Mathematical Research Letters

Volume 29 (2022)

Number 1

Complexity and support varieties for type $P$ Lie superalgebras

Pages: 59 – 100

DOI: https://dx.doi.org/10.4310/MRL.2022.v29.n1.a3

Authors

Brian D. Boe (Department of Mathematics, University of Georgia, Athens, Ga., U.S.A.)

Jonathan R. Kujawa (Department of Mathematics, University of Oklahoma, Norman, Ok., U.S.A.)

Abstract

We compute the complexity, $z$-complexity, and support varieties of the (thick) Kac modules for the Lie superalgebras of type $P$. We also show the complexity and the $z$-complexity have geometric interpretations in terms of support and associated varieties; these results are in agreement with formulas previously discovered for other classes of Lie superalgebras.

Our main technical tool is a recursive algorithm for constructing projective resolutions for the Kac modules. The indecomposable projective summands which appear in a given degree of the resolution are explicitly described using the combinatorics of weight diagrams. Surprisingly, the number of indecomposable summands in each degree can be computed exactly: we give an explicit formula for the corresponding generating function.

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Research of the second author was partially supported by NSA grant H98230-11-1-0127 and a Simons Collaboration Grant.

Received 16 February 2020

Accepted 10 August 2020

Published 6 September 2022