Delta and Theta Operator Expansions
We give an elementary symmetric function expansion for the expressions $M\Delta _{m_\gamma e_1}\Pi e_\lambda ^{\ast }$ and $M\Delta _{m_\gamma e_1}\Pi s_\lambda ^{\ast }$ when $t=1$ in terms of what we call $\gamma $ -parking functions and lattice $\gamma $ -parking...
| Principais autores: | , |
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| Formato: | Artigo |
| Idioma: | English |
| Publicado em: |
Cambridge University Press
2024-01-01
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| coleção: | Forum of Mathematics, Sigma |
| Assuntos: | |
| Acesso em linha: | https://www.cambridge.org/core/product/identifier/S2050509424000148/type/journal_article |
| _version_ | 1827324568601100288 |
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| author | Alessandro Iraci Marino Romero |
| author_facet | Alessandro Iraci Marino Romero |
| author_sort | Alessandro Iraci |
| collection | DOAJ |
| description | We give an elementary symmetric function expansion for the expressions
$M\Delta _{m_\gamma e_1}\Pi e_\lambda ^{\ast }$
and
$M\Delta _{m_\gamma e_1}\Pi s_\lambda ^{\ast }$
when
$t=1$
in terms of what we call
$\gamma $
-parking functions and lattice
$\gamma $
-parking functions. Here,
$\Delta _F$
and
$\Pi $
are certain eigenoperators of the modified Macdonald basis and
$M=(1-q)(1-t)$
. Our main results, in turn, give an elementary basis expansion at
$t=1$
for symmetric functions of the form
$M \Delta _{Fe_1} \Theta _{G} J$
whenever F is expanded in terms of monomials, G is expanded in terms of the elementary basis, and J is expanded in terms of the modified elementary basis
$\{\Pi e_\lambda ^\ast \}_\lambda $
. Even the most special cases of this general Delta and Theta operator expression are significant; we highlight a few of these special cases. We end by giving an e-positivity conjecture for when t is not specialized, proposing that our objects can also give the elementary basis expansion in the unspecialized symmetric function. |
| first_indexed | 2024-03-07T13:57:45Z |
| format | Article |
| id | doaj.art-016beb51a3f14c3885b2e41d352a181a |
| institution | Directory Open Access Journal |
| issn | 2050-5094 |
| language | English |
| last_indexed | 2024-03-07T13:57:45Z |
| publishDate | 2024-01-01 |
| publisher | Cambridge University Press |
| record_format | Article |
| series | Forum of Mathematics, Sigma |
| spelling | doaj.art-016beb51a3f14c3885b2e41d352a181a2024-03-07T07:18:30ZengCambridge University PressForum of Mathematics, Sigma2050-50942024-01-011210.1017/fms.2024.14Delta and Theta Operator ExpansionsAlessandro Iraci0https://orcid.org/0000-0002-3158-3929Marino Romero1https://orcid.org/0000-0002-7255-3179Dipartimento di Matematica, Università di Pisa, Largo Bruno Pontecorvo, 56127 Pisa, Italy; E-mail:Fakultät für Mathematik, Universität Wien, Oskar-Morgenstern-Platz 1, Vienna, 1090, Austria; E-mail:We give an elementary symmetric function expansion for the expressions $M\Delta _{m_\gamma e_1}\Pi e_\lambda ^{\ast }$ and $M\Delta _{m_\gamma e_1}\Pi s_\lambda ^{\ast }$ when $t=1$ in terms of what we call $\gamma $ -parking functions and lattice $\gamma $ -parking functions. Here, $\Delta _F$ and $\Pi $ are certain eigenoperators of the modified Macdonald basis and $M=(1-q)(1-t)$ . Our main results, in turn, give an elementary basis expansion at $t=1$ for symmetric functions of the form $M \Delta _{Fe_1} \Theta _{G} J$ whenever F is expanded in terms of monomials, G is expanded in terms of the elementary basis, and J is expanded in terms of the modified elementary basis $\{\Pi e_\lambda ^\ast \}_\lambda $ . Even the most special cases of this general Delta and Theta operator expression are significant; we highlight a few of these special cases. We end by giving an e-positivity conjecture for when t is not specialized, proposing that our objects can also give the elementary basis expansion in the unspecialized symmetric function.https://www.cambridge.org/core/product/identifier/S2050509424000148/type/journal_article05E0505E10 |
| spellingShingle | Alessandro Iraci Marino Romero Delta and Theta Operator Expansions Forum of Mathematics, Sigma 05E05 05E10 |
| title | Delta and Theta Operator Expansions |
| title_full | Delta and Theta Operator Expansions |
| title_fullStr | Delta and Theta Operator Expansions |
| title_full_unstemmed | Delta and Theta Operator Expansions |
| title_short | Delta and Theta Operator Expansions |
| title_sort | delta and theta operator expansions |
| topic | 05E05 05E10 |
| url | https://www.cambridge.org/core/product/identifier/S2050509424000148/type/journal_article |
| work_keys_str_mv | AT alessandroiraci deltaandthetaoperatorexpansions AT marinoromero deltaandthetaoperatorexpansions |