In this paper we study extensions between Cohen-Macaulay modules for algebras arising in the categorifications of Grassmannian cluster algebras. We prove that rank 1 modules are periodic, and we give explicit formulas for the computation of the period based solely on the rim of the rank 1 module in question. We determine Ext i (L I , L J ) for arbitrary rank 1 modules L I and L J . An explicit combinatorial algorithm is given for the computation of Ext i (L I , L J ) when i is odd, and when i even, we show that Ext i (L I , L J ) is cyclic over the centre, and we give an explicit formula for its computation. At the end of the paper we give a vanishing condition of Ext i (L I , L J ) for any i > 0.

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.

a b s t r a c tIn this paper we construct non-negative gradings on a basic Brauer tree algebra A Γ corresponding to an arbitrary Brauer tree Γ of type (m, e). We do this by transferring gradings via derived equivalence from a basic Brauer tree algebra A S , whose tree is a star with the exceptional vertex in the middle, to A Γ . The grading on A S comes from the tight grading given by the radical filtration. To transfer gradings via derived equivalence we use tilting complexes constructed by taking Green's walk around Γ (cf. Schaps and Zakay-Illouz (2001) [17]). By computing endomorphism rings of these tilting complexes we get graded algebras.We also compute Out K (A Γ ), the group of outer automorphisms that fix the isomorphism classes of simple A Γ -modules, where Γ is an arbitrary Brauer tree, and we prove that there is unique grading on A Γ up to graded Morita equivalence and rescaling.

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Rank 1 modules are the building blocks of the category CM(B k,n ) of Cohen-Macaulay modules over a quotient B k,n of a preprojective algebra of affine type A. Jensen, King and Su showed in [8] that the category CM(B k,n ) provides an additive categorification of the cluster algebra structure on the coordinate ring C[Gr(k, n)] of the Grassmannian variety of k-dimensional subspaces in C n . Rank 1 modules are indecomposable, they are known to be in bijection with k-subsets of {1, 2, . . . , n}, and their explicit construction has been given in [8]. In this paper, we give necessary and sufficient conditions for indecomposability of an arbitrary rank 2 module in CM(B k,n ) whose filtration layers are tightly interlacing. We give an explicit construction of all rank 2 decomposable modules that appear as extensions between rank 1 modules corresponding to tightly interlacing k-subsets I and J.