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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| Main Author: | |
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| Format: | Article |
| Language: | English |
| Published: |
Cambridge University Press
2024-01-01
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| Series: | Forum of Mathematics, Sigma |
| Subjects: | |
| Online Access: | https://www.cambridge.org/core/product/identifier/S2050509424000756/type/journal_article |
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| Summary: | 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. |
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| ISSN: | 2050-5094 |