Yanagawa, Kohji When is a Specht ideal Cohen-Macaulay? (English) Zbl 1481.13026 J. Commut. Algebra 13, No. 4, 589-608 (2021). Summary: For a partition \(\lambda\) of \(n\), let \(I_\lambda^{\mathrm{Sp}}\) be the ideal of \(R = K[x_1, \ldots, x_n]\) generated by all Specht polynomials of shape \(\lambda\). We show that if \(R / I_\lambda^{\mathrm{Sp}}\) is Cohen-Macaulay then \(\lambda\) is of the form either \((a, 1, \ldots ,1), (a, b)\), or \((a, a, 1)\). We also prove that the converse is true in the \(\mathrm{char}(K) = 0\) case. To show the latter statement, the radicalness of these ideals and a result of Etingof et al. are crucial. We also remark that \(R / I_{(n - 3,3)}^{\mathrm{Sp}}\) is not Cohen-Macaulay if and only if \(\mathrm{char}(K) = 2\). Cited in 1 ReviewCited in 8 Documents MSC: 13C14 Cohen-Macaulay modules 05E40 Combinatorial aspects of commutative algebra Keywords:Cohen-Macaulay ring; Specht ideal; Specht polynomial; subspace arrangement PDFBibTeX XMLCite \textit{K. Yanagawa}, J. Commut. Algebra 13, No. 4, 589--608 (2021; Zbl 1481.13026) Full Text: DOI arXiv Link References: [1] A. Brookner, D. Corwin, P. Etingof, and S. V. Sam, “On Cohen-Macaulayness of \[S_n\]-invariant subspace arrangements”, Int. Math. Res. Not. 2016:7 (2016), 2104-2126. · Zbl 1404.14063 · doi:10.1093/imrn/rnv200 [2] W. Bruns and J. Herzog, Cohen-Macaulay rings, Cambridge Studies in Advanced Mathematics 39, Cambridge University Press, 1993. · Zbl 0788.13005 [3] P. Etingof, E. Gorsky, and I. Losev, “Representations of rational Cherednik algebras with minimal support and torus knots”, Adv. Math. 277 (2015), 124-180. · Zbl 1321.16020 · doi:10.1016/j.aim.2015.03.003 [4] T. Harima, T. Maeno, H. Morita, Y. Numata, A. Wachi, and J. Watanabe, The Lefschetz properties, Lecture Notes in Mathematics 2080, Springer, 2013. · Zbl 1284.13001 · doi:10.1007/978-3-642-38206-2 [5] B. E. Sagan, The symmetric group, 2nd ed., Graduate Texts in Mathematics 203, Springer, 2001. · Zbl 0964.05070 · doi:10.1007/978-1-4757-6804-6 [6] R. P. Stanley, Catalan numbers, Cambridge University Press, New York, 2015. · Zbl 1317.05010 · doi:10.1017/CBO9781139871495 [7] J. Watanabe and K. Yanagawa, “Vandermonde determinantal ideals”, Math. Scand. 125:2 (2019), 179-184. · Zbl 1476.13016 · doi:10.7146/math.scand.a-114906 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.