Simone Severini
The support of a matrix M is the (0,1)-matrix with ij-th entry equal to 1 if the ij-th entry of M is non-zero, and equal to 0, otherwise. The digraph whose adjacency matrix is the support of M is said to be the digraph of M. This paper observes some structural properties of digraphs of unitary matrices.
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A digraph D is the pattern of a matrix M when D has an arc ij if and only if the ij-th entry of M is nonzero. Study the relationship between unitary matrices and their patterns is motivated by works in quantum chaology and quantum computation. In this note, we prove that if a Cayley digraph is a line digraph then it is the pattern of a unitary matrix. We prove that for any finite group with two generators there exists a set of generators such that the Cayley digraph with respect to such a set is a line digraph and hence the pattern of a unitary matrix.
We give an example of a BH(5; 5) matrix M satisfying that the eigenvalues of the unitary matrix $\frac{1}{\sqrt5}M$ are all primitive 10-th roots of unity and such that $\frac{1}{5}M^3$ is not a BH(5; 5) matrix. This example gives a negative answer to a conjecture proposed by R. Egan and P. O. Cathain.
Diagonalization, or eigenvalue decomposition, is very useful in many areas of applied mathematics, including signal processing and quantum physics. Matrix decomposition is also a useful tool for approximating matrices as the product of a matrix and its transpose, which relates to unitary diagonalization. As stated by the spectral theorem, only normal matrices are unitarily diagonalizable. However we show that all real square matrices are the real part of some unitarily diagonalizable matrix.
This paper concerns the enumeration of simultaneous conjugacy classes of tuples of commuting unitary matrices and of commuting symplectic matrices over a finite field $\mathbf{F}_q$ of odd size. For any given conjugacy class, the orbits for the action of its centralizer group on itself by conjugation are called branches. We determine the branching rules for the unitary groups $U_2(\mathbf{F}_q), U_3(\mathbf{F}_q)$, and for the symplectic groups $Sp_2(\mathbf{F}_q), Sp_4(\mathbf{F}_q)$.
The aim of this note is to prove that fluctuations of uniformly random alternating sign matrices (equivalently, configurations of the six-vertex model with domain wall boundary conditions) near the boundary are described by the Gaussian Unitary Ensemble and the GUE-corners process.
We provide a graphical calculus for computing averages of tensor network diagrams with respect to the distribution of random vectors containing independent uniform complex phases. Our method exploits the order structure of the partially ordered set of uniform block permutations. A similar calculus is developed for random vectors consisting of independent uniform signs, based on the combinatorics of the partially ordered set of even partitions. We employ our method to extend some of the results by Johnston and MacLean on the family of local diagonal unitary invariant matrices. Furthermore, our …
Let $U$ be a matrix chosen randomly, with respect to Haar measure, from the unitary group $U(d).$ We express the moments of the trace of any submatrix of $U$ as a sum over partitions whose terms count certain standard and semistandard Young tableaux. Using this combinatorial interpretation, we obtain a simple closed form for the moments of an individual entry of a random unitary matrix and use this to deduce that the entries converge in moments to standard complex Gaussian random variables. In addition, we recover a well-known theorem of E. Rains which shows that the moments of the trace of a …
In arXiv:1607.06679, Marcus proved that certain functions of multiple matrices, when summed over the symmetries of the cube, decompose into functions of the original matrices. In this note, we generalize the results from the Marcus paper to a larger class of functions of multiple matrices. We also answer a problem posed in the Marcus paper.
In this note we give a combinatorial and non-computational proof of the asymptotics of the integer moments of the moments of the characteristic polynomials of Haar distributed unitary matrices as the size of the matrix goes to infinity. This is achieved by relating these quantities to a lattice point count problem. Our main result is a new explicit expression for the leading order coefficient in the asymptotic as a volume of a certain region involving continuous Gelfand-Tsetlin patterns with constraints.
Asymptotic Freeness of Unitary Matrices in Tensor Product Spaces for Invariant States
本文扩展了Haar酉矩阵张量积的渐近自由性，证明其在一类基于对称群不变性的更广泛状态下成立，并推广至酉群一般表示序列及对称群与自由群同时作用的情境。
This paper is concerned with certain connections between the ensemble of n x n unitary matrices -- specifically the characteristic function of the random variable tr(U) -- and combinatorics -- specifically Ulam's problem concerning the distribution of the length of the longest increasing subsequence in permutation groups -- and the appearance of Painleve functions in the answers to apparently unrelated questions. Among the results is a representation in terms of a Painleve V function for the characteristic function of tr(U) and (using recent results of Baik, Deift and Johansson) an expression …
Twisted moments of characteristic polynomials of random matrices in the unitary group
本文提出酉群随机矩阵特征多项式扭曲矩的严格理论，证明相关 Schur 多项式恒等式，为黎曼 zeta 函数矩的启发式方法提供概念验证。
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