}(-∞)$?
• RQ5 Does the $*$-involution on $B(-∞)$ correspond to negation on the polytopes, and are $\varphi_i$ and $\varphi_i^*$ given by edge lengths?

Key findings

• The set of $ˇ{\mathrm{sl}}_2$ MV polytopes forms a crystal isomorphic to $B^{ˇ{\mathrm{sl}}_2}(-∞)$, with crystal operators $e_1$ and $e_0$ increasing the length of the top edge on the left and right side, respectively.
• Each MV polytope is uniquely determined by its left or right side, and any valid side data corresponds to a unique MV polytope.
• Kashiwara's $*$-involution on $B(-∞)$ corresponds to negation of the polytope in the root lattice.
• The crystal operators $e_1$ and $e_0$ increase the length of the top edge on the left and right side by 1, respectively, and $\varphi_1$, $\varphi_1^*$ are given by the lengths of the top and bottom edges.
• For $A_2^{(2)}$, the image of the $ˇ{\mathrm{sl}}_2$ MV polytopes under the scaling map $\gamma$ forms a set of integral $A_2^{(2)}$ MV polytopes isomorphic to $B^{A_2^{(2)}}(-∞)$.
• The construction confirms that the polytopes satisfy the defining crystal axioms, including the compatibility of $e_i$, $f_i$, and the weight map, via a strict morphism from the polytope set to $B(-∞)$.