2023/06/26 by Dong-Hyun Kim, Seung Jin Lee, Kim, Donghyun +3
Biochemistry, Genetics and Molecular Biology · Mathematics · #05E05 #05E10 #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Biochemical and Structural Characterization #Combinatorics (math.CO) #FOS: Mathematics #Representation Theory (math.RT)
paper · pdf · doi:10.48550/arxiv.2306.14371
openalex publication_date 2023/06/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01
We initiate the study of the Macdonald intersection polynomials Iμ(1),…,μ(k)[X;q,t], which are indexed by k-tuples of partitions μ(1),…,μ(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 Iμ(1),…,μ(k) and the character ∇ ek-1 of diagonal coinvariant algebra by employing the plethystic formula for the Macdonald polynomials of Garsia--Haiman--Tesler. Furthermore, we establish a connection between Iμ(1),…,μ(k) and the shuffle formula Dk-1[X;q,t], utilizing novel combinatorial tools such as the column exchange rule, a new fermionic formula for the shuffle formula, and the lightning bolt formula for Macdonald intersection polynomials. Notably, our findings provide a new proof for the shuffle theorem.