SL$ _n(\mathbb{Z}) $-normalizer of a principal congruence subgroup

<abstract><p>Let SL$ _n(\mathbb{Q}) $ be the set of matrices of order $ n $ over the rational numbers with determinant equal to 1. We study in this paper a subset $ \Lambda $ of SL$ _n(\mathbb{Q}) $, where a matrix $ B $ belongs to $ \Lambda $ if and only if the conjugate subgroup $ B\Gamma_q(n)B^{-1} $ of principal congruence subgroup $ \Gamma_q(n) $ of lever $ q $ is contained in modular group SL$ _n(\mathbb{Z}) $. The notion of least common denominator (LCD for convenience) of a rational matrix plays a key role in determining whether <italic>B</italic> belongs to $ \Lambda $. We show that LCD can be described by the prime decomposition of $ q $. Generally $ \Lambda $ is not a group, and not even a subsemigroup of SL$ _n(\mathbb{Q}) $. Nevertheless, for the case $ n = 2 $, we present two families of subgroups that are maximal in $ \Lambda $ in this paper.</p></abstract>

Matrix orthogonality in the plane versus scalar orthogonality in a Riemann surface

We consider a non-Hermitian matrix orthogonality on a contour in the complex plane. Given a diagonalizable and rational matrix valued weight, we show that the Christoffel–Darboux (CD) kernel, which is built in terms of matrix orthogonal polynomials, is equivalent to a scalar valued reproducing kernel of meromorphic functions in a Riemann surface. If this Riemann surface has genus $0$, then the matrix valued CD kernel is equivalent to a scalar reproducing kernel of polynomials in the plane. Interestingly, this scalar reproducing kernel is not necessarily a scalar CD kernel. As an application of our result, we show that the correlation kernel of certain doubly periodic lozenge tiling models admits a double contour integral representation involving only a scalar CD kernel. This simplifies a formula of Duits and Kuijlaars.

Generalized Fiedler pencils with repetition for rational matrix functions

We introduce generalized Fiedler pencil with repetition(GFPR) for an n x n rational matrix function G(?) relative to a realization of G(?). We show that a GFPR is a linearization of G(?) when the realization of G(?) is minimal and describe recovery of eigenvectors of G(?) from those of the GFPRs. In fact, we show that a GFPR allows operation-free recovery of eigenvectors of G(?). We describe construction of a symmetric GFPR when G(?) is symmetric. We also construct GFPR for the Rosenbrock system matrix S(?) associated with an linear time-invariant (LTI) state-space system and show that the GFPR are Rosenbrock linearizations of S(?). We also describe recovery of eigenvectors of S(?) from those of the GFPR for S(?). Finally, We analyze operation-free Symmetric/self-adjoint structure Fiedler pencils of system matrix S(?) and rational matrix G(?). We show that structure pencils are linearizations of G(?).