Henrique F. da Cruz

Let $A$ be an $np \times np$ positive semidefinite matrix partitioned into $p \times p$ blocks $A_{ij}$, and $P$ be the $n \times n$ matrix with $p_{ij} = {\rm per}(A_{ij})$. Marcus conjectures in 1965 that ${\rm per}(A) \geq {\rm per}(P)$, which remains open in general. In this paper, we prove a generalized Neuberger identity, which provides an expression for the norm of the symmetric product of tensors. The proof involves a careful analysis of double cosets of symmetric groups with respect to Young subgroups. Then, we leverage it to the Marcus's conjecture for block tridiagonal matrices.

Let $$\varvec{\mathcal {A}(n,k)}$$A ( n , k ) represent the collection of all $$\varvec{n\times n}$$n × n zero-and-one matrices, with the sum of all rows and columns equalling $$\varvec{k}$$k . This set can be ordered by an extension of the classical Bruhat order for permutations, seen as permutation matrices. The Bruhat order on $$\varvec{\mathcal {A}(n,k)}$$A ( n , k ) differs from the Bruhat order on permutations matrices not being, in general, graded, which results in some intriguing issues. In this paper, we focus on the maximum length of antichains in $$\varvec{\mathcal {A}(n,k)}$$A ( n , k ) with the Bruhat order. The crucial fact that allows us to obtain our main results is that two distinct matrices in $$\varvec{\mathcal {A}(n,k)}$$A ( n , k ) with an identical number of inversions cannot be compared using the Bruhat order. We construct sets of matrices in $$\varvec{\mathcal {A}(n,k)}$$A ( n , k ) so that each set consists of matrices with the same number of inversions. These sets are hence antichains in $$\varvec{\mathcal {A}(n,k)}$$A ( n , k ) . We use these sets to deduce lower bounds for the maximum length of antichains in these partially ordered sets.

Let $R$ and $S$ be two sequences of positive integers in nonincreasing order having the same sum. We denote by ${\cal A}(R,S)$ 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 ${\cal A}(R,S)$ 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 ${\cal A}(R,S)$. 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 ${\cal A}(R,S)$.

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.