TY - JOUR

T1 - Elementary construction of minimal free resolutions of the Specht ideals of shapes (n-2, 2) and (d, d, 1)

AU - Shibata, Kosuke

AU - Yanagawa, Kohji

N1 - Publisher Copyright:
© 2023 World Scientific Publishing Company.

PY - 2022

Y1 - 2022

N2 - For a partition λ of n ∈ N let IλSp be the ideal of R = K[x1,...,xn] generated by all Specht polynomials of shape λ. We assume that char(K) = 0. Then R/I(n-2,2)Sp is Gorenstein, and R/I(d,d,1)Sp is a Cohen-Macaulay ring with a linear free resolution. In this paper, we construct minimal free resolutions of these rings. Zamaere et al. [Jack polynomials as fractional quantum Hall states and the Betti numbers of the (k + 1)-equals ideal, Commun. Math. Phys. 330 (2014) 415-434] already studied minimal free resolutions of R/I(n-d,d)Sp, which are also Cohen-Macaulay, using highly advanced technique of the representation theory. However, we only use the basic theory of Specht modules, and explicitly describe the differential maps.

AB - For a partition λ of n ∈ N let IλSp be the ideal of R = K[x1,...,xn] generated by all Specht polynomials of shape λ. We assume that char(K) = 0. Then R/I(n-2,2)Sp is Gorenstein, and R/I(d,d,1)Sp is a Cohen-Macaulay ring with a linear free resolution. In this paper, we construct minimal free resolutions of these rings. Zamaere et al. [Jack polynomials as fractional quantum Hall states and the Betti numbers of the (k + 1)-equals ideal, Commun. Math. Phys. 330 (2014) 415-434] already studied minimal free resolutions of R/I(n-d,d)Sp, which are also Cohen-Macaulay, using highly advanced technique of the representation theory. However, we only use the basic theory of Specht modules, and explicitly describe the differential maps.

KW - Cohen-Macaulay ring

KW - minimal free resolution

KW - Specht ideal

KW - Specht polynomial

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U2 - 10.1142/S0219498823501992

DO - 10.1142/S0219498823501992

M3 - Article

AN - SCOPUS:85145808293

SN - 0219-4988

JO - Journal of Algebra and Its Applications

JF - Journal of Algebra and Its Applications

M1 - 2350199

ER -