Cyclic Sieving and Plethysm Coefficients
A combinatorial expression for the coefficient of the Schur function $s_{\lambda}$ in the expansion of the plethysm $p_{n/d}^d \circ s_{\mu}$ is given for all $d$ dividing $n$ for the cases in which $n=2$ or $\lambda$ is rectangular. In these cases, the coefficient $\langle p_{n/d}^d \circ s_{\mu},...
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| Format: | Article |
| Language: | English |
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Discrete Mathematics & Theoretical Computer Science
2015-01-01
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| Series: | Discrete Mathematics & Theoretical Computer Science |
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| Online Access: | https://dmtcs.episciences.org/2509/pdf |
| _version_ | 1797270189961641984 |
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| author | David B Rush |
| author_facet | David B Rush |
| author_sort | David B Rush |
| collection | DOAJ |
| description | A combinatorial expression for the coefficient of the Schur function $s_{\lambda}$ in the expansion of the plethysm $p_{n/d}^d \circ s_{\mu}$ is given for all $d$ dividing $n$ for the cases in which $n=2$ or $\lambda$ is rectangular. In these cases, the coefficient $\langle p_{n/d}^d \circ s_{\mu}, s_{\lambda} \rangle$ is shown to count, up to sign, the number of fixed points of an $\langle s_{\mu}^n, s_{\lambda} \rangle$-element set under the $d^e$ power of an order $n$ cyclic action. If $n=2$, the action is the Schützenberger involution on semistandard Young tableaux (also known as evacuation), and, if $\lambda$ is rectangular, the action is a certain power of Schützenberger and Shimozono's <i>jeu-de-taquin</i> promotion.This work extends results of Stembridge and Rhoades linking fixed points of the Schützenberger actions to ribbon tableaux enumeration. The conclusion for the case $n=2$ is equivalent to the domino tableaux rule of Carré and Leclerc for discriminating between the symmetric and antisymmetric parts of the square of a Schur function. |
| first_indexed | 2024-04-25T02:00:19Z |
| format | Article |
| id | doaj.art-1935ca62f0a84115a5f94616fd64c063 |
| institution | Directory Open Access Journal |
| issn | 1365-8050 |
| language | English |
| last_indexed | 2024-04-25T02:00:19Z |
| publishDate | 2015-01-01 |
| publisher | Discrete Mathematics & Theoretical Computer Science |
| record_format | Article |
| series | Discrete Mathematics & Theoretical Computer Science |
| spelling | doaj.art-1935ca62f0a84115a5f94616fd64c0632024-03-07T15:01:26ZengDiscrete Mathematics & Theoretical Computer ScienceDiscrete Mathematics & Theoretical Computer Science1365-80502015-01-01DMTCS Proceedings, 27th...Proceedings10.46298/dmtcs.25092509Cyclic Sieving and Plethysm CoefficientsDavid B Rush0Department of Mathematics [MIT]A combinatorial expression for the coefficient of the Schur function $s_{\lambda}$ in the expansion of the plethysm $p_{n/d}^d \circ s_{\mu}$ is given for all $d$ dividing $n$ for the cases in which $n=2$ or $\lambda$ is rectangular. In these cases, the coefficient $\langle p_{n/d}^d \circ s_{\mu}, s_{\lambda} \rangle$ is shown to count, up to sign, the number of fixed points of an $\langle s_{\mu}^n, s_{\lambda} \rangle$-element set under the $d^e$ power of an order $n$ cyclic action. If $n=2$, the action is the Schützenberger involution on semistandard Young tableaux (also known as evacuation), and, if $\lambda$ is rectangular, the action is a certain power of Schützenberger and Shimozono's <i>jeu-de-taquin</i> promotion.This work extends results of Stembridge and Rhoades linking fixed points of the Schützenberger actions to ribbon tableaux enumeration. The conclusion for the case $n=2$ is equivalent to the domino tableaux rule of Carré and Leclerc for discriminating between the symmetric and antisymmetric parts of the square of a Schur function.https://dmtcs.episciences.org/2509/pdfplethysmsschützenberger involution<i>jeu-de-taquin</i> promotioncanonical baseskashiwara crystalscyclic sieving phenomenon[info.info-dm] computer science [cs]/discrete mathematics [cs.dm] |
| spellingShingle | David B Rush Cyclic Sieving and Plethysm Coefficients Discrete Mathematics & Theoretical Computer Science plethysms schützenberger involution <i>jeu-de-taquin</i> promotion canonical bases kashiwara crystals cyclic sieving phenomenon [info.info-dm] computer science [cs]/discrete mathematics [cs.dm] |
| title | Cyclic Sieving and Plethysm Coefficients |
| title_full | Cyclic Sieving and Plethysm Coefficients |
| title_fullStr | Cyclic Sieving and Plethysm Coefficients |
| title_full_unstemmed | Cyclic Sieving and Plethysm Coefficients |
| title_short | Cyclic Sieving and Plethysm Coefficients |
| title_sort | cyclic sieving and plethysm coefficients |
| topic | plethysms schützenberger involution <i>jeu-de-taquin</i> promotion canonical bases kashiwara crystals cyclic sieving phenomenon [info.info-dm] computer science [cs]/discrete mathematics [cs.dm] |
| url | https://dmtcs.episciences.org/2509/pdf |
| work_keys_str_mv | AT davidbrush cyclicsievingandplethysmcoefficients |