Mathematical Research Letters

Volume 26 (2019)

Number 3

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

Pages: 643 – 710

DOI: https://dx.doi.org/10.4310/MRL.2019.v26.n3.a2

Authors

M. Balagovic (School of Mathematics, Statistics and Physics, Newcastle University, Newcastle, United Kingdom)

Z. Daugherty (Department of Mathematics, City College of New York, N.Y., U.S.A.)

I. Entova-Aizenbud (Department of Mathematics, Tel Aviv University, Tel Aviv, Israel)

I. Halacheva (School of Mathematics and Statistics, University of Melbourne, Victoria, Australia)

J. Hennig (Department of Mathematics and Statistical Sciences, University of Alberta, Edmonton, AB, Canada)

M.S. Im (Department of Mathematics Sciences, United States Military Academy, New York, U.S.A.)

G. Letzter (Department of Defense, Ft. George G. Meade, Maryland, U.S.A.)

E. Norton (Department of Mathematics, TU Kaiserslautern, Germany)

V. Serganova (Department of Mathematics, University of California at Berkeley)

C. Stroppel (Mathematisches Institut, Universität Bonn, Germany)

Abstract

We study the category $\mathcal{F}_n$ of finite-dimensional integrable representations of the periplectic Lie superalgebra $\mathfrak{p}(n)$. We define an action of the Temperley–Lieb algebra with infinitely many generators and defining parameter 0 on the category $\mathcal{F}_n$ by certain translation functors. We also introduce combinatorial tools, called weight diagrams and arrow diagrams for $\mathfrak{p}(n)$ resembling those for $\mathfrak{gl}(m \vert n)$. 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 $\mathcal{F}_n$. We also prove the surprising fact that indecomposable projective modules in this category are multiplicity-free.

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Z. Daugherty was partially supported by National Science Foundation Grant DMS-1162010.

I. Entova-Aizenbud was partially supported by the MPI, Bonn, and by UC Berkeley.

E. Norton was supported by the MPI, Bonn.

C. Stroppel was partially supported by the Hausdorff Center of Mathematics, Bonn.

Received 31 January 2017

Accepted 7 August 2018

Published 25 October 2019