We discuss arrangements of equal minors in totally positive matrices. More precisely, we would like to investigate the structure of possible equalities and inequalities between the minors. We show that arrangements of equals minors of largest value are in bijection with <i>sorted sets</i>, which earlier appeared in the context of <i>alcoved polytopes</i> and Gröbner bases. Maximal arrangements of this form correspond to simplices of the alcoved triangulation of the hypersimplex; and the number of such arrangements equals the <i>Eulerian number</i>. On the other hand, we conjecture and prove in many cases that arrangements of equal minors of smallest value are exactly the <i>weakly separated sets</i>. Weakly separated sets, originally introduced by Leclerc and Zelevinsky, are closely related to the \textitpositive Grassmannian and the associated <i>cluster algebra</i>.

Source : oai:HAL:hal-01207569v1

Volume: DMTCS Proceedings vol. AT, 26th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2014)

Section: Proceedings

Published on: January 1, 2014

Submitted on: November 21, 2016

Keywords: Schur positivity.,Gröbner bases,Eulerian numbers,hypersimplices,affine Coxeter arrangements,alcoved polytopes,thrackles,triangulations,sorted sets,plabic graphs,cluster algebras,Totally positive matrices,minors,the positive Grassmannian,Plücker coordinates,matrix completion problem,weakly separated sets,[INFO.INFO-DM] Computer Science [cs]/Discrete Mathematics [cs.DM],[MATH.MATH-CO] Mathematics [math]/Combinatorics [math.CO]

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