We associate a monoidal category H B , defined in terms of planar diagrams, to any graded Frobenius superalgebra B. This category acts naturally on modules over the wreath product algebras associated to B. To B we also associate a (quantum) lattice Heisenberg algebra h B . We show that, provided B is not concentrated in degree zero, the Grothendieck group of H B is isomorphic, as an algebra, to h B . For specific choices of Frobenius algebra B, we recover existing results, including those of Khovanov and Cautis-Licata. We also prove that certain morphism spaces in the category H B contain generalizations of the degenerate affine Hecke algebra. Specializing B, this proves an open conjecture of Cautis-Licata.

We classify all solutions (p, q) to the equation p(u)q(u) = p(u + β)q(u + α) where p and q are complex polynomials in one indeterminate u, and α and β are fixed but arbitrary complex numbers. This equation is a special case of a system of equations which ensures that certain algebras defined by generators and relations are non-trivial. We first give a necessary condition for the existence of non-trivial solutions to the equation. Then, under this condition, we use combinatorics of generalized Dyck paths to describe all solutions and a canonical way to factor each solution into a product of irreducible solutions.

Abstract. In this paper we first generalize to the case of partial flags a result proved both by Spaltenstein and by Steinberg that relates the relative position of two complete flags and the irreducible components of the flag variety in which they lie, using the Robinson-Schensted-Knuth correspondence. Then we use this result to generalize the mirabolic Robinson-Schensted-Knuth correspondence defined by Travkin, to the case of two partial flags and a line.

We show that the Grothendieck groups of the categories of finitely-generated graded supermodules and finitely-generated projective graded supermodules over a tower of graded superalgebras satisfying certain natural conditions give rise to twisted Hopf algebras that are twisted dual. Then, using induction and restriction functors coming from such towers, we obtain a categorification of the twisted Heisenberg double and its Fock space representation. We show that towers of wreath product algebras (in particular, the tower of Sergeev superalgebras) and the tower of nilcoxeter graded superalgebras satisfy our axioms. In the latter case, one obtains a categorification of the quantum Weyl algebra.Comment: 29 pages; v2: Minor changes (corrected typos, etc.) throughou

Kato introduced the exotic nilpotent cone to be a substitute for the ordinary nilpotent cone of type C with cleaner properties. Here we describe the irreducible components of exotic Springer fibres (the fibres of the resolution of the exotic nilpotent cone), and prove that they are naturally in bijection with standard bitableaux. As a result, we deduce the existence of an exotic Robinson-Schensted bijection, which is a variant of the type C Robinson-Schensted bijection between pairs of same-shape standard bitableaux and elements of the Weyl group; this bijection is described explicitly in the sequel to this paper. Note that this is in contrast with ordinary type C Springer fibres, where the parametrisation of irreducible components, and the resulting geometric Robinson-Schensted bijection, are more complicated. As an application, we explicitly describe the structure in the special cases where the irreducible components of the exotic Springer fibre have dimension 2, and show that in those cases one obtains Hirzebruch surfaces. Contents