Translation functors and decomposition numbers for the periplectic Lie superalgebra $\mathfrak{p}(n)$

Abstract

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. Contents

The Primitive Spectrum and Category for the Periplectic Lie Superalgebra

“…By [B+9, Lemma 4.2.1(1)], we have Ω(v ⊗ e i ) = Ω 0 (v ⊗ e i ). By [B+9,Lemma 4.2.1(2)], we thus find Ω(v ⊗ e i ) ∈ (λ i + 1 − i)(v ⊗ e i ) + N i−1 . Similarly, with some additional straightforward computations, we have…”

“…This operator Ω also appeared in [Co2,Section 8.4] and [CP,Section 2]. For the explicit realisation of Ω ∈ g ⊗ gl(n|n), we refer to [B+9,Section 4.1]. It decomposes as…”

The affine VW supercategory

“…We recall some facts from the representation theory of the Lie superalgebra p(n). For more details on Lie superalgebras see for instance [31], [39], and for p(n) see also [3].…”

“…(In fact, the centre of U (p(n)) is trivial.) We can however use the supertrace form on gl(n n) to define a fake Casimir in p(n) ⊗ gl(n n) as follows (see also [3]). Let…”

“…This allows one to apply the Cherednik [13] and Okounkov-Vershik [10], [33] approaches in this context. First steps in this direction were already successfully taken in [3] and [14] from different perspectives to determine the blocks and decomposition numbers in the category of finite dimensional representations of p(n) and of the Brauer superalgebra, and further developed in [15]. A thorough treatment of the corresponding category O is so far missing and will be deferred to subsequent work.…”