$c$-Birkhoff polytopes

TL;DR

The paper defines the $c$-Birkhoff polytope ${ m Birk}(c)$ for a Coxeter element $c$ in type $A_n$ and proves it is unimodularly equivalent to the order polytope ${ m O}({ m Heap}({ m sort}_c(w_0)))$ of the heap of the $c$-sorting word of the longest permutation. This establishes a direct bridge between pattern-avoiding Birkhoff subpolytopes and poset geometry, tying their volumes to the number of linear extensions of the heap and, equivalently, to longest chains in the type $A$ $c$-Cambrian lattice. The construction uses a lattice-preserving projection ${ abla}_c$ and a unitriangular transformation ${U}_c$ to realize the unimodular equivalence explicitly, with special cases recovering Davis and Sagan's results for the Tamari orientation. The work yields both a conceptual framework and concrete tools for computing volumes and understanding the combinatorial structure underlying these polytopes, and it points toward further exploration of reduced words and other types.

Abstract

In a 2018 paper, Davis and Sagan studied several pattern-avoiding polytopes. They found that a particular pattern-avoiding Birkhoff polytope had the same normalized volume as the order polytope of a certain poset, leading them to ask if the two polytopes were unimodularly equivalent. Motivated by Davis and Sagan's question, in this paper we define a pattern-avoiding Birkhoff polytope called a $c$-Birkhoff polytope for each Coxeter element $c$ of the symmetric group. We then show that the $c$-Birkhoff polytope is unimodularly equivalent to the order polytope of the heap poset of the $c$-sorting word of the longest permutation. When $c=s_1s_2\dots s_{n}$, this result recovers an affirmative answer to Davis and Sagan's question. Another consequence of this result is that the normalized volume of the $c$-Birkhoff polytope is the number of the longest chains in the (type A) $c$-Cambrian lattice.

Figures (12)

• Figure 1: Left: Hasse diagram of the underlying poset of ${\sf Heap}([1214321432])$. Right: the heap diagram of ${\sf Heap}([1214321432])$, with each label $j$ replaced by $s_j$.
• Figure 2: Left: The zero relations of \ref{['prop:ZeroRelationsOnMatrix']} in the permutation matrix of a $c$-singleton for $c = [1432657]$. Right: The permutation matrix for $s_1 s_4 s_3 s_2$, a $c$-singleton for this $c$.
• Figure 3: Left: We depict the projection $\Pi_c$ for $c = [1432657]$. We also place red X's in the entries which are guaranteed to be zero by \ref{['prop:ZeroRelationsOnMatrix']}. Right: We draw the permutation matrix for $s_1s_4s_3s_2$ and circle the entries which are recorded by $\Pi_c$.
• Figure 4: Left: Algorithm for constructing the heap diagram for $H_c={\sf Heap}(\mathop{\mathrm{{\mathcal{R}}_c}}\nolimits)$ for $c=\left[{4321657}\right]=(1~ \underline{5}~\underline{7} ~ 8 ~ \overline{6}~\overline{4}~\overline{3}~ \overline{2})$ (equivalently, the lower-barred letters are $\underline{5}$, $\underline{7}$, and the upper-barred letters are $\overline{6}, \overline{4}, \overline{3}, \overline{2}$) in $A_7$. Right: The heap diagram for $H_c$.
• Figure 5: Left: Heap of $s_1s_2s_3s_4$-sorting word of the longest element $w_0$ in $A_4$. Right: Heap of $s_1s_2s_3s_4 \dots s_n$-sorting word of the longest element $w_0$ in $A_n$.

Theorems & Definitions (125)

• Definition 2.1
• Example 2.2
• Remark 2.3
• Definition 2.4
• Proposition 2.5
• Example 2.6
• Remark 2.8
• Example 2.9
• Proposition 2.10: Corollary 4.4 of reading07-clusters-paper4
• Example 2.11