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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| Veröffentlicht in: | Forum of Mathematics, Sigma Jg. 12 |
|---|---|
| 1. Verfasser: | |
| Format: | Journal Article |
| Sprache: | Englisch |
| Veröffentlicht: |
Cambridge, UK
Cambridge University Press
11.11.2024
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| Schlagworte: | |
| ISSN: | 2050-5094, 2050-5094 |
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| Abstract | 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. |
|---|---|
| AbstractList | 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. 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. 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. |
| ArticleNumber | e97 |
| Author | Rhoades, Brendon |
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| Cites_doi | 10.4153/CJM-1961-015-3 10.1007/s00026-017-0365-x 10.1093/imrn/rnad033 10.1016/0001-8708(92)90034-I 10.1090/tran/8237 10.1007/s00209-021-02800-z 10.1112/plms/s2-26.1.531 10.1090/S0002-9947-1979-0526314-6 10.1007/BFb0090011 10.1090/S0002-9904-1963-10980-5 10.1016/j.aim.2018.01.028 10.1007/978-1-4757-6804-6 |
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| Copyright | The Author(s), 2024. Published by Cambridge University Press The Author(s), 2024. Published by Cambridge University Press. This work is licensed under the Creative Commons Attribution License This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (https://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited. (the “License”). Notwithstanding the ProQuest Terms and Conditions, you may use this content in accordance with the terms of the License. |
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| DOI | 10.1017/fms.2024.75 |
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| References | 1963; 69 1927; 26 2023 2023; 5 2022 2017; 21 1979; 249 2018 2018; 329 2021; 374 1961; 13 1992; 94 2022; 300 Beullens (S2050509424000756_r1) 2018 Fulton (S2050509424000756_r6) 1997; 35 S2050509424000756_r4 S2050509424000756_r3 S2050509424000756_r5 S2050509424000756_r8 S2050509424000756_r19 S2050509424000756_r18 S2050509424000756_r7 S2050509424000756_r17 S2050509424000756_r9 S2050509424000756_r16 S2050509424000756_r15 S2050509424000756_r14 S2050509424000756_r13 S2050509424000756_r12 S2050509424000756_r11 Hamaker (S2050509424000756_r10) 2022 Briaud (S2050509424000756_r2) 2023; 5 S2050509424000756_r20 |
| References_xml | – volume: 13 start-page: 179 year: 1961 end-page: 191 article-title: Longest increasing and decreasing subsequences publication-title: Canad. J. Math. – year: 2023 article-title: Zonotopal algebras, orbit harmonics, and Donaldson–Thomas invariants of symmetric quivers publication-title: Int. Math. Res. Not. IMRN – year: 2022 article-title: The characters of local and regular permutation statistics publication-title: Preprint – volume: 26 start-page: 531 year: 1927 end-page: 555 article-title: Some properties of enumeration in the theory of modular systems publication-title: Proc. London Math. Soc. – volume: 374 start-page: 2609 year: 2021 end-page: 2660 article-title: Ordered set partitions, Garsia–Procesi modules, and rank varieties publication-title: Trans. Amer. Math. Soc. – volume: 5 start-page: 391 year: 2023 end-page: 422 article-title: A new algebraic approach to the regular syndrome decoding problem and implications for PCG constructions publication-title: EUROCRYPT – volume: 300 start-page: 639 year: 2022 end-page: 660 article-title: Cyclic sieving and orbit harmonics publication-title: Math. Z. – year: 2018 article-title: PKP-based signature scheme publication-title: Cryptology – volume: 329 start-page: 851 year: 2018 end-page: 915 article-title: Ordered set partitions, generalized coinvariant algebras, and the Delta conjecture publication-title: Adv. Math. – volume: 249 start-page: 139 issue: 1 year: 1979 end-page: 157 article-title: Balanced Cohen–Macaulay complexes publication-title: Trans. Amer. Math. Soc. – volume: 69 start-page: 518 issue: 4 year: 1963 end-page: 526 article-title: Lie group representations on polynomial rings publication-title: Bull. Amer. Math. Soc. – volume: 94 start-page: 82 year: 1992 end-page: 138 article-title: On certain graded ${S}_n$ -modules and the $q$ -Kostka polynomials publication-title: Adv. Math. – volume: 21 start-page: 535 year: 2017 end-page: 549 article-title: Longest increasing subsequences and log concavity publication-title: Ann. Comb. – ident: S2050509424000756_r4 – ident: S2050509424000756_r18 doi: 10.4153/CJM-1961-015-3 – year: 2018 ident: S2050509424000756_r1 article-title: PKP-based signature scheme publication-title: Cryptology – ident: S2050509424000756_r3 doi: 10.1007/s00026-017-0365-x – ident: S2050509424000756_r16 doi: 10.1093/imrn/rnad033 – ident: S2050509424000756_r7 doi: 10.1016/0001-8708(92)90034-I – ident: S2050509424000756_r8 doi: 10.1090/tran/8237 – ident: S2050509424000756_r15 doi: 10.1007/s00209-021-02800-z – ident: S2050509424000756_r13 doi: 10.1112/plms/s2-26.1.531 – ident: S2050509424000756_r19 doi: 10.1090/S0002-9947-1979-0526314-6 – ident: S2050509424000756_r20 doi: 10.1007/BFb0090011 – ident: S2050509424000756_r12 – year: 2022 ident: S2050509424000756_r10 article-title: The characters of local and regular permutation statistics publication-title: Preprint – ident: S2050509424000756_r14 – ident: S2050509424000756_r5 – ident: S2050509424000756_r11 doi: 10.1090/S0002-9904-1963-10980-5 – ident: S2050509424000756_r9 doi: 10.1016/j.aim.2018.01.028 – ident: S2050509424000756_r17 doi: 10.1007/978-1-4757-6804-6 – volume: 5 start-page: 391 year: 2023 ident: S2050509424000756_r2 article-title: A new algebraic approach to the regular syndrome decoding problem and implications for PCG constructions publication-title: EUROCRYPT – volume: 35 volume-title: Young Tableaux: With Applications to Representation Theory and Geometry year: 1997 ident: S2050509424000756_r6 |
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| Snippet | 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... 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... 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... |
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| SubjectTerms | 05E10 13P10 Avatars Combinatorics Discrete Mathematics Matrices (mathematics) Permutations Polynomials Rings (mathematics) Series (mathematics) Shadows Variables Vector space Vector spaces |
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| Title | Increasing subsequences, matrix loci and Viennot shadows |
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