IPAC: A raising operator formula for Macdonald polynomials
I gave a series of three virtual talks in the Important Papers in Algebraic Combinatorics Seminar about the paper “A raising operator formula for Macdonald polynomials,” joint with Jonah Blasiak, Mark Haiman, Jennifer Morse, and Anna Pun. In the first two lectures, I gave background and set the scene for the tools that we used to prove this formula. In the third, I presented the actual formula and gave some ideas about how to think about it.
Abstract: In the theory of symmetric functions, it is customary to give both combinatorial and algebraic descriptions of various important bases. Perhaps the most notable example is the Schur function basis, which can be described as a weight generating function of semistandard tableaux of a given shape, but also via the Weyl Character Formula. Equally important are rules governing how to write certain distinguished bases in terms of other bases. Many classical examples of these change of basis rules appear in combinatorial forms, often using variations of semistandard tableaux formulas. However, another old but useful form of these change of basis rules occurs using the lesser known concept of “raising operators.” In particular, there are raising operator formulas relating Schur functions and complete homogeneous polynomials, but also Schur functions and Hall-Littlewood polynomials. Indeed, raising operator formulas for Hall-Littlewood polynomials in terms of Schur polynomials can expose some nice properties that may be difficult to see via other methods.
Modified Macdonald polynomials serve as a generalization of many important bases of symmetric polynomials, including Hall-Littlewood polynomials. However, despite much study over the last 30 years, raising operator formulas for modified Macdonald polynomials remained absent. In this series of talks, I will discuss an explicit raising operator formula for Macdonald polynomials my collaborators and I posted in July 2023. The talks will start with putting raising operators on a rigorous footing and demonstrate how to work with them via examples. Next, we will discuss modified Macdonald polynomials purely in combinatorial terms via the Haglund-Haiman-Loehr formula and introduce LLT polynomials from a combinatorial perspective. Finally, we will discuss how to combine the Haglund-Haiman-Loehr formula with a raising operator formula for a the Macdonald \(\nabla\) operator applied to LLT polynomials in order to recover the desired raising operator formula for Macdonald polynomials. Time permitting, we will also further discuss nice properties this raising operator formula exposes and how it also gives a new generalization of Macdonald polynomials.
Date published: Tuesday, June 25, 2024