[Paper Review] Analogue of the Duistermaat-van der Kallen Theorem for Group Algebras

Wenhua Zhao, Roel M. Willems|arXiv (Cornell University)|Sep 29, 2010

Advanced Differential Equations and Dynamical Systems16 references4 citations

TL;DR

This paper establishes that for finite groups $G$ over integral domains $R$ of characteristic zero or $p > |G|$, the subspace $V_G$ of group algebra $R[G]$ with zero constant term forms a Mathieu subspace. However, for free abelian groups $\mathbb{Z}^n$ over fields of positive characteristic, $V_G$ fails to be a Mathieu subspace, showing the Duistermaat–van der Kallen theorem cannot be generalized to positive characteristic.

ABSTRACT

Let $G$ be a group, $R$ an integral domain, and $V_G$ the subspace of the group algebra $R[G]$ consisting of all the elements of $R[G]$ whose coefficient of the identity element $1_G$ of $G$ is equal to zero. Motivated by the Mathieu conjecture [M], the Duistermaat-van der Kallen theorem [DK], and also by recent studies on the notion of Mathieu subspaces introduced in [Z4] and [Z6], we show that for finite groups $G$, $V_G$ under certain conditions also forms a Mathieu subspace of the group algebra $R[G]$. We also show that for the free abelian groups $G=\Bbb Z^n$ $(n\ge 1)$ and any integral domain $R$ of positive characteristic, $V_G$ fails to be a Mathieu subspace of $R[G]$, which is equivalent to saying that the Duistermaat-van der Kallen theorem [DK] cannot be generalized to any field or integral domain of positive characteristic.

Motivation & Objective

• To determine under what conditions the subspace $V_G \subset R[G]$ (elements with zero constant term) forms a Mathieu subspace of the group algebra $R[G]$.
• To extend the Duistermaat–van der Kallen theorem to group algebras over integral domains, particularly addressing its validity in positive characteristic.
• To resolve the open problem of whether $V_G$ is a Mathieu subspace for finite groups and free abelian groups over various base rings.
• To clarify the role of group structure and characteristic of the base ring in determining the Mathieu subspace property of $V_G$.

Proposed method

• Define $V_G$ as the $R$-subspace of $R[G]$ consisting of elements with zero coefficient at the identity element $1_G$.
• Use the notion of Mathieu subspaces: a subspace $M \subset \mathcal{A}$ is a Mathieu subspace if $a^m \in M$ for all $m \gg 0$ implies $ba^m c \in M$ for all $m \gg 0$, for any $b,c \in \mathcal{A}$.
• Prove that for finite $G$ and $R$ with $\text{char}(R) = 0$ or $p > |G|$, $V_G$ is a Mathieu subspace via structural analysis of group algebras and properties of binomial coefficients modulo $p$.
• For $G = \mathbb{Z}^n$ and $\text{char}(R) = p > 0$, construct a counterexample using Laurent polynomials: $f = z^{-1} + z^{p-1}$, showing $f^m$ has zero constant term for all $m$, but $z^{-1}f^{p^k - 1}$ has constant term $(-1)^{p^{k-1}} \neq 0$.
• Leverage Lucas’ theorem via binomial coefficient identities modulo $p$, specifically $\binom{p^k - 1}{a} \equiv (-1)^a \pmod{p}$ and $\binom{bp}{b} \equiv 0 \pmod{p}$.
• Reduce the general case to $n=1$ via subgroup embedding and use the identification $R[\mathbb{Z}] \simeq R[z^{-1}, z]$ to analyze Laurent polynomial behavior.

Experimental results

Research questions

• RQ1 Under what conditions on a group $G$ and an integral domain $R$ is the subspace $V_G \subset R[G]$ (with zero constant term) a Mathieu subspace?
• RQ2 Can the Duistermaat–van der Kallen theorem, which holds over $\mathbb{C}$, be generalized to group algebras over fields of positive characteristic?
• RQ3 Does the failure of $V_G$ to be a Mathieu subspace in positive characteristic depend on the group structure, particularly for free abelian groups $\mathbb{Z}^n$?
• RQ4 What role does the characteristic of $R$ play in determining whether $V_G$ is a Mathieu subspace, especially when $p \leq |G|$?
• RQ5 Are there structural or combinatorial obstructions (e.g., binomial coefficient behavior modulo $p$) that prevent $V_G$ from being a Mathieu subspace in positive characteristic?

Key findings

• For any finite group $G$ and integral domain $R$ with $\text{char}(R) = 0$ or $p > |G|$, the subspace $V_G \subset R[G]$ is a Mathieu subspace.
• When $R$ is an algebraically closed field of characteristic $p > 0$, $V_G$ is a Mathieu subspace if and only if $p > |G|$.
• For the free abelian group $G = \mathbb{Z}^n$ with $n \geq 1$, and any integral domain $R$ of positive characteristic $p$, the subspace $V_G$ is not a Mathieu subspace of $R[G]$.
• The counterexample $f = z^{-1} + z^{p-1} \in \mathbb{F}_p[z^{-1}, z]$ satisfies $\text{Const}(f^m) = 0$ for all $m \geq 1$, but $\text{Const}(z^{-1}f^{p^k - 1}) = (-1)^{p^{k-1}} \neq 0$ for all $k \geq 1$, violating the Mathieu subspace condition.
• The failure of the Duistermaat–van der Kallen theorem in positive characteristic is established: the theorem does not extend to any field of positive characteristic.
• The binomial coefficient identity $\binom{p^k - 1}{a} \equiv (-1)^a \pmod{p}$ is crucial in proving the non-vanishing of the constant term in $z^{-1}f^{p^k - 1}$.