Mathematical Research Letters
Volume 29 (2022)
Complexity and support varieties for type $P$ Lie superalgebras
Pages: 59 – 100
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.
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