Complexity and Support Varieties for Type P Lie Superalgebras

2020/01/30 by Boe, Brian D., Kujawa, Jonathan R.
#17B10 (Primary) 17B56 (Secondary) #17B55 #FOS: Mathematics #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.2001.11310

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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