2020/06/10 by Alex Fink, Karola Mészáros, Avery St. Dizier · 1 citation
Mathematics · Chemistry · #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Molecular spectroscopy and chirality
paper · pdf · doi:10.1007/s00209-020-02544-2
Abstract We prove that if σ ∈ Sm <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>σ</mml:mi> <mml:mo>∈</mml:mo> <mml:msub> <mml:mi>S</mml:mi> <mml:mi>m</mml:mi> </mml:msub> </mml:mrow> </mml:math> is a pattern of w ∈ Sn <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>w</mml:mi> <mml:mo>∈</mml:mo> <mml:msub> <mml:mi>S</mml:mi> <mml:mi>n</mml:mi> </mml:msub> </mml:mrow> </mml:math> , then we can express the Schubert polynomial \mathfrak Sw <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>S</mml:mi> <mml:mi>w</mml:mi> </mml:msub> </mml:math> as a monomial times \mathfrak Sσ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>S</mml:mi> <mml:mi>σ</mml:mi> </mml:msub> </mml:math> (in reindexed variables) plus a polynomial with nonnegative coefficients. This implies that the set of permutations whose Schubert polynomials have all their coefficients equal to either 0 or 1 is closed under pattern containment. Using Magyar’s orthodontia, we characterize this class by a list of twelve avoided patterns. We also give other equivalent conditions on \mathfrak Sw <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>S</mml:mi> <mml:mi>w</mml:mi> </mml:msub> </mml:math> being zero-one. In this case, the Schubert polynomial \mathfrak Sw <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>S</mml:mi> <mml:mi>w</mml:mi> </mml:msub> </mml:math> is equal to the integer point transform of a generalized permutahedron.