We show that bigrassmannian permutations determine the socle of the cokernel of an inclusion of Verma modules in type A. All such socular constituents turn out to be indexed by Weyl group elements from the penultimate two-sided cell. Combinatorially, the socular constituents in the cokernel of the inclusion of a Verma module indexed by $$w\in S_n$$ w ∈ S n into the dominant Verma module are shown to be determined by the essential set of w and their degrees in the graded picture are shown to be computable in terms of the associated rank function. As an application, we compute the first extension from a simple module to a Verma module.

Abstract. The paper concerns a certain subcategory of the category of representations for a semisimple algebraic group G in characteristic p, which arise from the semisimple modules for the corresponding quantum group at a p-th root of unity. The subcategory, thus, records the cohomological difference between quantum groups and algebraic groups. We define translation functors in this category and use them to obtain information on the irreducible characters for G when the Lusztig character formula does not hold.

We define a new category of quantum polynomial functors extending the quantum polynomials introduced by Hong and Yacobi. We show that our category has many properties of the category of Hong and Yacobi and is the natural setting in which one can define composition of quantum polynomial functors. Throughout the paper we highlight several key differences between the theory of classical and quantum polynomial functors.Proposition 2.2. The following diagram commutes:from which the commutativity of the diagram follows. Let (V, R V ) and (W, R W ) be Yang-Baxter spaces. We define the generalized (q-)Schur algebra S(V, W ; d) := (A(W, V ) d ) * as in [HY17]. The following is proved in [HY17]: Proposition 2.3. Let V, W be Yang-Baxter spaces. Then there is a natural isomorphism S(V, W ; d) ∼ = Hom B d (V ⊗d , W ⊗d ) Proof. See Proposition 2.7 in [HY17].By taking the dual of ∆ V,W,U we obtain a mapThere is a natural map m ′ U,W,V :The following Proposition shows they are the same map under the isomorphism in Proposition 2.3.Proposition 2.4. Given three Yang-Baxter spaces V, U, W , the following diagram commutes:Remark 2.5. If W = V , the quadratic relation (4) becomes the RTT relation due to Faddeev, Reshetikhin and Taktajan. The algebra A(V, V ) is then just the algebra denoted by A R V in [FRT88].

We observe that the join operation for the Bruhat order on a Weyl group agrees with the intersections of Verma modules in type A. The statement is not true in other types, and we propose a conjectural statement of a weaker correspondence. Namely, we introduce distinguished subsets of the Weyl group on which the join operation conjecturally agrees with the intersections of Verma modules. We also relate our conjecture with a statement about the socles of the cokernels of inclusions between Verma modules. The latter determines the first Ext spaces between a simple module and a Verma module. We give a conjectural complete description of such socles, which we verify in a number of cases. Along the way, we determine the poset structure of the join-irreducible elements in Weyl groups and obtain closed formulae for certain families of Kazhdan-Lusztig polynomials.