Affine Mirković-Vilonen polytopes

Abstract

Each integrable lowest weight representation of a symmetrizable Kac-Moody Lie algebra g has a crystal in the sense of Kashiwara, which describes its combinatorial properties. For a given g, there is a limit crystal, usually denoted by B(−∞), which contains all the other crystals. When g is finite dimensional, a convex polytope, called the Mirković-Vilonen polytope, can be associated to each element in B(−∞). This polytope sits in the dual space of a Cartan subalgebra of g, and its edges are parallel to the roots of g. In this paper, we generalize this construction to the case where g is a symmetric affine Kac-Moody algebra. The datum of the polytope must however be complemented by partitions attached to the edges parallel to the imaginary root δ. We prove that these decorated polytopes are characterized by conditions on their normal fans and on their 2faces. In addition, we discuss how our polytopes provide an analog of the notion of Lusztig datum for affine Kac-Moody algebras. Our main tool is an algebro-geometric model for B(−∞) constructed by Lusztig and by Kashiwara and Saito, based on representations of the completed preprojective algebra Λ of the same type as g. The underlying polytopes in our construction are described with the help of Buan, Iyama, Reiten and Scott's tilting theory for the category Λ-mod. The partitions we need come from studying the category of semistable Λ-modules of dimension-vector a multiple of δ.One of these tools is Buan, Iyama, Reiten and Scott's tilting ideals for Λ [14]. Let S i be the simple Λ-module of dimension-vector α i and let I i be its annihilator, a one-codimensional two-sided ideal of Λ. The products of these ideals I i are known to satisfy the braid relations, so to each w in the Weyl group of g, we can attach a two-sided ideal I w of Λ by the rule I w = I i 1 · · · I i ℓ , where s i 1 · · · s i ℓ is any reduced decomposition of w. Given a finite-dimensional Λ-module T , we denote the image of the evaluation map I w ⊗ Λ Hom Λ (I w , T ) → T by T w .Recall that the dominant Weyl chamber C 0 and the Tits cone C T are the convex cones in the dual of RI defined asWe will show the equality T min θ = T max θ = T w for any finite dimensional Λ-module T , any w ∈ W and any linear form θ ∈ wC 0 . This implies that dim T w is a vertex of Pol(T ) and that the normal cone to Pol(T ) at this vertex contains wC 0 . This also implies that Pol(T ) is contained inWhen θ runs over the Tits cone, it generically belongs to a chamber, and we have just seen that in this case, the face P θ is a vertex. When θ lies on a facet, P θ is an edge (possibly degenerate). More precisely, if θ lies on the facet that separates the chambers wC 0 and ws i C 0 , with say ℓ(ws i ) > ℓ(w), then (T min θ , T max θ ) = (T ws i , T w ). Results in [1] and [27] moreover assert that T w /T ws i is the direct sum of a finite number of copies of the Λ-module I w ⊗ Λ S i .There is a similar description when θ is in −C T ; here the submodules T w of T that come into play are the kernels of the coevaluation maps T → ...

Toda Systems, Cluster Characters, and Spectral Networks

“…Indeed, that these objects are connected is already well-known. We recall in particular the result of [BK12,BKT11] that the convex hull of the dimension vectors of submodules of a projective-injective Λ n -module is a Weyl polytope, the convex hull of the weights of a fundamental representation of SL n pCq. ,Wil13b].…”

Rank 2 affine MV polytopes

“…We believe that the polytopes defined here essentially agree with the sl 2 case of that construction, and we plan to address this issue in future work. As discussed in [2], this would give a combinatorial description of MV polytopes in all symmetric affine cases 1 . Finally, we would like to mention the work of Naito, Sagaki and Saito [17] (see also Muthiah [15]) giving a version of MV polytopes for sl n , so in particular, sl 2 .…”

“…The second is the construction by the other three authors [2] of MV polytopes (for symmetric finite and affine Lie algebras) associated to components of Lusztig's nilpotent varieties. We believe that the polytopes defined here essentially agree with the sl 2 case of that construction, and we plan to address this issue in future work.…”

“…In the finite type, rank 2 MV polytopes were characterized combinatorially on a case-by-case basis in [9] using the tropical Plucker relations. In the present paper, we define MV polytopes for sl 2 and A (2) 2 using diagonals (Definition 3.4). A careful examination of the tropical Plucker relations shows that this definition also characterizes MV polytopes in the finite type case (at least for A 2 and B 2 , as G 2 is still unclear).…”

“…The polygons we use have combinatorial properties suggesting they are the sl 2 analogues of the Mirković-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 other rank 2 affine Kac-Moody algebra. …”