On a conjecture concerning the Bruhat order

We introduce a generalization of alternating sign matrices (ASMs) called multiASMs and develop some of their properties. Classes of multiASMs with specified row and column sum vectors $R$ and $S$ extend the classes of $(0,1)$-matrices with specified $R$ and $S$. The special case when $R=S$ is a constant vector, in particular all 2's, is treated in more detail. We also investigate the polytope spanned by a class of multiASMs. Finally, we discuss the possibility of defining a Bruhat order on a class of multiASMs.

show abstract

“…But note that this is not true for A(R, S) in general. See [12] and some of its references for further information on this issue.…”

Section:  mentioning

confidence: 99%

Multi-alternating sign matrices

Brualdi

Dahl

2023

ELA

View full text Add to dashboard Cite

show abstract

“…The Bruhat order on (0, 1)-matrices is receiving the attention of many researchers, [4,6,10,11,12,13,15,16,17,18,19]. In the recent years several authors have taken Brualdi and Hwang's ideas, and extended the Bruhat order to other classes of matrices than (0, 1)-matrices: the Bruhat order has been studied on the class of tournament matrices with a given score vector, [8], on the class of alternating sign matrices, [9], and on the class of doubly stochastic matrices, [7].…”

Section: Introductionmentioning

confidence: 99%

The Bruhat order on classes of isotopic Latin squares

2020

Self Cite

View full text Add to dashboard Cite

In a previous paper, the authors introduced and studied the Bruhat order in the class of Latin squares of order n. In this paper, we investigate the restriction of the Bruhat order in a class of isotopic Latin squares. We present equivalent conditions for two Latin squares be related by the Bruhat order when one of them is obtained from the other by interchanging rows or columns or symbols. The cover relation is also addressed, and we present orthogonal isotopic Latin squares related by the Bruhat order.

show abstract

“…Doing this they gave rise to two distinct partial order relations on A(R, S). In the last 16 years, many research have focused on several topics of these two partial order relations: conjectures [15], minimal elements [2,4,16], coincidence [12], chains and antichains [9,10,20,21], restrictions of the Bruhat order on subclasses of A(R, S) [8,11], or extensions of one of these orders to other classes of matrices distinct of A(R, S) [5,6,7,13,14].…”

mentioning

confidence: 99%

On the little secondary Bruhat order

Fernandes

Cruz

Salomão³

2021

ELA

Self Cite

View full text Add to dashboard Cite

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 ${\cal A}(R,S)$.

show abstract

On a conjecture concerning the Bruhat order

Cited by 6 publications

References 10 publications

Multi-alternating sign matrices

Multi-alternating sign matrices

The Bruhat order on classes of isotopic Latin squares

On the little secondary Bruhat order

Contact Info

Product

Resources

About