Finite free position of maximal abelian $\ast$-subalgebras of the matrix algebra

Abstract

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.