Regularity of Cohen-Macaulay Specht ideals

2020/02/06 by Shibata, Kosuke, Yanagawa, Kohji
#Commutative Algebra (math.AC) #FOS: Mathematics

paper · doi:10.48550/arxiv.2002.02221

Abstract

For a partition λ of n ∈ \mathbb N, let I^{\rm Sp}_λ be the ideal of R=K[x₁,…,x_n] generated by all Specht polynomials of shape λ. In the previous paper, the second author showed that if R/I^{\rm Sp}_λ is Cohen-Macaulay, then λ is either (n-d,1,…,1),(n-d,d), or (d,d,1), and the converse is true if \rm char(K)=0. In this paper, we compute the Hilbert series of R/I^{\rm Sp}_λ for λ=(n-d,d) or (d,d,1). Hence, we get the Castelnuovo-Mumford regularity of R/I^{\rm Sp}_λ, when it is Cohen-Macaulay. In particular, I^{\rm Sp}_(d,d,1) has a (d+2)-linear resolution in the Cohen-Macaulay case.

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