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We study analogues of Kronecker coefficients for symmetric inverse semigroups, for dual symmetric inverse semigroups and for the inverse semigroups of bijections between subquotients of finite sets. In all cases, we reduce the problem of determination of such coefficients to some group-theoretic and combinatorial problems. For symmetric inverse semigroups, we provide an explicit formula in terms of the classical Kronecker and Littlewood-Richardson coefficients for symmetric groups.

We describe Calabi-Yau objects in the regular block of the (parabolic) BGG category O associated to a semi-simple finite dimensional complex Lie algebra. Each such object comes with a natural transformation from the Serre functor to a shifted identity whose evaluation at that object is an isomorphism.

We prove that applying a projective functor to a holonomic simple module over a semisimple finite-dimensional complex Lie algebra produces a module that has an essential semisimple submodule of finite length. This implies that holonomic simple supermodules over certain Lie superalgebras are quotients of modules that are induced from simple modules over the even part. We also provide some further insight into the structure of Lie algebra modules that are obtained by applying projective functors to simple modules.

In this paper, we investigate extensions between graded Verma modules in the Bernstein- Gelfand-Gelfand category O . In particular, we determine exactly which information about extensions between graded Verma modules is given by the coefficients of the R-polynomials. We also give some upper bounds for the dimensions of graded extensions between Verma modules in terms of Kazhdan-Lusztig combinatorics. We completely determine all extensions between Verma module in the regular block of category O for sl₄ and construct various "unexpected" higher extensions between Verma modules.

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 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 space 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.

We give a complete combinatorial answer to Kostant's problem for simple highest weight modules indexed by fully commutative permutations. We also propose a reformulation of Kostant's problem in the context of fiab bicategories and classify annihilators of simple objects in the principal birepresentations of such bicategories generalizing the Barbasch-Vogan theorem for Lie algebras.

We introduce and study a multiparameter colored partition category CPar(x) by extending the construction of the partition category, over an algebraically closed field k of characteristic zero and for a multiparameter x∈k^r. The morphism spaces in CPar(x) have bases in terms of partition diagrams whose parts are colored by elements of the multiplicative cyclic group Cr. We show that the endomorphism spaces of CPar(x) and additive Karoubi envelope of CPar(x) are generically semisimple. The category CPar(x) is rigid symmetric strict monoidal and we give a presentation of CPar(x) as a monoidal category. The path algebra of CPar(x) admits a triangular decomposition with Cartan subalgebra being equal to the direct sum of the group algebras of complex reflection groups G(r,n). We compute the structure constants for the classes of simple modules in the split Grothendieck ring of the category of modules over the path algebra of the downward partition subcategory of CPar(x) in two ways. Among other things, this gives a formula for the product of the reduced Kronecker coefficients in terms of the Littlewood–Richardson coefficients for G(r,n) and certain Kronecker coefficients for the wreath product (C_r×C_r)≀S_n. For r=1, this formula reduces to a formula for the reduced Kronecker coefficients given by Littlewood. We also give two analogues of the Robinson–Schensted correspondence for colored partition diagrams and, as an application, we classify the equivalence classes of Green's left, right and two-sided relations for the colored partition monoid in terms of these correspondences.

We construct large families of simple modules for untwisted affine Lie algebras using induction from one-dimensional modules over nilpotent loop subalgebras. We also show that the vector space of the first self-extensions for these module has uncountable dimension and that generic tensor products of these modules are simple.

For l, n is an element of N we define tonal partition algebra P-l (n) over Z[delta]. We construct modules {Delta mu} mu for P-l (n) over Z[delta], and hence over any integral domain containing Z[delta] (such as C[delta]), that pass to a complete set of irreducible modules over the field of fractions. We show that P-l (n) is semisimple there. That is, we construct for the tonal partition algebras a modular system in the sense of Brauer. Using a "geometrical" index set for the Delta-modules, we give an order with respect to which the decomposition matrix over C (with d. C-x) is upper-unitriangular. We establish several crucial properties of the Delta-modules. These include a tower property, with respect to n, in the sense of Green and Cox-Martin-Parker-Xi; contravariant forms with respect to a natural involutive antiautomorphism; a highest weight category property; and branching rules.

We determine for which Coxeter types the associated small quo-tient of the 2-category of Soergel bimodules is finitary and, for such a small quotient, classify the simple transitive 2-representations (sometimes under the additional assumption of gradability). We also describe the underlying cat-egories of the simple transitive 2-representations. For the small quotients of general Coxeter types, we give a description for the cell 2-representations.