Journal of Lie Theory 31 (2021), No. 3, 797--809
Copyright Heldermann Verlag 2021
Hilbert Series of Typical Representations for Lie Superalgebras
Alexander Heaton
Max Planck Institute for Mathematics in the Sciences, Leipzig, Germany
and: Technische Universität, Berlin, Germany
heaton@mis.mpg.de
Songpon Sriwongsa
Dept. of Mathematics, and: Center in Theoretical and Comp. Science, Faculty of Science, King Mongkut's University of Technology, Thonburi, Bangkok, Thailand
songpon.sri@kmutt.ac.th
Let g be a basic classical Lie superalgebra over the complex numbers C. In the case of a typical weight whose every nonnegative integer multiple is also typical, we compute a closed form for the Hilbert series whose coefficients encode the dimensions of finite-dimensional irreducible typical g-representations. We give a formula for this Hilbert series in terms of elementary symmetric polynomials and Eulerian polynomials. Additionally, we show a simple closed form in terms of differential operators.
Keywords: Hilbert series, projective embedding, typical representations.
MSC: 17B10, 05E10, 20C35.
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