Flips in symmetric separated set-systems

TL;DR

This paper extends known connectivity results of maximal separated collections to symmetric set-systems, introducing symmetric flips and exploring related geometric structures, with implications for symmetric higher Bruhat orders.

Contribution

It establishes that all maximal symmetric separated collections are connected via symmetric flips and explores their geometric representations, extending prior non-symmetric results.

Findings

Maximal symmetric strongly separated collections are connected via symmetric flips.

Maximal symmetric r-separated collections are connected when n and r are even.

The results relate to symmetric higher Bruhat orders and geometric tilings.

Abstract

For a positive integer $n$, a collection $S$ of subsets of $[n]=\{1,\ldots,n\}$ is called symmetric if $X\in S$ implies $X^\ast\in S$, where $X^\ast:=\{i\in [n]\colon n-i+1\notin X\}$ (the involution $\ast$ was introduced by Karpman). Leclerc and Zelevinsky showed that the set of maximal strongly (resp. weakly) separated collections in $2^{[n]}$ is connected via flips, or mutations, ``in the presence of six (resp. four) witnesses''. We give a symmetric analog of those results, by showing that each maximal symmetric strongly (weakly) separated collection in $2^{[n]}$ can be obtained from any other one by a series of special symmetric local transformations, so-called symmetric flips. Also we establish the connectedness via symmetric flips for the class of maximal symmetric $r$-separated collections in $2^{[n]}$ when $n,r$ are even (where sets $A,B\subseteq [n]$ are called $r$-separated if there are no elements $i_0<i_1< \cdots <i_{r+1}$ in $[n]$ which alternate in $A\setminus B$ and $B\setminus A$). This is related to a symmetric version of higher Bruhat orders. These results are obtained as consequences of our study of related geometric objects: symmetric rhombus and combined tilings and symmetric cubillages.