Pairs of matrices that preserve the value of a generalized matrix function on the set of the upper triangular matrices

AbstractLet H be a subgroup of the symmetric group of degree m and let χ be an irreducible character of H. In this paper we give conditions that characterize the pairs of matrices that leave invariant the value of a generalized matrix function associated with H and χ, on the set of the upper triangular matrices

Equality of immanantal decomposable tensors, II

We state a necessary and sufficient condition for equality of nonzero decomposable symmetrized tensors when the symmetrizer is associated to an irreducible character of the symmetric group of degree m.info:eu-repo/semantics/publishedVersio

Convertible subspaces that arise from different numberings of the vertices of a graph

In this paper, we describe subspaces of generalized Hessenberg matrices where the determinant is convertible into the permanent by affixing ± signs. These subspaces can arise from different numberings of the vertices of a graph. With this numbering process, we obtain some well-known sequences of integers. For instance, in the case of a path of length n, we prove that the number of these subspaces is the (n + 1)th Fibonacci number.info:eu-repo/semantics/publishedVersio

Classes of (0,1)-matrices Where the Bruhat Order and the Secondary Bruhat Order Coincide

FCT (UID/MAT/00212/2019) FCT (UID/MAT/00297/2019)Given two nonincreasing integral vectors R and S, with the same sum, we denote by A(R, S) the class of all (0,1)-matrices with row sum vector R, and column sum vector S. The Bruhat order and the Secondary Bruhat order on A(R, S) are both extensions of the classical Bruhat order on Sn, the symmetric group of degree n. These two partial orders on A(R, S) are, in general, different. In this paper we prove that if R = (2,2,…,2) or R = (1,1,…,1), then the Bruhat order and the Secondary Bruhat order on A(R, S) coincide.authorsversionpublishe

On the little secondary bruhat order

CMA and Departamento de Matematica da Faculdade de Ciencias Tecnologia, Universidade Nova de Lisboa, 2829-516 Caparica, Portugal.([email protected]).Partially supported by the Fundacao para a Ciencia e a Tecnologia through the project UID/MAT/04721/2019. Departamento de Matematica da Universidade da Beira Interior, Rua Marques D'Avila and Bolama, 6201-001 Covilha, Portugal ([email protected]).Partially supported by the Fundacao para a Ciencia e a Tecnologia through the project UIDB/MAT/00212/2020.Let R and S be two sequences of positive integers in nonincreasing order having the same sum. We denote by A(R, S) the class of all (0, 1)-matrices having row sum vector R and column sum vector S. Brualdi and Deaett (More on the Bruhat order for (0, 1)-matrices, Linear Algebra Appl., 421:219{232, 2007) suggested the study of the secondary Bruhat order on A(R, S) but with some constraints. In this paper, we study the cover relation and the minimal elements for this partial order relation, which we call the little secondary Bruhat order, on certain classes A(R, S). Moreover, we show that this order is different from the Bruhat order and the secondary Bruhat order. We also study a variant of this order on certain classes of symmetric matrices of A(R, S).publishersversionpublishe

Latin Squares and their Bruhat Order

In this paper we investigate the Bruhat order on the class of Latin squares. We study its cover relation and minimal elements. We prove that the class of Latin squares of order $n$, with $n\not\in\{1,2,4\}$, has at least two minimal elements, and we present a process to construct some minimal Latin squares for this relation

Matrices in A(R,S) with minimum t-term ranks

Fundacao para a Ciencia e a Tecnologia through the projects UID/MAT/00297/2019 and UID/MAT/00212/2019.Let R and S be two sequences of nonnegative integers in nonincreasing order which have the same sum, and let A(R,S) be the class of all (0,1)-matrices which have row sums given by R and column sums given by S. For a positive integer t, the t-term rank of a (0,1)-matrix A is defined as the maximum number of 1's in A with at most one 1 in each column and at most t 1's in each row. In this paper, we address conditions for the existence of a matrix in A(R,S) that realizes all the minimum t-term ranks, for t≥1.authorsversionpublishe

Decomposable λ-critical tensors 1This work was partially supported by Fundação para a Ciência e Tecnologia and was done within the activities of the Centro de Estruturas Lineares e Combinatórias.1

AbstractLet λ=(λ1,…,λs) be a partition of m and let V be a finite dimensional vector space over C. We also denote by λ the irreducible character of Sm associated with the partition λ and by Vλ we denote the symmetry class of tensors associated with λ and V. Let j∈{1,…,λ1} and z∈Vλ. The concept of j-reach of z is defined. Using this concept we introduce the concept of λ-critical element of Vλ. Generalized Plücker polynomials are constructed in a way that the set of their common roots contains the set of the families of components of decomposable λ-critical elements of Vλ. The concepts and results are generalizations of those defined and proved in [4]