We provide a nontrivial upper bound for the nonnegative rank of rank-three matrices which allows us to prove that [6(n+1)/7] linear inequalities suffice to describe a convex n-gon up to a linear projection.
We discuss the problem of counting vertices in Gelfand--Zetlin polytopes. Namely, we deduce a partial differential equation with constant coefficients on the exponential generating function for these numbers. For some particular classes of Gelfand-Zetlin polytopes, the number of vertices can be given by explicit formulas.
The goal of the present paper is to extend the mitosis algorithm, originally developed by Ezra Miller and Allen Knutson for the case of Schubert polynomials, to the case of Grothendieck polynomials. In addition we will also use this algorithm to construct a short combinatorial proof of Fomin–Kirillov's formula for the coefficients of Grothendieck polynomials.
The Cherednik–Orr conjecture expresses the t →∞limit of the nonsymmetric Macdonald polynomials in terms of the PBW twisted characters of the affine level one Demazure modules. We prove this conjecture in several special cases.
We consider partial sum rules for the homogeneous limit of the solution of the q-deformed Knizhnik–Zamolodchikov equation with reflecting boundaries in the Dyck path representation of the Temperley–Lieb algebra. We show that these partial sums arise in a solution of the discrete Hirota equation, and prove that they are the generating functions of τ 2-weighted punctured cyclically symmetric transpose complement plane partitions where τ =−(q+q−1). In the cases of no or minimal punctures, we prove that these generating functions coincide with τ 2-enumerations of vertically symmetric alternating sign matrices and modifications thereof.