We study the category F n of finite-dimensional integrable representations of the periplectic Lie superalgebra p(n). We define an action of the Temperley-Lieb algebra with infinitely many generators and defining parameter 0 on the category F n by certain translation functors. We also introduce combinatorial tools, called weight diagrams and arrow diagrams for p(n) resembling those for gl(m n). We discover two natural highest weight structures. Using the Temperley-Lieb algebra action and the combinatorics of weight and arrow diagrams, we then calculate the multiplicities of irreducibles in standard and costandard modules and classify the blocks of F n . We also prove the surprising fact that indecomposable projective modules in this category are multiplicity-free.
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We define the affine VW supercategory s⩔, which arises from studying the action of the periplectic Lie superalgebra p(n) on the tensor product M ⊗ V ⊗a of an arbitrary representation M with several copies of the vector representation V of p(n). It plays a role analogous to that of the degenerate affine Hecke algebras in the context of representations of the general linear group; the main obstacle was the lack of a quadratic Casimir element in p(n) ⊗ p(n). When M is the trivial representation, the action factors through the Brauer supercategory sBr . Our main result is an explicit basis theorem for the morphism spaces of s⩔ and, as a consequence, of sBr . The proof utilises the close connection with the representation theory of p(n). As an application we explicitly describe the centre of all endomorphism algebras, and show that it behaves well under the passage to the associated graded and under deformation.
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