Combinatorial Expansions of Macdonald and LLT Polynomials / Alexander Vetter.

Format:

Book

Thesis/Dissertation

Author/Creator:

Vetter, Alexander, author.

Contributor:

University of Pennsylvania. Mathematics, degree granting institution.

Language:

English

Subjects (All):

Mathematics.

Applied mathematics.

Mathematics--Penn dissertations.

Penn dissertations--Mathematics.

Local Subjects:

Mathematics.

Applied mathematics.

Mathematics--Penn dissertations.

Penn dissertations--Mathematics.

Physical Description:

1 online resource (128 pages)

Contained In:

Dissertations Abstracts International 85-12B.

Place of Publication:

[Philadelphia, Pennsylvania] : University of Pennsylvania, 2022.

Ann Arbor : ProQuest Dissertations & Theses, 2024

Language Note:

English

Summary:

In 1987, Ian Macdonald introduced a special family of symmetric polynomials Hµ(X; q, t). These are now known as Macdonald polynomials, written as Hµ(X; q, t) = ∑λ⊢n Kλ,µ(q, t)sλ(X), a sum over Schur functions sλ(X), a basis for the ring of symmetric functions. Macondald conjectured that Kλ,µ(q, t) ∈ ℕ[q, t], i.e., have positive coefficients. Shortly after, a more natural form of these polynomials was introduced, H˜µ(X; q, t). Written in the Schur basis, H˜µ(X; q, t) = ∑λ⊢n K˜λ,µ(q, t)sλ(X) where K˜λ,µ(q, t) = t n(µ)Kλ,µ(q, 1/t). In 2001, using algebraic geometry, Mark Haiman showed K˜λ,µ(q, t) ∈ ℕ[q, t]. Since then, it has been a major open problem to find a combinatorial interpretation for K˜λ,µ(q, t). We prove a new formula for K˜λ,µ(q, t) when µ = (n−k−1, 2, 1k−1 ) in terms of statistic on Standard Young Tableau. Using this formula, we then prove a special case of a conjecture due to Lynne Butler in 1994 on the change of Schur coefficients from a hook shape to an augmented hook shape. In 1997, Alain Lascoux, Bernard Leclerc, and Jean-Yves Thibon introduced a new family of symmetric polynomials, now known as LLT polynomials. In 2005, Jim Haglund, Mark Haiman, and Nick Loehr showed how to write Macdonald polynomials as a sum of LLT polynomials. Thus, a combinatorial formula for Macdonald polynomials can be derived from a combinatorial formula for LLT polynomials. In 2020, Alex Abreu and Antonio Nigro showed that if G is an indifference graph, then LLTG(q) = ∑ σ≤m(q − 1)n−ℓ(λ(σ))q wtG(σ) eλ(σ) . Using this expansion of the LLT polynomials into the e-basis, we prove a combinatorial formula for the coefficients of sλ when λ = (n − k, 1k ) or λ = (n − k − 1, 2, 1k−1 ).

Notes:

Source: Dissertations Abstracts International, Volume: 85-12, Section: B.

Advisors: Haglund, Jim; Committee members: Hartmann, Julia; Blasiak, Jonah.

Department: Mathematics.

Ph.D. University of Pennsylvania 2024.

Local Notes:

School code: 0175

ISBN:

9798382830971

Access Restriction:

Restricted for use by site license.