Dual Equivalence Graphs Revisited with Applications to LLT and Macdonald Polynomials
Abstract. 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...
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ftciteseerx:oai:CiteSeerX.psu:10.1.1.377.6253 2023-05-15T18:12:21+02:00 Dual Equivalence Graphs Revisited with Applications to LLT and Macdonald Polynomials Austin Roberts The Pennsylvania State University CiteSeerX Archives application/pdf http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.377.6253 http://www.dmtcs.org/pdfpapers/dmAS0178.pdf en eng http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.377.6253 http://www.dmtcs.org/pdfpapers/dmAS0178.pdf Metadata may be used without restrictions as long as the oai identifier remains attached to it. http://www.dmtcs.org/pdfpapers/dmAS0178.pdf text ftciteseerx 2016-09-18T00:10:42Z Abstract. 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 Text sami Unknown |
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Abstract. 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 |
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Austin Roberts |
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Austin Roberts Dual Equivalence Graphs Revisited with Applications to LLT and Macdonald Polynomials |
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Austin Roberts |
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Austin Roberts |
title |
Dual Equivalence Graphs Revisited with Applications to LLT and Macdonald Polynomials |
title_short |
Dual Equivalence Graphs Revisited with Applications to LLT and Macdonald Polynomials |
title_full |
Dual Equivalence Graphs Revisited with Applications to LLT and Macdonald Polynomials |
title_fullStr |
Dual Equivalence Graphs Revisited with Applications to LLT and Macdonald Polynomials |
title_full_unstemmed |
Dual Equivalence Graphs Revisited with Applications to LLT and Macdonald Polynomials |
title_sort |
dual equivalence graphs revisited with applications to llt and macdonald polynomials |
url |
http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.377.6253 http://www.dmtcs.org/pdfpapers/dmAS0178.pdf |
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http://www.dmtcs.org/pdfpapers/dmAS0178.pdf |
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http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.377.6253 http://www.dmtcs.org/pdfpapers/dmAS0178.pdf |
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