Gelfand--Kirillov dimensions of highest weight modules for basic classical Lie superalgebras

Abstract

In this paper, we develop a combinatorial algorithm to compute the Gelfand--Kirillov (GK) dimension of simple highest weight modules for type I basic classical Lie superalgebras. Building upon the results for classical Lie algebras via Lusztig's {\bf a}-function and the Robinson--Schensted (RS) insertion algorithm, we extend these techniques to the super setting, providing explicit formulas for types $\mathfrak{sl}(m|n)$ and $\mathfrak{osp}(2|2n)$. Our results show that for type I basic classical Lie superalgebras, the GK dimension of a simple highest weight module is determined entirely by the even part of the Lie superalgebra.

29 pages