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 $I_n \subseteq {\mathbb {F}}[{\mathbf {x}}_{n \times n}]$ generated by all row and...
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| Vydané v: | Forum of Mathematics, Sigma Ročník 12 |
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| Hlavný autor: | |
| Médium: | Journal Article |
| Jazyk: | English |
| Vydavateľské údaje: |
Cambridge, UK
Cambridge University Press
11.11.2024
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| Predmet: | |
| ISSN: | 2050-5094, 2050-5094 |
| On-line prístup: | Získať plný text |
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| Shrnutí: | 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. |
|---|---|
| Bibliografia: | ObjectType-Article-1 SourceType-Scholarly Journals-1 ObjectType-Feature-2 content type line 14 |
| ISSN: | 2050-5094 2050-5094 |
| DOI: | 10.1017/fms.2024.75 |