Preprint
Flagged LLT polynomials, nonsymmetric plethysm, and nonsymmetric Macdonald polynomials
10 Jun 2025
Abstract
The plethystic transformation $f[X] \mapsto f[X/(1-t)]$ and LLT polynomials
are central to the theory of symmetric Macdonald polynomials. In this work, we
introduce and study nonsymmetric flagged LLT polynomials. We show that these
admit both an algebraic and a combinatorial description, that they Weyl
symmetrize to the usual symmetric LLT polynomials, and we conjecture that they
expand positively in terms of Demazure atoms. Additionally, we construct a
nonsymmetric plethysm operator $\Pi_{t,x}$ on $\mathfrak{K}[x_1,\ldots,x_n]$,
which serves as an analogue of $f[X] \mapsto f[X/(1-t)]$. We prove that
$\Pi_{t,x}$ remarkably maps flagged LLT polynomials defined over a signed
alphabet to ones over an unsigned alphabet.
Our main application of this theory is to formulate a nonsymmetric version of
Macdonald positivity, similar in spirit to conjectures of Knop and Lapointe,
but with several new features. To do this, we recast the Haglund-Haiman-Loehr
formula for nonsymmetric Macdonald polynomials $\mathcal{E}_{\mu }(x;q,t)$ as a
positive sum of signed flagged LLT polynomials. Then, after taking a suitable
stable limit of both $\mathcal{E}_{\mu }(x;q,t)$ and $\Pi_{t,x}$, we obtain
modified nonsymmetric Macdonald polynomials which are positive sums of flagged
LLT polynomials and thus are conjecturally atom positive, strengthening the
Macdonald positivity conjecture.
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Details
- Title
- Flagged LLT polynomials, nonsymmetric plethysm, and nonsymmetric Macdonald polynomials
- Creators
- Jonah BlasiakMark HaimanJennifer MorseAnna PunGeorge H Seelinger
- Resource Type
- Preprint
- Language
- English
- Academic Unit
- Mathematics
- Other Identifier
- 991022057000604721