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Dens, nests and the Loehr-Warrington conjecture
arXiv (Cornell University)
13 Dec 2021
Abstract
In a companion paper, we introduced raising operator series called
Catalanimals. Among them are Schur Catalanimals, which represent Schur
functions inside copies $\Lambda (X^{m,n})\subset \mathcal{E} $ of the algebra
of symmetric functions embedded in the elliptic Hall algebra $\mathcal{E} $ of
Burban and Schiffmann.
Here we obtain a combinatorial formula for symmetric functions given by a
class of Catalanimals that includes the Schur Catalanimals. Our formula is
expressed as a weighted sum of LLT polynomials, with terms indexed by
configurations of nested lattice paths called nests, having endpoints and
bounding constraints controlled by data called a den.
Applied to Schur Catalanimals for the alphabets $X^{m,1}$ with $n=1$, our
`nests in a den' formula proves the combinatorial formula conjectured by Loehr
and Warrington for $\nabla^m s_{\mu }$ as a weighted sum of LLT polynomials
indexed by systems of nested Dyck paths. When $n$ is arbitrary, our formula
establishes an $(m,n)$ version of the Loehr-Warrington conjecture.
In the case where each nest consists of a single lattice path, the nests in a
den formula reduces to our previous shuffle theorem for paths under any line.
Both this and the $(m,n)$ Loehr-Warrington formula generalize the $(km,kn)$
shuffle theorem proven by Carlsson and Mellit (for $n=1$) and Mellit. Our
formula here unifies these two generalizations.
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Details
- Title
- Dens, nests and the Loehr-Warrington conjecture
- Creators
- Jonah BlasiakMark HaimanJennifer MorseAnna PunGeorge Seelinger
- Publication Details
- arXiv (Cornell University)
- Resource Type
- Preprint
- Language
- English
- Academic Unit
- Mathematics
- Other Identifier
- 991021862256404721