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Shuffle formula in science fiction for Macdonald polynomials

Part of: Algebraic combinatorics

Published online by Cambridge University Press:  28 August 2025

Donghyun Kim ,
Seung Jin Lee  and
Jaeseong Oh
Donghyun Kim
Affiliation:
Department of Mathematical Sciences, Seoul National University, Seoul 08826, Republic of Korea hyun920310@snu.ac.kr
Seung Jin Lee
Affiliation:
Department of Mathematical Sciences, Research institute of Mathematics, Seoul National University, Seoul 08826, Republic of Korea lsjin@snu.ac.kr
Jaeseong Oh
Affiliation:
June E Huh Center for Mathematical Challenges, Korea Institute for Advanced Study, Seoul 02455, Republic of Korea jsoh@kias.re.kr
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Abstract

We study the Macdonald intersection polynomials $\operatorname {I}_{\mu ^{(1)},\dots ,\mu ^{(k)}}[X;q,t]$, which are indexed by $k$-tuples of partitions $\mu ^{(1)},\dots ,\mu ^{(k)}$. These polynomials are conjectured to be equal to the bigraded Frobenius characteristic of the intersection of Garsia–Haiman modules, as proposed by the science fiction conjecture of Bergeron and Garsia. In this work, we establish the vanishing identity and the shape independence of the Macdonald intersection polynomials. Additionally, we unveil a remarkable connection between $\operatorname {I}_{\mu ^{(1)},\dots ,\mu ^{(k)}}$ and the character $\nabla e_{k-1}$ of diagonal coinvariant algebra by employing the plethystic formula for the Macdonald polynomials of Garsia, Haiman, and Tesler. Furthermore, we establish a connection between $\operatorname {I}_{\mu ^{(1)},\dots ,\mu ^{(k)}}$ and the shuffle formula $D_{k-1}[X;q,t]$, utilizing novel combinatorial tools such as the column exchange rule and the lightning bolt formula for Macdonald intersection polynomials. Notably, our findings provide a new proof for the shuffle theorem.

Keywords

Macdonald polynomialsMacdonald intersection polynomialsscience fiction conjectureshuffle theoremKreweras numbers

MSC classification

Primary: 05E05: Symmetric functions and generalizations 05E10: Combinatorial aspects of representation theory

Information

Type
Research Article
Information
Compositio Mathematica , Volume 161 , Issue 5 , May 2025 , pp. 1075 - 1127
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Copyright
© The Author(s), 2025. The publishing rights in this article are licensed to Foundation Compositio Mathematica under an exclusive licence

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