Increasing subsequences, matrix loci and Viennot shadows
Let ${\mathbf {x}}_{n \times n}$ be an $n \times n$ matrix of variables, and let ${\mathbb {F}}[{\mathbf {x}}_{n \times n}]$ be the polynomial ring in these variables over a field ${\mathbb {F}}$ . We study the ideal $I_n \subseteq {\mathbb {F}}[{\mathbf {x}}_{n \times...
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Cambridge University Press
2024-01-01
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| Series: | Forum of Mathematics, Sigma |
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| Online Access: | https://www.cambridge.org/core/product/identifier/S2050509424000756/type/journal_article |
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| author | Brendon Rhoades |
| author_facet | Brendon Rhoades |
| author_sort | Brendon Rhoades |
| collection | DOAJ |
| description | Let
${\mathbf {x}}_{n \times n}$
be an
$n \times n$
matrix of variables, and let
${\mathbb {F}}[{\mathbf {x}}_{n \times n}]$
be the polynomial ring in these variables over a field
${\mathbb {F}}$
. We study the ideal
$I_n \subseteq {\mathbb {F}}[{\mathbf {x}}_{n \times n}]$
generated by all row and column variable sums and all products of two variables drawn from the same row or column. We show that the quotient
${\mathbb {F}}[{\mathbf {x}}_{n \times n}]/I_n$
admits a standard monomial basis determined by Viennot’s shadow line avatar of the Schensted correspondence. As a corollary, the Hilbert series of
${\mathbb {F}}[{\mathbf {x}}_{n \times n}]/I_n$
is the generating function of permutations in
${\mathfrak {S}}_n$
by the length of their longest increasing subsequence. Along the way, we describe a ‘shadow junta’ basis of the vector space of k-local permutation statistics. We also calculate the structure of
${\mathbb {F}}[{\mathbf {x}}_{n \times n}]/I_n$
as a graded
${\mathfrak {S}}_n \times {\mathfrak {S}}_n$
-module. |
| format | Article |
| id | doaj-art-e5d8cfce1267460684fb4f33aea47e4e |
| institution | OA Journals |
| issn | 2050-5094 |
| language | English |
| publishDate | 2024-01-01 |
| publisher | Cambridge University Press |
| record_format | Article |
| series | Forum of Mathematics, Sigma |
| spelling | doaj-art-e5d8cfce1267460684fb4f33aea47e4e2025-08-20T02:13:56ZengCambridge University PressForum of Mathematics, Sigma2050-50942024-01-011210.1017/fms.2024.75Increasing subsequences, matrix loci and Viennot shadowsBrendon Rhoades0Department of Mathematics, University of California, San Diego, 9500 Gilman Dr., La Jolla, 92093-0112, USA;Let ${\mathbf {x}}_{n \times n}$ be an $n \times n$ matrix of variables, and let ${\mathbb {F}}[{\mathbf {x}}_{n \times n}]$ be the polynomial ring in these variables over a field ${\mathbb {F}}$ . We study the ideal $I_n \subseteq {\mathbb {F}}[{\mathbf {x}}_{n \times n}]$ generated by all row and column variable sums and all products of two variables drawn from the same row or column. We show that the quotient ${\mathbb {F}}[{\mathbf {x}}_{n \times n}]/I_n$ admits a standard monomial basis determined by Viennot’s shadow line avatar of the Schensted correspondence. As a corollary, the Hilbert series of ${\mathbb {F}}[{\mathbf {x}}_{n \times n}]/I_n$ is the generating function of permutations in ${\mathfrak {S}}_n$ by the length of their longest increasing subsequence. Along the way, we describe a ‘shadow junta’ basis of the vector space of k-local permutation statistics. We also calculate the structure of ${\mathbb {F}}[{\mathbf {x}}_{n \times n}]/I_n$ as a graded ${\mathfrak {S}}_n \times {\mathfrak {S}}_n$ -module.https://www.cambridge.org/core/product/identifier/S2050509424000756/type/journal_article05E1013P10 |
| spellingShingle | Brendon Rhoades Increasing subsequences, matrix loci and Viennot shadows Forum of Mathematics, Sigma 05E10 13P10 |
| title | Increasing subsequences, matrix loci and Viennot shadows |
| title_full | Increasing subsequences, matrix loci and Viennot shadows |
| title_fullStr | Increasing subsequences, matrix loci and Viennot shadows |
| title_full_unstemmed | Increasing subsequences, matrix loci and Viennot shadows |
| title_short | Increasing subsequences, matrix loci and Viennot shadows |
| title_sort | increasing subsequences matrix loci and viennot shadows |
| topic | 05E10 13P10 |
| url | https://www.cambridge.org/core/product/identifier/S2050509424000756/type/journal_article |
| work_keys_str_mv | AT brendonrhoades increasingsubsequencesmatrixlociandviennotshadows |