The Bruhat rank of binary symmetric staircase pattern Carlos M. da - PowerPoint PPT Presentation

The Bruhat rank of binary symmetric staircase pattern Carlos M. da Fonseca with Zhibin Du Department of Mathematics Kuwait University Kuwait Shanghai Jiao Tong University May 20, 2015 C.M. da Fonseca (Kuwait University) The Bruhat rank of a staircase pattern May 20, 2015 1 / 18

Abstract In this talk we discuss the Bruhat rank of a symmetric ( 0 , 1 ) -matrix of order n with a staircase pattern, total support, and containing I n . Several other related questions are also discussed. Some illustrative examples are presented. C.M. da Fonseca (Kuwait University) The Bruhat rank of a staircase pattern May 20, 2015 2 / 18

The Bruhat shadow Setting � � � � 0 1 1 0 L 2 = and I 2 = , 1 0 0 1 the standard inversion-reducing interchange process applied to a permutation matrix P , replaces a 2 × 2 submatrix equals to L 2 by I 2 , for short, L 2 → I 2 . C.M. da Fonseca (Kuwait University) The Bruhat rank of a staircase pattern May 20, 2015 3 / 18

The Bruhat shadow Setting � � � � 0 1 1 0 L 2 = and I 2 = , 1 0 0 1 the standard inversion-reducing interchange process applied to a permutation matrix P , replaces a 2 × 2 submatrix equals to L 2 by I 2 , for short, L 2 → I 2 . Given two permutation matrices P and Q of the same order, Q is below P in the Bruhat order and written as Q � B P , if Q can be obtained from P by a sequence of L 2 → I 2 interchanges. C.M. da Fonseca (Kuwait University) The Bruhat rank of a staircase pattern May 20, 2015 3 / 18

The Bruhat shadow Setting � � � � 0 1 1 0 L 2 = and I 2 = , 1 0 0 1 the standard inversion-reducing interchange process applied to a permutation matrix P , replaces a 2 × 2 submatrix equals to L 2 by I 2 , for short, L 2 → I 2 . Given two permutation matrices P and Q of the same order, Q is below P in the Bruhat order and written as Q � B P , if Q can be obtained from P by a sequence of L 2 → I 2 interchanges. The Bruhat order in terms of permutation matrices has attracted considerable attention recently. C.M. da Fonseca (Kuwait University) The Bruhat rank of a staircase pattern May 20, 2015 3 / 18

References R.A. Brualdi, G. Dahl, The Bruhat shadow of a permutation matrix, Mathematical papers in honour of E. Marques de S´ a, 25-38, Textos Mat. S´ er. B, 39, University of Coimbra, 2006 C.M. da Fonseca (Kuwait University) The Bruhat rank of a staircase pattern May 20, 2015 4 / 18

References R.A. Brualdi, G. Dahl, The Bruhat shadow of a permutation matrix, Mathematical papers in honour of E. Marques de S´ a, 25-38, Textos Mat. S´ er. B, 39, University of Coimbra, 2006 R.A. Brualdi, L. Deaett, More on the Bruhat order for ( 0 , 1 ) -matrices , Linear Algebra Appl. 421 (2007), no.2-3, 219-232 C.M. da Fonseca (Kuwait University) The Bruhat rank of a staircase pattern May 20, 2015 4 / 18

References R.A. Brualdi, G. Dahl, The Bruhat shadow of a permutation matrix, Mathematical papers in honour of E. Marques de S´ a, 25-38, Textos Mat. S´ er. B, 39, University of Coimbra, 2006 R.A. Brualdi, L. Deaett, More on the Bruhat order for ( 0 , 1 ) -matrices , Linear Algebra Appl. 421 (2007), no.2-3, 219-232 R.A. Brualdi, S.-G. Hwang, A Bruhat order for the class of ( 0 , 1 ) -matrices with row sum vector R and column sum vector S , Electron. J. Linear Algebra 12 (2004/2005), 6-16 C.M. da Fonseca (Kuwait University) The Bruhat rank of a staircase pattern May 20, 2015 4 / 18

References R.A. Brualdi, G. Dahl, The Bruhat shadow of a permutation matrix, Mathematical papers in honour of E. Marques de S´ a, 25-38, Textos Mat. S´ er. B, 39, University of Coimbra, 2006 R.A. Brualdi, L. Deaett, More on the Bruhat order for ( 0 , 1 ) -matrices , Linear Algebra Appl. 421 (2007), no.2-3, 219-232 R.A. Brualdi, S.-G. Hwang, A Bruhat order for the class of ( 0 , 1 ) -matrices with row sum vector R and column sum vector S , Electron. J. Linear Algebra 12 (2004/2005), 6-16 A. Conflitti, C.M. da Fonseca, R. Mamede, The maximal length of a chain in the Bruhat order for a class of binary matrices , Linear Algebra Appl. 436 (2012), no.3, 753-757 C.M. da Fonseca (Kuwait University) The Bruhat rank of a staircase pattern May 20, 2015 4 / 18

