Invariants of Superalgebras as Complex Integrals

TL;DR

This work proves Budzik's conjecture by showing that the difference of multiplicities across hooks, $m_λ(k,ℓ)-m_λ(k-1,ℓ-1)$, equals a hook Schur inner product that reduces to $m_λ'$; this yields exact, rational Poincaré series for invariants and concomitants of GL supergroups via hook Schur function theory. The authors develop a new two-variable inner product, leverage Rosas's and Berele–Remmel identities, and provide explicit generating functions for key cases, including $P'(k,ℓ;0,1)$ in terms of typical/self-conjugate partitions. The results unify invariant-theoretic and combinatorial perspectives on U(k|ℓ) invariants, with explicit product-form identities and clear asymptotic structure. Collectively, the paper delivers exact formulas, rationality results, and combinatorial interpretations for the invariants and concomitants of general linear Lie superalgebras.

Abstract

In [A. Berele, Computing super matrix invariants, {\it Advances in Applied Math. \bf48} (2012), 273--289.] we defined integrals that approximated the Poincaré 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 Poincaré series. The purpose of this paper is to prove that Budzik's ideas are correct. As a consequence we prove that the Poincaré series are rational functions.

Figures (2)

• Figure 1: A Typical Partition
• Figure 2: Construction of $\lambda_0$, $\mu$ and $\nu$

Theorems & Definitions (26)

• Theorem 1
• Theorem 2: The Hook Theorem
• Theorem 3: Józefiak and Pragacz
• Theorem 4: Berele and Regev
• Theorem 5: Rosas
• Theorem 6: Berele and Remmel
• Lemma 7
• Lemma 8
• proof
• Lemma 9