The product of any key polynomial with a single-row Schur polynomial expands into key polynomials with coefficients in {-1,0,1}, via a weight-preserving bijection on Kohnert diagrams.
Weak dual equivalence for polynomials
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abstract
We use dual equivalence to give a short, combinatorial proof that Stanley symmetric functions are Schur positive. We introduce weak dual equivalence, and use it to give a short, combinatorial proof that Schubert polynomials are key positive. To demonstrate further the utility of this new tool, we use weak dual equivalence to prove a nonnegative Littlewood--Richardson rule for the key expansion of the product of a key polynomial and a Schur polynomial, and to introduce skew key polynomials that, when skewed by a partition, expand nonnegatively in the key basis.
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2019 1verdicts
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A Pieri rule for Demazure characters of the general linear group
The product of any key polynomial with a single-row Schur polynomial expands into key polynomials with coefficients in {-1,0,1}, via a weight-preserving bijection on Kohnert diagrams.