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When is a Specht ideal Cohen-Macaulay? (English) Zbl 1481.13026

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 \(\mathrm{char}(K) = 2\).

MSC:

13C14 Cohen-Macaulay modules
05E40 Combinatorial aspects of commutative algebra
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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
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