arXiv subjects

George Seelinger

Publications and source records attributed to George Seelinger.

4 recordsLinked to original sources

arXiv2023

A raising operator formula for Macdonald polynomials

We give an explicit raising operator formula for the modified Macdonald polynomials $\tilde{H}_{μ}(X;q,t)$, which follows from our recent formula for $\nabla$ on an LLT polynomial and the Haglund-Haiman-Loehr formula expressing modified Macdonald polynomials as sums of LLT polynomials. Our method just as easily yields a formula for a family of symmetric functions $\tilde{H}^{1,n}(X;q,t)$ that we call $1,n$-Macdonald polynomials, which reduce to a scalar multiple of $\tilde{H}_μ(X;q,t)$ when $n=1$. We conjecture that the coefficients of $1,n$-Macdonald polynomials in terms of Schur functions belong to $\mathbb{N}[q,t]$, generalizing Macdonald positivity.

math.CO↗

arXiv2021

LLT polynomials in the Schiffmann algebra

We identify certain combinatorially defined rational functions which, under the shuffle to Schiffmann algebra isomorphism, map to LLT polynomials in any of the distinguished copies $Λ(X^{m,n})\subset \mathcal{E}$ of the algebra of symmetric functions embedded in the elliptic Hall algebra $\mathcal{E}$ of Burban and Schiffmann. As a corollary, we deduce an explicit raising operator formula for the $\nabla$ operator applied to any LLT polynomial. In particular, we obtain a formula for $\nabla ^m s_λ$ which serves as a starting point for our proof of the Loehr-Warrington conjecture in a companion paper to this one.

math.CO↗

arXiv2021

Dens, nests and the Loehr-Warrington conjecture

In a companion paper, we introduced raising operator series called Catalanimals. Among them are Schur Catalanimals, which represent Schur functions inside copies $Λ(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 $\nabla^m s_{μ}$ 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.

math.CO↗

arXiv2014

Level Matrices

Let $n>1$ and $k>0$ be fixed integers. A matrix is said to be level if all its column sums are equal. A level matrix with $m$ rows is called reducible if we can delete $j$ rows, $0 \ell$, any $m\times n$ level matrix with entries in $\{0,\ldots,k\}$ is reducible. It is known that $\ell(2,k)=2k-1$. In this paper, we establish the existence of $\ell(n,k)$ for $n\geq 3$ by giving upper and lower bounds for it. We then apply this result to bound the number of certain types of vector space multipartitions.

math.CO↗