AI 中文总结
本文研究矩阵代数中极大交换$\ast$-子代数的有限自由位置，通过行列式条件刻画其成立当且仅当$n\le3$且$\sqrt{n}U$为复Hadamard矩阵，并引入均匀主子式偏差量化$n\ge4$时的偏离，给出对称性、单调性及下界。
AI 中文摘要
有限自由卷积是通过对Haar酉共轭的特征多项式取平均得到的。我们研究何时复矩阵代数${\sf M}_n$的两个极大交换$\ast$-子代数可以置于有限自由位置：即，当它们的相对位置对每一对元素（分别来自两个子代数）恰好实现这种平均时。将这样的一对写成${\sf D}_n$和$U{\sf D}_nU^*$，其中$U$是酉矩阵，我们通过条件$|\det U[I,J]|^2=\binom{n}{r}^{-1}$（对所有$1\le r\le n$以及所有满足$|I|=|J|=r$的$I,J$）来刻画有限自由位置（无论是加法卷积还是乘法卷积）。我们证明该条件成立当且仅当$n\le3$且$\sqrt{n} U$是复Hadamard矩阵。为了量化$n\ge4$时精确实现的失败程度，我们引入了均匀主子式偏差$\delta_r(U)$。我们将其与两个对角矩阵（其对角元独立且均匀分布在单位圆上）的有限自由乘法卷积的第$r$个系数的均方误差等同起来。我们建立了对称性$\delta_r(U)=\delta_{n-r}(U)$和单调性$\delta_1(U)\le\delta_2(U)\le\cdots\le \delta_{\lfloor n/2\rfloor}(U)$。对于平坦酉矩阵，我们推导出$\delta_2$的显式公式，从而得出对$2\le r\le n-2$有$\delta_r(U)\ge\frac{n-3}{2n}$。当$r=2$时等号成立当且仅当$\sqrt{n}U$的逐项平方也是复Hadamard矩阵。
英文摘要
Finite free convolution is obtained by averaging characteristic polynomials over Haar unitary conjugation. We ask when two maximal abelian $\ast$-subalgebras of the complex matrix algebra ${\sf M}_n$ can be placed in finite free position: that is, when their relative position realizes this averaging exactly for every pair of elements, one from each subalgebra. Writing such a pair as ${\sf D}_n$ and $U{\sf D}_nU^*$ with $U$ unitary, we characterize finite free position, for either additive or multiplicative convolution, by the condition $|\det U[I,J]|^2=\binom{n}{r}^{-1}$ for every $1\le r\le n$ and all $I,J$ with $|I|=|J|=r$. We show that this condition holds if and only if $n\le3$ and $\sqrt{n} U$ is a complex Hadamard matrix. To quantify the failure of exact realization for $n\ge 4$, we introduce the uniform-minor discrepancy $δ_r(U)$. We identify it with the mean-square error in the $r$-th coefficient of finite free multiplicative convolution for two diagonal matrices whose diagonal entries are independent and uniformly distributed on the unit circle. We establish the symmetry $δ_r(U)=δ_{n-r}(U)$ and the monotonicity $δ_1(U)\leδ_2(U)\le\cdots\le δ_{\lfloor n/2\rfloor}(U)$. For flat unitaries, we derive an explicit formula for $δ_2$, yielding $δ_r(U)\ge\frac{n-3}{2n}$ for $2\le r\le n-2$. Equality for $r=2$ holds precisely when the entrywise square of $\sqrt{n}U$ is also complex Hadamard.
Comments18 pages
