AI 中文总结
本文证明紧致连通李群G的核ker I_G为Mathieu-Zhao子空间当且仅当G为环面，并对非交换情形构造满足精确Pascal行恒等式的矩消失例子，直接证明不依赖Mathieu猜想。
AI 中文摘要
设$G$为紧致连通李群，令$\mathcal R(G)$表示其代表函数代数，并令$\mathcal I_G(f)=\int_G f(g)\\,\mathrm d g$为归一化Haar积分。我们证明$\ker \mathcal I_G$是$\mathcal R(G)$的Mathieu--Zhao子空间当且仅当$G$是环面。更强地，对于每个非交换紧致连通李群$G$，我们构造$A,P,Q\in\mathcal R(G)$，其中$A\geq 0$且$A\not\equiv 0$，使得$P$的所有纯矩消失，且标记矩满足精确的Pascal行恒等式$$ \mathcal I_G(Q^sP^m) =c_m\binom{m-1}{s-1}\mathcal I_G(A^{4m+s})>0 \qquad (1\leq s\leq m), $$其中$c_m=4^m(m!)^2/(2m+1)!$；对于$s>m$，相同的矩消失。该构造始于$\mathbb C^2$上的齐次、相位平衡多项式对。Hopf球面$S^3$上的精确系数恒等式通过轨道平均转移到$\mathbb C^2$上每个紧支撑的$SU(2)$-不变测度。最高权表示和可见的单根双合子在每个紧致单李群上产生这样的测度。相位平衡给出通过每个中心商的下降，而从伴随单商拉回处理任意非交换紧致连通群。该证明是直接的，不依赖于从Mathieu猜想蕴含Jacobian猜想。环面方向是Duistermaat和van der Kallen的定理。
英文摘要
Let $G$ be a compact connected Lie group, let $\mathcal R(G)$ denote its algebra of representative functions, and let $\mathcal I_G(f)=\int_G f(g)\,\mathrm d g$ be normalized Haar integration. We prove that $\ker \mathcal I_G$ is a Mathieu--Zhao subspace of $\mathcal R(G)$ if and only if $G$ is a torus. More strongly, for every nonabelian compact connected Lie group $G$ we construct $A,P,Q\in\mathcal R(G)$, with $A\geq 0$ and $A\not\equiv 0$, such that all pure moments of $P$ vanish and the marked moments satisfy an exact Pascal-row identity $$ \mathcal I_G(Q^sP^m) =c_m\binom{m-1}{s-1}\mathcal I_G(A^{4m+s})>0 \qquad (1\leq s\leq m), $$ where $c_m=4^m(m!)^2/(2m+1)!$; the same moments vanish for $s>m$. The construction begins with a homogeneous, phase-balanced polynomial pair on $\mathbb C^2$. An exact coefficient identity on the Hopf sphere $S^3$ is transferred by orbit averaging to every compactly supported $SU(2)$-invariant measure on $\mathbb C^2$. A highest-weight representation and a visible simple-root doublet produce such a measure on every compact simple Lie group. Phase balance gives descent through every central quotient, and pullback from an adjoint simple quotient handles arbitrary nonabelian compact connected groups. The proof is direct and independent of the implication from the Mathieu conjecture to the Jacobian conjecture. The torus direction is the theorem of Duistermaat and van der Kallen.
Comments14 pages, 1 table; Python program is on GitHub
