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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 ofk-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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The Hilbert series of the superspace coinvariant ring
Abstract Let$$\Omega _n$$be the ring of polynomial-valued holomorphic differential forms on complexn-space, referred to in physics as the superspace ring of rankn. The symmetric group$${\mathfrak {S}}_n$$acts diagonally on$$\Omega _n$$by permuting commuting and anticommuting generators simultaneously. We let$$SI_n \subseteq \Omega _n$$be the ideal generated by$${\mathfrak {S}}_n$$-invariants with vanishing constant term and study the quotient$$SR_n = \Omega _n / SI_n$$of superspace by this ideal. We calculate the doubly-graded Hilbert series of$$SR_n$$and prove an ‘operator theorem’, which characterizes the harmonic space$$SH_n \subseteq \Omega _n$$attached to$$SR_n$$in terms of the Vandermonde determinant and certain differential operators. Our methods employ commutative algebra results that were used in the study of Hessenberg varieties. Our results prove conjectures of N. Bergeron, Colmenarejo, Li, Machacek, Sulzgruber, Swanson, Wallach and Zabrocki.
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- Award ID(s):
- 2246846
- PAR ID:
- 10591091
- Publisher / Repository:
- Forum of Mathematics, Pi
- Date Published:
- Journal Name:
- Forum of Mathematics, Pi
- Volume:
- 12
- ISSN:
- 2050-5086
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1017/fmp.2024.14
