Inner functions of matrix argument and conjugacy classes in unitary groups

This work arose from the following question asked by a student on an electronics technical course in 2012: “Is the Sar-rus rule only valid for determinants of square matrices of order three?” . Responding to this challenging question and questions such as Thomas Muir's and R. Osborn's objection, we give the reader unprecedented material that presents in a didactic way (and with demonstration) a generalization for Sarrus' method applying it to matrices of arbitrary order. We will call the generalization method the generalized sarrus method or the STAF method (acronym for Sarrus-Trovão-Almeida-Ferreira).

In this chapter we treat the problem of factorizing a non-proper rational matrix function. The realization used in the earlier chapters is replaced by $$ V(\lambda ) = I + C(\lambda G - A)^{ - 1} B. $$ Here I=I m is the m × m identity matrix, A and G are square matrices of order n say, and the matrices C and B are of sizes m × n and n × m, respectively. Any rational m × m matrix function W, proper or non-proper, admits such a representation. The representation (4.1) allows us to extend the results obtained in the previous chapter to arbitrary rational matrix functions. As an application we treat the problem to invert a singular integral operator with a rational matrix symbol.

Editorial

The class of square matrices of order n n having a positive determinant and all their minors up to order n − 1 n-1 nonpositive is considered. A characterization of these matrices based on the Cauchon algorithm is presented, which provides an easy test for their recognition. Furthermore, it is shown that all matrices lying between two matrices of this class with respect to the checkerboard ordering are contained in this class, too. For both results, we require that the entry in the bottom-right position is negative.

One considers the hypergroups associated with the HX-groups Z=nZ and with the set of square matrices of order 2, with coefficients in Z=2Z and one calculates their fuzzy grade.

Let R be an integral domain, K its field of fractions, and M n (R) the R-algebra of square matrices of order n and entries from R. In this paper we present two ways of constructing space-time block codes as submodules of M n (R) for n ≥ 1: first, by embedding free associative R-algebras of finite rank n with no zero divisors into M n (R); second, by injecting free R-modules of finite rank n into M n (R) ∩ GL n (K) ∪{ 0}. Some examples of such space-time block codes are given.

The goal of this chapter is to discuss the theory of some special linear maps of \(\mathbb{R}^{2}\) (of the plane) and to bring into light the connection between various problems in plane geometry and the theory of square matrices of order 2.

We consider an irreducible discrete-time Markov process with states represented as (k, i) where k is an M-dimensional vector with non-negative integer entries, and i indicates the state (phase) of the external environment. The number n of phases may be either finite or infinite. One-step transitions of the process from a state (k, i) are limited to states (n, j) such that n ≥ k−1, where 1 represents the vector of all 1s. We assume that for a vector k ≥ 1, the one-step transition probability from a state (k, i) to a state (n, j) may depend on i, j, and n − k, but not on the specific values of k and n. This process can be classified as a Markov chain of M/G/1 type, where the minimum entry of the vector n defines the level of a state (n, j). It is shown that the first passage distribution matrix of such a process, also known as the matrix G, can be expressed through a family of nonnegative square matrices of order n, which is a solution to a system of nonlinear matrix equations.

Matrices originating from certain difference equations

In his seminal Lecture Notes in Mathematics published in 1981, Andrey Zelevinsky introduced a new family of Hopf algebras which he called PSH-algebras. These algebras were designed to capture the representation theory of the symmetric groups and of classical groups over finite fields. The gist of this construction is to translate representation-theoretic operations such as induction and restriction and their parabolic variants to algebra and coalgebra operations such as multiplication and comultiplication. The Mackey formula, for example, is then reincarnated as the Hopf axiom on the algebra side. In this paper we take substantial steps to adapt these ideas for general linear groups over compact discrete valuation rings. We construct an analogous bialgebra that contains a large PSH-algebra that extends Zelevinsky's algebra for the case of general linear groups over finite fields. We prove several base change results relating algebras over extensions of discrete valuation rings.

For a proper holomorphic map F from a ball B n into another ball B N , we give a criterion when a rational map F is equivalent to a polynomial one.As application, we show that proper rational holomorphic maps from B 2 into B N of degree two are equivalent to polynomial maps, and we also give an example of a rational holomorphic maps of degree 3 that are 'almost' linear but are not equivalent to holomorphic polynomial maps.

We study the continuity of archimedean zeta integrals, as well as the lower bound of their operator norms, with respect to a natural family of Sobolev norms arising from the functional calculus of harmonic oscillator. The relevant notion and results will be generalized to archimedean Rankin–Selberg integrals of the tempered principal series representations of general linear groups.

We prove the first direction of a recently posed conjecture by Gan, Gross and Prasad, which predicts branching laws that govern restriction from p -adic GL_n to GL_{n-1} of irreducible smooth representations within the Arthur-type class. We extend this prediction to the full class of unitarizable representations, by exhibiting a combinatorial relation that must be satisfied for any pair of irreducible representations, in which one appears as a quotient of the restriction of the other. We settle the full conjecture for the cases in which either one of the representations in the pair is generic. The method of proof involves a transfer of the problem, using the Bernstein decomposition and the quantum affine Schur–Weyl duality, into the realm of quantum affine algebras. This restatement of the problem allows for an application of the combined power of a result of Hernandez on cyclic modules together with the Lapid–Mínguez criterion from the p -adic setting.

Let $K/F$ be a quadratic extension of $p$-adic fields, $\sigma$ the nontrivial element of the Galois group of $K$ over $F$, and $\pi$ a quasi-square-integrable representation of $GL(n,K)$. Denoting by $\pi^{\vee}$ the smooth contragredient of $\pi$, and by $\pi^{\sigma}$ the representation $\pi\circ \sigma$, we show that the representation of $GL(2n, K)$ obtained by normalized parabolic induction of the representation $\pi^\vee \otimes \pi^\sigma$ is distinguished with respect to $GL(2n,F)$. This is a step towards the classification of distinguished generic representations of general linear groups over $p$-adic fields.

We assume that K is an infinite field, so that there is no difference between formal polynomials and the functions that they give rise to.