Finite free position of maximal abelian ∗-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 ∗-subalgebras of the complex matrix algebra sf Mn 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 Dn and Usf DnU* with U unitary, we characterize finite free position, for either additive or multiplicative convolution, by the condition |det U[I,J]|2=binomnr−1 for every 1≤ r≤ n and all I,J with |I|=|J|=r. We show that this condition holds if and only if n≤3 and √n U is a complex Hadamard matrix. To quantify the failure of exact realization for n≥ 4, we introduce the uniform-minor discrepancy δr(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 δr(U)=δn−r(U) and the monotonicity δ1(U)≤δ2(U)≤⋯≤ δ⌊ n/2⌋(U). For flat unitaries, we derive an explicit formula for δ2, yielding δr(U)≥(n−3)/2n for 2≤ r≤ n−2. Equality for r=2 holds precisely when the entrywise square of √nU is also complex Hadamard.
Related
- Finite-dimensional Lie subalgebras of the universal enveloping algebra of sl(2,ℂ)
- Maximal subalgebras in the three exceptional simple finite-dimensional Lie superalgebras
- Maximal ranks of free abelian subgroups of generalized cactus groups associated with finite Coxeter systems
- Maximal graded subalgebras of simple vectorial Lie superalgebras with polynomial coefficients
- Identities of Representation of Lie subalgebras of the first Weyl algebra
BibTeX
@misc{ueda2026finite,
title = {{Finite free position of maximal abelian $\ast$-subalgebras of the matrix algebra}},
author = {Yuki Ueda},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.39677},
eprint = {2609.39677},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.39677},
}
RIS
TY - UNPB
TI - Finite free position of maximal abelian ∗-subalgebras of the matrix algebra
AU - Yuki Ueda
PY - 2026
DA - 2026/09/30
PB - arXiv
DO - 10.48550/arxiv.2609.39677
UR - https://arxiv.org/abs/2609.39677
AB - Finite free convolution is obtained by averaging characteristic polynomials over Haar unitary conjugation. We ask when two maximal abelian ∗-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 Usf D_nU^* with U unitary, we characterize finite free position, for either additive or multiplicative convolution, by the condition |det U[I,J]|^2=binomnr^−1 for every 1≤ r≤ n and all I,J with |I|=|J|=r. We show that this condition holds if and only if n≤3 and √n U is a complex Hadamard matrix. To quantify the failure of exact realization for n≥ 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)≤δ_2(U)≤⋯≤ δ_⌊ n/2⌋(U). For flat unitaries, we derive an explicit formula for δ_2, yielding δ_r(U)≥(n−3)/2n for 2≤ r≤ n−2. Equality for r=2 holds precisely when the entrywise square of √nU is also complex Hadamard.
ER -