Abstract
We show that several families of polynomials defined via fillings of diagrams satisfy linear recurrences under a natural operation on the shape of the diagram. We focus on key polynomials, (also known as Demazure characters), and Demazure atoms. The same technique can be applied to Hall-Littlewood polynomials and dual Grothendieck polynomials.The motivation behind this is that such recurrences are strongly connected with other nice properties, such as interpretations in terms of lattice points in polytopes and divided difference operators.
Highlights
Using a similar technique as in [Ale14], we provide a framework for showing that under certain conditions, polynomials encoding statistics on certain tableaux, or fillings of diagrams, satisfy a linear recurrence
We prove that several of the classical polynomials from representation theory fall into this category, such as Schur polynomials and Hall–Littlewood polynomials
There are several reasons why one would be interested in showing that a such sequence satisfies a linear recurrence: 1. To obtain hints about the existence or non-existence of formulas of certain type
Summary
Using a similar technique as in [Ale14], we provide a framework for showing that under certain conditions, polynomials encoding statistics on certain tableaux, or fillings of diagrams, satisfy a linear recurrence. The skew Schur polynomials can be obtained as lattice points in certain marked order polytopes, called Gelfand–Tsetlin polytopes. Such a polytope interpretation implies the existence of a linear recurrence relation. We sketch two additional proofs in the case of key polynomials, to illustrate that several nice properties imply the existence of a linear recurrence relation. These methods are based on a lattice-point representation and an operator characterization of the key polynomials. Proving the existence or non-existence of linear recurrence relations is an informative step towards alternative combinatorial descriptions of the family of polynomials
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