The Fubini Theorem for Normal Lie Subgroups of Index $2n$

TL;DR

This work extends the Fubini theorem for Haar integration from index-2 subgroups to normal Lie subgroups of index $2n$ by proving a semidirect-product decomposition that allows iterative handling of $n$ involution cosets. It shows that, for a normal subgroup $\Gamma_+\subset\Gamma$ with index $2n$, the Haar integral satisfies $\int_{\Gamma} f(\gamma) = \frac{1}{2^n} \sum_{i \in N_2^n} \int_{\Gamma_+} f(\lambda^i \gamma)$, enabling practical computation by reducing to $\Gamma_+$; the approach rests on a key decomposition lemma generalizing the index-2 case. The paper applies this generalized Fubini theorem to invariant theory via Molien series, including a concrete Lorentz-group–like example that yields the Molien series and a Hilbert basis, illustrating counting invariants for larger index subgroups. Overall, the results provide a systematic method for integrating over larger quotient extensions and have implications for invariant polynomials and symmetry analyses in physics and geometry.

Abstract

Let $Γ_+$ be a normal subgroup of index $2n$ of a group $Γ$ and $γ_i \in Γ\setminus Γ_+$ be involutions. We first prove that if $Γ= Γ_+ \rtimes (\mathbb{Z}_2(γ_1) \times \cdots \times \mathbb{Z}_2(γ_n))$ then $Γ= (Γ_+ \rtimes \mathbb{Z}_2(γ_1) \rtimes \cdots \rtimes \mathbb{Z}_2(γ_{i-1})) \rtimes (\mathbb{Z}_2(γ_{i}) \times \cdots \times \mathbb{Z}_2(γ_n))$, where $i=2,\cdots,n$. Second, we use this result to prove the well-known Fubini theorem for a subgroup of index $2n$ of a compact Lie group. Finally, we present an application to invariant theory.

Theorems & Definitions (10)

• Definition 1: golubitsky2012singularities
• Theorem 1: The Fubini theorem
• Lemma 1: oliveiraaspectos
• Lemma 2
• Proposition 1
• proof
• Theorem 2: Fubini Theorem for normal Lie subgroups of index $2n$
• proof
• Theorem 3: Molien theorem
• proof