Journal article
A raising operator formula for Macdonald polynomials
Forum of mathematics. Sigma, v 13, e47
01 Jan 2025
Abstract
We give an explicit raising operator formula for the modified Macdonald polynomials
$\tilde {H}_{\mu }(X;q,t)$
, 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
$\tilde {H}_{\mu }(X;q,t)$
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.
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Details
- Title
- A raising operator formula for Macdonald polynomials
- Creators
- J. BlasiakM. HaimanJ. MorseA. PunG. H. Seelinger - University of Michigan
- Publication Details
- Forum of mathematics. Sigma, v 13, e47
- Publisher
- Cambridge University Press; CAMBRIDGE
- Number of pages
- 18
- Grant note
- NSF: DMS-1855784, DMS-1855804, DMS-2303175 Simons Foundation: 821999
Authors were supported by NSF Grants DMS-1855784 (J. B.), DMS-1855804 (J. M., A. P. and G. S.), DMS-2303175 (G. S.) and Simons Foundation-821999 (J. M.)
- Resource Type
- Journal article
- Language
- English
- Academic Unit
- Mathematics
- Web of Science ID
- WOS:001435763900001
- Scopus ID
- 2-s2.0-85218779514
- Other Identifier
- 991022028226204721
InCites Highlights
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- Collaboration types
- Domestic collaboration
- Web of Science research areas
- Mathematics
- Mathematics, Applied