Cluster Categories From Grassmannians and Root Combinatorics

Abstract

The category of Cohen-Macaulay modules of an algebra B k,n is used in [JKS16] to give an additive categorification of the cluster algebra structure on the homogeneous coordinate ring of the Grassmannian of k-planes in n-space. In this paper, we find canonical Auslander-Reiten sequences and study the Auslander-Reiten translation periodicity for this category. Furthermore, we give an explicit construction of Cohen-Macaulay modules of arbitrary rank. We then use our results to establish a correspondence between rigid indecomposable modules of rank 2 and real roots of degree 2 for the associated Kac-Moody algebra in the tame cases.Proposition 5.2. Let I and J be r-interlacing. Then there exist 0 ≤ a 1 ≤ a 2 ≤ . . . a r−1 such that, as Z-modules,Proof. We assume that we have drawn the rims I and J one above the other, say I above J, as in the proof of Theorem 3.1 in [BB16, Section 3]. For every i 2s ∈ J \ I we have that rims I and J are not parallel between points i 2s−1 and i 2s , yielding a left trapezium. Similarly, for every i 2s+1 ∈ I \ J we have that rims I and J are not parallel between points i 2s and i 2s+1 , yielding a right trapezium. Since I and J are r-interlacing, we have, in alternating order r-left and r-right trapezia, giving us in total r boxes. The statement now follows from the proof of Theorem 3.1 in [BB16] which says that Ext 1 (L I , L J ) is a product of r − 1 cyclic Z-modules. Corollary 5.3. Let k = 3 and I and J be rims. Then Ext 1 (L I , L J ) ∼ = C × C if and only if I and J are 3-interlacing.Proof. If I and J are 3-interlacing, then they are both unions of three one-element sets. Hence, all the lateral sides in the boxes from the above proof are of length 1 and the statement follows since a i from the previous proposition are strictly positive, but at most equal to the lengths of the boxes involved. Corollary 5.4. Let k = 3. If I and J are crossing but not 3-interlacing, then Ext 1 (L I , L J ) ∼ = C.Note that if in Corollary 5.4 we have L J = τ (L I ), then we are in the situation of Theorem 3.12.Proposition 5.5. Assume that (k, n) = (3, 9) or (k, n) = (4, 8). Let M ∈ CM(B k,n ) be a rigid indecomposable rank 2 module. Then M ∼ = L I | L J where I and J are 3-interlacing.

Friezes satisfying higher SLk-determinants

“…is E 8 . The categories C(3, 9) and C(4, 8) are tame, these categories are known to be of tubular type, see [BBGE19], where the non-homogenous tubes are described.…”

Grassmannians and Cluster Structures

“…The above construction in fact can be given a bit more generally without specifying the entries in the upper right corner of the 2 Question 2 When is M(I , J ) indecomposable? First answers are given in [3]. In particular, it is known that if a rank two module with filtration M I /M J is indecomposable, then I and J have to be tightly 3-interlacing.…”

“…For (k, n) ∈ { (3,9), (4, 8)}, the category F k,n is of infinite type but tame (i.e., these are the first infinite cases to study). These are tubular categories: their Auslander-Reiten quiver is formed by tubes.…”

“…These are tubular categories: their Auslander-Reiten quiver is formed by tubes. Using Theorem 3.11 and computing syzygies for rank 2 modules, we are able to determine the tubes with low rank rigid modules [3]. The tubes for F 3,9 are of rank 2, 3, 6 and F 4,8 of rank 2, 4, 4.…”

Self-Injective Jacobian Algebras from Postnikov Diagrams

“…Even though CM(B) is triangulated, we remark that B is not a triangulated subcategory of CM(B). However, we can explicitly describe the Serre functor [2] of CM(B) (see also [BBGE18,Proposition 3.15]). It turns out that inside CM Z/nZ (R) there is an isomorphism of functors [2] ∼ = (−k), which in our setting translates into the following: Remark 5.8.…”