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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 In. 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 L2 =

• 1

1

• and

I2 =

• 1

1

• ,

the standard inversion-reducing interchange process applied to a permutation matrix P, replaces a 2 × 2 submatrix equals to L2 by I2, for short, L2 → I2 .

C.M. da Fonseca (Kuwait University) The Bruhat rank of a staircase pattern May 20, 2015 3 / 18

The Bruhat shadow

Setting L2 =

• 1

1

• and

I2 =

• 1

1

• ,

the standard inversion-reducing interchange process applied to a permutation matrix P, replaces a 2 × 2 submatrix equals to L2 by I2, for short, L2 → I2 . 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 L2 → I2 interchanges.

C.M. da Fonseca (Kuwait University) The Bruhat rank of a staircase pattern May 20, 2015 3 / 18

The Bruhat shadow

Setting L2 =

• 1

1

• and

I2 =

• 1

1

• ,

the standard inversion-reducing interchange process applied to a permutation matrix P, replaces a 2 × 2 submatrix equals to L2 by I2, for short, L2 → I2 . 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 L2 → I2 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 Sn denotes the set of all permutation matrices of order n, a nonempty subset I of Sn 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 Sn denotes the set of all permutation matrices of order n, a nonempty subset I of Sn 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 Sn denotes the set of all permutation matrices of order n, a nonempty subset I of Sn 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 Sn denotes the set of all permutation matrices of order n, a nonempty subset I of Sn 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 P =

                         

1 1 1 1 1 1

                         

and Q =

                         

1 1 1 1 1 1

                          ,

we have

S(P, Q) =                          

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 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 P =

                         

1 1 1 1 1 1

                         

and Q =

                         

1 1 1 1 1 1

                          ,

we have

S(P, Q) =                          

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

                          .

Observe that S(P, Q) has a staircase pattern with I6, 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 = i1 < i2 < · · · < ip n, such that ri1 = ri1+1 = · · · = ri2−1 < ri2 = ri2+1 = · · · = ri3−1 < · · · < rip = rip+1 = · · · = rn is a sequence with integers in the set {1, . . . , n}, and ri 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 = i1 < i2 < · · · < ip n, such that ri1 = ri1+1 = · · · = ri2−1 < ri2 = ri2+1 = · · · = ri3−1 < · · · < rip = rip+1 = · · · = rn is a sequence with integers in the set {1, . . . , n}, and ri i. This sequence r = r1, . . . , rn 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

r- and ℓ-sequences

Suppose now that we have a set of indices 1 = i1 < i2 < · · · < ip n, such that ri1 = ri1+1 = · · · = ri2−1 < ri2 = ri2+1 = · · · = ri3−1 < · · · < rip = rip+1 = · · · = rn is a sequence with integers in the set {1, . . . , n}, and ri i. This sequence r = r1, . . . , rn is called a right-sequence or, for short, r-sequence. Analogously, for a given set of indices 1 j1 < j2 < · · · < jq = n, the sequence ℓ = ℓ1, . . . , ℓn with integers in the set {1, . . . , n}, and ℓi i, such that

ℓ1 = ℓ2 = · · · = ℓj1 < ℓj1+1 = · · · = ℓj2 < · · · < ℓjq−1+1 = · · · = ℓjq

is called a left-sequence or, briefly, ℓ-sequence.

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 = i1 < i2 < · · · < ip n, such that ri1 = ri1+1 = · · · = ri2−1 < ri2 = ri2+1 = · · · = ri3−1 < · · · < rip = rip+1 = · · · = rn is a sequence with integers in the set {1, . . . , n}, and ri i. This sequence r = r1, . . . , rn is called a right-sequence or, for short, r-sequence. Analogously, for a given set of indices 1 j1 < j2 < · · · < jq = n, the sequence ℓ = ℓ1, . . . , ℓn with integers in the set {1, . . . , n}, and ℓi i, such that

ℓ1 = ℓ2 = · · · = ℓj1 < ℓj1+1 = · · · = ℓj2 < · · · < ℓjq−1+1 = · · · = ℓjq

is called a left-sequence or, briefly, ℓ-sequence. When aij = 1, for 1 j ri, where r1 · · · rn = n, and aij = 0,

• therwise, we call A = (aij) a full staircase pattern.

C.M. da Fonseca (Kuwait University) The Bruhat rank of a staircase pattern May 20, 2015 7 / 18

Bruhat rank

Brualdi and Dahl defined the Bruhat rank of an n × n (0, 1)-matrix M, denoted by rB(M), as the smallest number of permutation matrices P, such that if I is the Bruhat ideal generated by the P’s, then S(I) = M.

C.M. da Fonseca (Kuwait University) The Bruhat rank of a staircase pattern May 20, 2015 8 / 18

Total support

Recall that of an n × n (0, 1)-matrix M has total support provided that each 1 of M is on a permutation matrix P with P M.

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The problem

Let M be an n × n (0, 1)-matrix, with a staircase pattern, with total support with In M. Determine either a formula for the Bruhat rank of M or an algorithm for evaluating it.

C.M. da Fonseca (Kuwait University) The Bruhat rank of a staircase pattern May 20, 2015 10 / 18

Characterizing the Bruhat shadow

Theorem [Brualdi & Dahl]

Let A be a (0, 1)-matrix. Then A is the Bruhat shadow of a permutation matrix if and only if A has a staircase pattern and the matrix A′ of order m

• btained from A by striking out the rows and columns of its extreme

positions satisfies Im A′.

