Invariants of Superalgebras as Complex Integrals

TL;DR

This paper confirms that a new method accurately computes the Poincaré series of invariants in superalgebras, demonstrating their rationality and advancing understanding of superalgebra invariants.

Contribution

It proves the correctness of Budzik's method for exact Poincaré series computation in superalgebras, showing these series are rational functions.

Findings

Budzik's method yields exact Poincaré series

Poincaré series are rational functions

Validation of superalgebra invariants computation

Abstract

In [A. Berele, Computing super matrix invariants, {\it Advances in Applied Math. \bf48} (2012), 273--289.] we defined integrals that approximated the Poincar\'e series of the invariants and concomitants of the general linear Lie supergroup or superalgebra. Budzik suggested in [K. Budzik, Supergroup Invariants and the Brane/Negative Brane Expansion, (preprint) arXiv:2509.20451] a way to adapt this method to get the exact Poincar\'e series. The purpose of this paper is to prove that Budzik's ideas are correct. As a consequence we prove that the Poincar\'e series are rational functions.