The characterization of the Bruhat shadow of a permutation matrix was first established by Brualdi and Dahl in [2, Theorem 3.5].

*[2] 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 obtained from A by striking out the rows and columns of its extreme positions satisfies I*_{m} ⩽ *A*'.

Taking into account Theorem 2.1, given an *r*-sequence (2) and an 𝓵-sequence (3), we may characterize explicitly these sequences such that both define the Bruhat shadow of an indecomposable permutation matrix.

*The two sequences* (2) *and* (3) *are, respectively, the r**- and* 𝓁*-sequences of an indecomposable permutation matrix if and only if*

(i)

{*i*_{1}, ...,*i*_{p}} ∩ {*j*_{1}, ...,*j*_{q}} = ∅,

(ii)

{*r*_{i1}, ...,*r*_{ip}} ∩ {𝓁_{j1}, ...,𝓁_{jq}} = ∅ *, and*

(iii)

*if κ*_{x} *is the x**-th lowest integer in* {1, ...,*n*} *not in* {*r*_{i1}, ...,*r*_{ip}, 𝓁_{j1}, ...,𝓁_{jq}} *, and τ*_{x} *is the x**-th lowest integer in* {1, ..., *n*} *not in* {*i*_{1}, ..., *i*_{p}, *j*_{1}, ..., *j*_{q}} *, then* 𝓁_{τx} < *κ*_{x} < *r*_{τx}.

Let us start by assuming that (2) and (3) are, respectively, the *r*- and 𝓁-sequences of a permutation matrix *P*. Since *P* is indecomposable and contains exactly one entry 1 in each row and each column and 0’s elsewhere, the assertion (i) and (ii) are immediate. According to Theorem 2.1, *m* = *n* − *p* − *q*, the diagonal entries of *A*' are equal to 1. Note that the diagonal entries of *A*' are the (*τ*_{t}, *κ*_{t})-entry in *A*, for *t* = 1, ..., *m*, thus the inequalities in (iii) follow clearly.

Conversely, condition (i) guarantees that there exists exactly one 1 in each row, while condition (ii) says that there is exactly one 1 in each column. If {*r*_{i1}, ..., *r*_{ip}, *𝓁*_{j1}, ..., *𝓁*_{jq}} = {1, ..., *n*}, then the result comes immediately. Otherwise, let *κ*_{1} be the lowest positive integer not in {*r*_{i1}, ..., *r*_{ip}, *𝓁*_{j1}, ..., *𝓁*_{jq}} and let *τ*_{1} be the lowest positive integer not in {*i*_{1}, ..., *i*_{p}, *j*_{1}, ..., *j*_{q}}. From condition (iii), we have *𝓁*_{τ1} < *κ*_{1} < *r*_{τ1}, which means that the (*τ*_{1}, *κ*_{1})-entry of *A* is 1. If {*r*_{i1}, ..., *r*_{ip}, *𝓁*_{j1}, ..., *𝓁*_{jq}, *κ*_{1}} = {1, ..., *n*}, then in Theorem 2.1, *m* = 1 and *A*' = *A*[*τ*_{1}, *κ*_{1}], the submatrix of *A* resulting from the retention of the row and column indexed by *τ*_{1} and *κ*_{1}, respectively. Otherwise, we proceed analogously to choose *κ*_{2} as the lowest positive integer not in {*r*_{i1}, ..., *r*_{ip}, *𝓁*_{j1}, ..., *𝓁*_{jq}, *κ*_{1}}, and let *τ*_{2} be the lowest positive integer not in {*i*_{1}, ..., *i*_{p}, *j*_{1}, ..., *j*_{q}, *τ*_{1}}. From condition (iii), we have *𝓁*_{τ2} < *κ*_{2} < *r*_{τ2}, i.e., the (*τ*_{2}, *κ*_{2})-entry of *A* is 1. In the end we would get the submatrix *A*' = *A*[*τ*_{1}, ..., *τ*_{m}, *κ*_{1}, ..., *κ*_{m}] with *I*_{m} ⩽ *A*'. □

We remark that in the condition (iii) of the previous theorem, we are considering the integers in {1, ..., *n*} ordered increasingly. □

*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, *i*_{1} = 1, *i*_{2} = 2*, j*_{1} = 3, *j*_{2} = 5*, and j*_{3} = 7*. Now, we have* {1, 2} ∩ {3, 5; 7} = ∅ *and* {5, 7} ∩ {1, 2, 4} = ∅. *Moreover*, *κ*_{1} = 3 *and κ*_{2} = 6. *On the other hand, τ*_{1} = 4 *and* *τ*_{2} = 6. *It is straightforward to see now that* *𝓁*_{4} < 3 < *r*_{4} *and* 𝓁_{6} < 6 < *r*_{6}. *In another words, following the notation of [2], interchanging the roles of r and* 𝓁 *, as we pointed out previously, we have*

