Quasiinvariants of S3

Jason Bandlow; Gregg Musiker

doi:10.1016/j.jcta.2004.09.004

Quasiinvariants of S₃

Jason Bandlow, Gregg Musiker

School of Mathematics

Research output: Contribution to journal › Article › peer-review

3 Scopus citations

Abstract

Let s_ij represent a transposition in S_n. A polynomial P in ℚ[X_n] is said to be m-quasiinvariant with respect to S_n if (χ_i - χ_j) ^2m+1 divides (1 - s_ij) P for all 1 ≤ i, j ≤ n. We call the ring of m-quasiinvariants, QI_m[X_n]. We describe a method for constructing a basis for the quotient QI_m[X₃]/(e₁, e₂, e₃). This leads to the evaluation of certain binomial determinants that are interesting in their own right.

Original language	English (US)
Pages (from-to)	281-298
Number of pages	18
Journal	Journal of Combinatorial Theory. Series A
Volume	109
Issue number	2
DOIs	https://doi.org/10.1016/j.jcta.2004.09.004
State	Published - Feb 2005

Keywords

Binomial coefficients
Determinant evaluations
Non-intersecting lattice paths
Symmetric functions
Symmetric group
m-quasiinvariants

Publisher link

10.1016/j.jcta.2004.09.004

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title = "Quasiinvariants of S3",

abstract = "Let sij represent a transposition in Sn. A polynomial P in ℚ[Xn] is said to be m-quasiinvariant with respect to Sn if (χi - χj) 2m+1 divides (1 - sij) P for all 1 ≤ i, j ≤ n. We call the ring of m-quasiinvariants, QIm[Xn]. We describe a method for constructing a basis for the quotient QIm[X3]/(e1, e2, e3). This leads to the evaluation of certain binomial determinants that are interesting in their own right.",

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AB - Let sij represent a transposition in Sn. A polynomial P in ℚ[Xn] is said to be m-quasiinvariant with respect to Sn if (χi - χj) 2m+1 divides (1 - sij) P for all 1 ≤ i, j ≤ n. We call the ring of m-quasiinvariants, QIm[Xn]. We describe a method for constructing a basis for the quotient QIm[X3]/(e1, e2, e3). This leads to the evaluation of certain binomial determinants that are interesting in their own right.

KW - Binomial coefficients

KW - Determinant evaluations

KW - Non-intersecting lattice paths

KW - Symmetric functions

KW - Symmetric group

KW - m-quasiinvariants

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