The Bruhat shadow If S n denotes the set of all permutation matrices of order n , a nonempty subset I of S n is called a Bruhat ideal if P ∈ I and Q � B P imply that Q ∈ I . C.M. da Fonseca (Kuwait University) The Bruhat rank of a staircase pattern May 20, 2015 5 / 18

The Bruhat shadow If S n denotes the set of all permutation matrices of order n , a nonempty subset I of S n is called a Bruhat ideal if P ∈ I and Q � B P imply that Q ∈ I . A principal Bruhat ideal � P � is an ideal generated by a single permutation matrix P . C.M. da Fonseca (Kuwait University) The Bruhat rank of a staircase pattern May 20, 2015 5 / 18

The Bruhat shadow If S n denotes the set of all permutation matrices of order n , a nonempty subset I of S n is called a Bruhat ideal if P ∈ I and Q � B P imply that Q ∈ I . A principal Bruhat ideal � P � is an ideal generated by a single permutation matrix P . Denoting the Boolean sum of two ( 0 , 1 ) -matrices A and B by A + b B , the Bruhat shadow of I is the matrix S ( I ) = + b { Q ∈ I} . C.M. da Fonseca (Kuwait University) The Bruhat rank of a staircase pattern May 20, 2015 5 / 18

The Bruhat shadow If S n denotes the set of all permutation matrices of order n , a nonempty subset I of S n is called a Bruhat ideal if P ∈ I and Q � B P imply that Q ∈ I . A principal Bruhat ideal � P � is an ideal generated by a single permutation matrix P . Denoting the Boolean sum of two ( 0 , 1 ) -matrices A and B by A + b B , the Bruhat shadow of I is the matrix S ( I ) = + b { Q ∈ I} . If I = � P � , then we simply write S ( P ) : the Bruhat shadow of P . C.M. da Fonseca (Kuwait University) The Bruhat rank of a staircase pattern May 20, 2015 5 / 18

Example I As an example, setting  0 0 1 0 0 0   1 0 0 0 0 0              0 0 0 1 0 0 0 1 0 0 0 0                      1 0 0 0 0 0   0 0 0 0 1 0          P = Q = and ,          0 0 0 0 0 1   0 0 1 0 0 0                      0 1 0 0 0 0 0 0 0 1 0 0                         0 0 0 0 1 0 0 0 0 0 0 1 we have  1 1 1 0 0 0        1 1 1 1 0 0            1 1 1 1 1 0      S ( � P , Q � ) = .      0 1 1 1 1 1            0 1 1 1 1 1            0 0 0 0 1 1  C.M. da Fonseca (Kuwait University) The Bruhat rank of a staircase pattern May 20, 2015 6 / 18

Example I As an example, setting  0 0 1 0 0 0   1 0 0 0 0 0              0 0 0 1 0 0 0 1 0 0 0 0                      1 0 0 0 0 0   0 0 0 0 1 0          P = Q = and ,          0 0 0 0 0 1   0 0 1 0 0 0                      0 1 0 0 0 0 0 0 0 1 0 0                         0 0 0 0 1 0 0 0 0 0 0 1 we have  1 1 1 0 0 0        1 1 1 1 0 0            1 1 1 1 1 0      S ( � P , Q � ) = .      0 1 1 1 1 1            0 1 1 1 1 1            0 0 0 0 1 1  Observe that S ( � P , Q � ) has a staircase pattern with I 6 , P , Q � S ( � P , Q � ) . C.M. da Fonseca (Kuwait University) The Bruhat rank of a staircase pattern May 20, 2015 6 / 18

r - and ℓ -sequences Suppose now that we have a set of indices 1 = i 1 < i 2 < · · · < i p � n , such that r i 1 = r i 1 + 1 = · · · = r i 2 − 1 < r i 2 = r i 2 + 1 = · · · = r i 3 − 1 < · · · < r i p = r i p + 1 = · · · = r n is a sequence with integers in the set { 1 , . . . , n } , and r i � i . C.M. da Fonseca (Kuwait University) The Bruhat rank of a staircase pattern May 20, 2015 7 / 18

r - and ℓ -sequences Suppose now that we have a set of indices 1 = i 1 < i 2 < · · · < i p � n , such that r i 1 = r i 1 + 1 = · · · = r i 2 − 1 < r i 2 = r i 2 + 1 = · · · = r i 3 − 1 < · · · < r i p = r i p + 1 = · · · = r n is a sequence with integers in the set { 1 , . . . , n } , and r i � i . This sequence r = r 1 , . . . , r n is called a right-sequence or, for short, r-sequence . C.M. da Fonseca (Kuwait University) The Bruhat rank of a staircase pattern May 20, 2015 7 / 18