*and* σ

*gives rise to the permutation matrix whose Bruhat shadow is*
$$\left(\begin{array}{ccccccc}1\hfill & 1\hfill & 1\hfill & 1\hfill & 1\hfill & 0\hfill & 0\hfill \\ 1\hfill & 1\hfill & 1\hfill & 1\hfill & 1\hfill & 1\hfill & 1\hfill \\ 1\hfill & 1\hfill & 1\hfill & 1\hfill & 1\hfill & 1\hfill & 1\hfill \\ 0\hfill & 1\hfill & 1\hfill & 1\hfill & 1\hfill & 1\hfill & 1\hfill \\ 0\hfill & 1\hfill & 1\hfill & 1\hfill & 1\hfill & 1\hfill & 1\hfill \\ 0\hfill & 0\hfill & 0\hfill & 1\hfill & 1\hfill & 1\hfill & 1\hfill \\ 0\hfill & 0\hfill & 0\hfill & 1\hfill & 1\hfill & 1\hfill & 1\hfill \end{array}\right)\phantom{\rule{thinmathspace}{0ex}}.$$*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*, *i*_{1} = 1, *i*_{2} = 3, *j*_{1} = 2, *j*_{2} = *4*, *j*_{3} = *5, and j*_{4} = 7 (* We have* {1, 3} ∩ {2, 4, 5, 7} = ∅ *and* {6, 7} ∩ {1, 2, 4, 5} = ∅. * Moreover, κ*_{1} = 3 *and τ*_{1} = 6(* However*, 𝓁_{6} < 3 < *r*_{6} *is obviously false*.

Example 2.4 shows that the matrix defined in [2, pp.31]
$$\left(\begin{array}{ccccccc}1\hfill & 1\hfill & 1\hfill & 1\hfill & 1\hfill & 1\hfill & 0\hfill \\ 1\hfill & 1\hfill & 1\hfill & 1\hfill & 1\hfill & 1\hfill & 0\hfill \\ 0\hfill & 1\hfill & 1\hfill & 1\hfill & 1\hfill & 1\hfill & 1\hfill \\ 0\hfill & 1\hfill & 1\hfill & 1\hfill & 1\hfill & 1\hfill & 1\hfill \\ 0\hfill & 0\hfill & 0\hfill & 1\hfill & 1\hfill & 1\hfill & 1\hfill \\ 0\hfill & 0\hfill & 0\hfill & 0\hfill & 1\hfill & 1\hfill & 1\hfill \\ 0\hfill & 0\hfill & 0\hfill & 0\hfill & 1\hfill & 1\hfill & 1\hfill \end{array}\right)\phantom{\rule{thinmathspace}{0ex}}$$
is not the Bruhat shadow of any permutation matrix.

When a permutation matrix is symmetric, its Bruhat shadow will clearly be symmetric. In this case, any characterization depends on one of the sequences defined previously. We will consider, for example, the integral sequence *r* introduced in (2). Then, the 𝓁-sequence of the symmetric permutation matrix is
$$\underset{{r}_{{i}_{1}}}{\underset{\u23df}{{i}_{1},\dots ,{i}_{1}}},\phantom{\rule{thinmathspace}{0ex}}\underset{{r}_{{i}_{2}}\phantom{\rule{thinmathspace}{0ex}}-\phantom{\rule{thinmathspace}{0ex}}{r}_{{i}_{1}}}{\underset{\u23df}{{i}_{2},\dots ,{i}_{2}}},...,\underset{{r}_{{i}_{p}}\phantom{\rule{thinmathspace}{0ex}}-\phantom{\rule{thinmathspace}{0ex}}{r}_{{i}_{p-1}}}{\underset{\u23df}{{i}_{p},\dots ,{i}_{p}}}$$

Note that, in this case, *j*_{t} = *ri*_{t} and 𝓁*j*_{t} = *i*_{t}, for *t* = 1, ..., *p*.

As a consequence of Theorem 2.2, we have the following:

*The integral sequence* (2) *is the r**-sequence of a symmetric permutation matrix if and only if* {*i*_{1}, ..., *i*_{p}} ∩ {*r*_{i1}, ..., *r*_{ip}} = ∅.

We only need to observe that 𝓁_{κ} < *κ* < *r*_{κ}, in particular for each *κ* ∉ {*i*_{1}, ..., *i*_{p}, *r*_{i1}, ..., *r*_{ip}}. □

This corollary considerably simplifies [2, Theorem 3.6].

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