On Maximum Chains in the Bruhat Order of A(n,2)

Abstract

Let $\mathcal{A}(R,S)$ denote the class of all matrices of zeros and ones with row sum vector $R$ and column sum vector~$S$. We introduce the notion of an inversion in a $(0,1)$--matrix. This definition extends the standard notion of an inversion of a permutation, in the sense that both notions agree on the class of permutation matrices. We prove that the number of inversions in a $(0,1)$--matrix is monotonic with respect to the secondary Bruhat order of the class $\mathcal{A}(R,S)$. We apply this result in establishing the maximum length of a chain in the Bruhat order of the class $\mathcal{A}(n,2)$ of $(0,1)$--matrices of order $n$ in which every row and every column has a sum of~$2$. We give algorithmic constructions of chains of maximum length in the Bruhat order of $\mathcal{A}(n,2)$.

Figures (5)

• Figure 1: A chain of length $6$ from $P_5$ to $Z$ in the Bruhat order of $\mathcal{A}(5,2)$. Dots represent zeros.
• Figure 2: A chain of length $23$ from $Z$ to $Q_5$ in the Bruhat order of $\mathcal{A}(5,2)$. Dots represent zeros.
• Figure 3: The algorithm in the proof of Proposition \ref{['prop:chainEven']} for $n=8$. Here $J$ stands for $J_2$ and each dot represents a $2\times 2$ block of zeros. The numbers above arrows indicate the length of a chain connecting the two matrices.
• Figure 4: The algorithm in the proof of Proposition \ref{['prop:chainOdd']} for $n=9$. Here $J$ and $F$ stand for $J_2$ and $F_3$ respectively, and each dot represents a block of zeros of the appropriate size. The numbers above arrows indicate the length of a chain connecting the two matrices.
• Figure 5: The Bruhat graph of the class $\mathcal{A}({R,R})$ where $R=(2,2,1)$. There is an arc from $A_i$ to $A_j$ when $A_i\preceq_{B} A_j$.

Theorems & Definitions (16)

• Definition 1
• Lemma 2
• proof
• Lemma 3
• Lemma 5
• Lemma 6
• proof
• Lemma 7
• proof
• Proposition 8