C.M. da Fonseca (Kuwait University) The Bruhat rank of a staircase pattern May 20, 2015 11 / 18

Characterizing the Bruhat shadow

Theorem [Brualdi & Dahl]

Let A be a (0, 1)-matrix. Then A is the Bruhat shadow of a permutation matrix if and only if A has a staircase pattern and the matrix A′ of order m

• btained from A by striking out the rows and columns of its extreme

positions satisfies Im A′.

Theorem

The two sequences are the r- and ℓ-sequences of an indecomposable permutation matrix if and only if (i) {i1, . . . , ip} ∩ {j1, . . . , jq} = ∅, (ii) {ri1, . . . , rip} ∩ {ℓj1, . . . , ℓjq} = ∅, and (iii) if κx is the x-th lowest integer in {1, . . . , n} not in {ri1, . . . , rip, ℓj1, . . . , ℓjq}, and τx is the x-th lowest integer in {1, . . . , n} not in

{i1, . . . , ip, j1, . . . , jq}, then ℓτx < κx < rτx.

C.M. da Fonseca (Kuwait University) The Bruhat rank of a staircase pattern May 20, 2015 11 / 18

Example II

Let us consider the r-sequence r = 5, 7, 7, 7, 7, 7, 7 and the ℓ-sequence

ℓ = 1, 1, 1, 2, 2, 4, 4. Then p = 2, q = 3, i1 = 1, i2 = 2, j1 = 3, j2 = 5, and

j3 = 7. Now, we have {1, 2} ∩ {3, 5, 7} = ∅ and {5, 7} ∩ {1, 2, 4} = ∅. Moreover, κ1 = 3 and κ2 = 6. On the other hand, τ1 = 4 and τ2 = 6. It is straightforward to see now that ℓ4 < 3 < r4 and ℓ6 < 6 < r6.

C.M. da Fonseca (Kuwait University) The Bruhat rank of a staircase pattern May 20, 2015 12 / 18

Example II

Let us consider the r-sequence r = 5, 7, 7, 7, 7, 7, 7 and the ℓ-sequence

ℓ = 1, 1, 1, 2, 2, 4, 4. Then p = 2, q = 3, i1 = 1, i2 = 2, j1 = 3, j2 = 5, and

j3 = 7. Now, we have {1, 2} ∩ {3, 5, 7} = ∅ and {5, 7} ∩ {1, 2, 4} = ∅. Moreover, κ1 = 3 and κ2 = 6. On the other hand, τ1 = 4 and τ2 = 6. It is straightforward to see now that ℓ4 < 3 < r4 and ℓ6 < 6 < r6. In another words, we have

ℓ

1 1 1 2 2 4 4

σ

5 7 1 3 2 6 4 r 5 7 7 7 7 7 7 and σ gives rise to the permutation matrix whose Bruhat shadow is

C.M. da Fonseca (Kuwait University) The Bruhat rank of a staircase pattern May 20, 2015 12 / 18

Example II

                              

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

                               .

C.M. da Fonseca (Kuwait University) The Bruhat rank of a staircase pattern May 20, 2015 13 / 18

Example III

Let us consider now the r-sequence r = 6, 6, 7, 7, 7, 7, 7 and the

ℓ-sequence ℓ = 1, 1, 2, 2, 4, 5, 5. Here p = 2, q = 4, i1 = 1, i2 = 3, j1 = 2,

j2 = 4, j3 = 5, and j4 = 7. We have {1, 3} ∩ {2, 4, 5, 7} = ∅ and

{6, 7} ∩ {1, 2, 4, 5} = ∅. Moreover, κ1 = 3 and τ1 = 6. However, ℓ6 < 3 < r6

is obviously false.

C.M. da Fonseca (Kuwait University) The Bruhat rank of a staircase pattern May 20, 2015 14 / 18

Example III

Let us consider now the r-sequence r = 6, 6, 7, 7, 7, 7, 7 and the

ℓ-sequence ℓ = 1, 1, 2, 2, 4, 5, 5. Here p = 2, q = 4, i1 = 1, i2 = 3, j1 = 2,

j2 = 4, j3 = 5, and j4 = 7. We have {1, 3} ∩ {2, 4, 5, 7} = ∅ and

{6, 7} ∩ {1, 2, 4, 5} = ∅. Moreover, κ1 = 3 and τ1 = 6. However, ℓ6 < 3 < r6

is obviously false. This example shows that the matrix defined by Brualdi and Dahl

                              

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

                              

is not the Bruhat shadow of any permutation matrix.

C.M. da Fonseca (Kuwait University) The Bruhat rank of a staircase pattern May 20, 2015 14 / 18

Bruhat rank: the symmetric case

Theorem

The Bruhat rank of any symmetric (0, 1)-matrix of order n, with a staircase pattern and In M, is at most 2.

C.M. da Fonseca (Kuwait University) The Bruhat rank of a staircase pattern May 20, 2015 15 / 18

Bruhat rank: the symmetric case II

M =

                                               

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

                                                .

C.M. da Fonseca (Kuwait University) The Bruhat rank of a staircase pattern May 20, 2015 16 / 18

Bruhat rank: the symmetric case III

                                              1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1                                               +b                                               1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1                                              

C.M. da Fonseca (Kuwait University) The Bruhat rank of a staircase pattern May 20, 2015 17 / 18

A nonsymmetric example

M =

                         

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

                          =

C.M. da Fonseca (Kuwait University) The Bruhat rank of a staircase pattern May 20, 2015 18 / 18

A nonsymmetric example

M =

                         

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

                          =                          

1 1 1 1 1 1 1 1 1 1 1 1 1

                          +b                          

1 1 1 1 1 1 1 1 1 1 1 1 1 1

                          +b                          

1 1 1 1 1 1 1 1 1 1 1

                         

C.M. da Fonseca (Kuwait University) The Bruhat rank of a staircase pattern May 20, 2015 18 / 18