Dual Equivalence Graphs Revisited
International audience In 2007 Sami Assaf introduced dual equivalence graphs as a method for demonstrating that a quasisymmetric function is Schur positive. The method involves the creation of a graph whose vertices are weighted by Ira Gessel's fundamental quasisymmetric functions so that the s...
Main Author: | |
---|---|
Other Authors: | , , |
Format: | Conference Object |
Language: | English |
Published: |
HAL CCSD
2013
|
Subjects: | |
Online Access: | https://inria.hal.science/hal-01229693 https://inria.hal.science/hal-01229693/document https://inria.hal.science/hal-01229693/file/dmAS0178.pdf https://doi.org/10.46298/dmtcs.2354 |
id |
ftccsdartic:oai:HAL:hal-01229693v1 |
---|---|
record_format |
openpolar |
spelling |
ftccsdartic:oai:HAL:hal-01229693v1 2023-12-24T10:24:38+01:00 Dual Equivalence Graphs Revisited Roberts, Austin Department of Mathematics Seattle University of Washington Seattle Alain Goupil and Gilles Schaeffer Paris, France 2013 https://inria.hal.science/hal-01229693 https://inria.hal.science/hal-01229693/document https://inria.hal.science/hal-01229693/file/dmAS0178.pdf https://doi.org/10.46298/dmtcs.2354 en eng HAL CCSD Discrete Mathematics and Theoretical Computer Science DMTCS info:eu-repo/semantics/altIdentifier/doi/10.46298/dmtcs.2354 hal-01229693 https://inria.hal.science/hal-01229693 https://inria.hal.science/hal-01229693/document https://inria.hal.science/hal-01229693/file/dmAS0178.pdf doi:10.46298/dmtcs.2354 info:eu-repo/semantics/OpenAccess ISSN: 1462-7264 EISSN: 1365-8050 Discrete Mathematics and Theoretical Computer Science Discrete Mathematics and Theoretical Computer Science (DMTCS) 25th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2013) https://inria.hal.science/hal-01229693 25th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2013), 2013, Paris, France. pp.921-932, ⟨10.46298/dmtcs.2354⟩ Dual equivalence graph LLT polynomial Macdonald polynomial Schur expansion quasisymmetric function [INFO.INFO-DM]Computer Science [cs]/Discrete Mathematics [cs.DM] info:eu-repo/semantics/conferenceObject Conference papers 2013 ftccsdartic https://doi.org/10.46298/dmtcs.2354 2023-11-26T01:47:51Z International audience In 2007 Sami Assaf introduced dual equivalence graphs as a method for demonstrating that a quasisymmetric function is Schur positive. The method involves the creation of a graph whose vertices are weighted by Ira Gessel's fundamental quasisymmetric functions so that the sum of the weights of a connected component is a single Schur function. In this paper, we improve on Assaf's axiomatization of such graphs, giving locally testable criteria that are more easily verified by computers. We then demonstrate the utility of this result by giving explicit Schur expansions for a family of Lascoux-Leclerc-Thibon polynomials. This family properly contains the previously known case of polynomials indexed by two skew shapes, as was described in a 1995 paper by Christophe Carré and Bernard Leclerc. As an immediate corollary, we gain an explicit Schur expansion for a family of modified Macdonald polynomials in terms of Yamanouchi words. This family includes all polynomials indexed by shapes with less than four cells in the first row and strictly less than three cells in the second row, a slight improvement over the known two column case described in 2005 by James Haglund, Mark Haiman, and Nick Loehr. Conference Object sami Archive ouverte HAL (Hyper Article en Ligne, CCSD - Centre pour la Communication Scientifique Directe) Haglund ENVELOPE(12.180,12.180,65.320,65.320) |
institution |
Open Polar |
collection |
Archive ouverte HAL (Hyper Article en Ligne, CCSD - Centre pour la Communication Scientifique Directe) |
op_collection_id |
ftccsdartic |
language |
English |
topic |
Dual equivalence graph LLT polynomial Macdonald polynomial Schur expansion quasisymmetric function [INFO.INFO-DM]Computer Science [cs]/Discrete Mathematics [cs.DM] |
spellingShingle |
Dual equivalence graph LLT polynomial Macdonald polynomial Schur expansion quasisymmetric function [INFO.INFO-DM]Computer Science [cs]/Discrete Mathematics [cs.DM] Roberts, Austin Dual Equivalence Graphs Revisited |
topic_facet |
Dual equivalence graph LLT polynomial Macdonald polynomial Schur expansion quasisymmetric function [INFO.INFO-DM]Computer Science [cs]/Discrete Mathematics [cs.DM] |
description |
International audience In 2007 Sami Assaf introduced dual equivalence graphs as a method for demonstrating that a quasisymmetric function is Schur positive. The method involves the creation of a graph whose vertices are weighted by Ira Gessel's fundamental quasisymmetric functions so that the sum of the weights of a connected component is a single Schur function. In this paper, we improve on Assaf's axiomatization of such graphs, giving locally testable criteria that are more easily verified by computers. We then demonstrate the utility of this result by giving explicit Schur expansions for a family of Lascoux-Leclerc-Thibon polynomials. This family properly contains the previously known case of polynomials indexed by two skew shapes, as was described in a 1995 paper by Christophe Carré and Bernard Leclerc. As an immediate corollary, we gain an explicit Schur expansion for a family of modified Macdonald polynomials in terms of Yamanouchi words. This family includes all polynomials indexed by shapes with less than four cells in the first row and strictly less than three cells in the second row, a slight improvement over the known two column case described in 2005 by James Haglund, Mark Haiman, and Nick Loehr. |
author2 |
Department of Mathematics Seattle University of Washington Seattle Alain Goupil and Gilles Schaeffer |
format |
Conference Object |
author |
Roberts, Austin |
author_facet |
Roberts, Austin |
author_sort |
Roberts, Austin |
title |
Dual Equivalence Graphs Revisited |
title_short |
Dual Equivalence Graphs Revisited |
title_full |
Dual Equivalence Graphs Revisited |
title_fullStr |
Dual Equivalence Graphs Revisited |
title_full_unstemmed |
Dual Equivalence Graphs Revisited |
title_sort |
dual equivalence graphs revisited |
publisher |
HAL CCSD |
publishDate |
2013 |
url |
https://inria.hal.science/hal-01229693 https://inria.hal.science/hal-01229693/document https://inria.hal.science/hal-01229693/file/dmAS0178.pdf https://doi.org/10.46298/dmtcs.2354 |
op_coverage |
Paris, France |
long_lat |
ENVELOPE(12.180,12.180,65.320,65.320) |
geographic |
Haglund |
geographic_facet |
Haglund |
genre |
sami |
genre_facet |
sami |
op_source |
ISSN: 1462-7264 EISSN: 1365-8050 Discrete Mathematics and Theoretical Computer Science Discrete Mathematics and Theoretical Computer Science (DMTCS) 25th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2013) https://inria.hal.science/hal-01229693 25th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2013), 2013, Paris, France. pp.921-932, ⟨10.46298/dmtcs.2354⟩ |
op_relation |
info:eu-repo/semantics/altIdentifier/doi/10.46298/dmtcs.2354 hal-01229693 https://inria.hal.science/hal-01229693 https://inria.hal.science/hal-01229693/document https://inria.hal.science/hal-01229693/file/dmAS0178.pdf doi:10.46298/dmtcs.2354 |
op_rights |
info:eu-repo/semantics/OpenAccess |
op_doi |
https://doi.org/10.46298/dmtcs.2354 |
_version_ |
1786199629302333440 |