2025/07/24 by Colin Defant, Defant, Colin, Yuhan Jiang +11
Engineering · Medicine · #05E16 #05E18 #06A07 #06B10 #16E30 #Combinatorics (math.CO) #FOS: Mathematics #Mechanical stress and fatigue analysis #Sports injuries and prevention
paper · pdf · doi:10.48550/arxiv.2507.18230
openalex publication_date 2025/07/24 · openalex created_date 2025/10/16 · openalex updated_date 2026/07/28
Given a linear extension σ of a finite poset R, we consider the permutation matrix indexing the Schubert cell containing the Cartan matrix of R with respect to σ. This yields a bijection Echσ\colon R→ R that we call echelonmotion; it is the inverse of the Coxeter permutation studied by Klász, Marczinzik, and Thomas. Those authors proved that echelonmotion agrees with rowmotion when R is a distributive lattice. We generalize this result to semidistributive lattices. In addition, we prove that every trim lattice has a linear extension with respect to which echelonmotion agrees with rowmotion. We also show that echelonmotion on an Eulerian poset (with respect to any linear extension) is an involution. Finally, we initiate the study of echelon-independent posets, which are posets for which echelonmotion is independent of the chosen linear extension. We prove that a lattice is echelon-independent if and only if it is semidistributive. Moreover, we show that echelon-independent connected posets are bounded and have semidistributive MacNeille completions.