Regularity of Cohen-Macaulay Specht ideals (Q2029238)

Let \(n\) be a positive integer, a partition of \(n\) is a non increasing sequence of positive integers whose sum gives \(n\). A partition \(\lambda=(\lambda_1,\ldots, \lambda_t)\) of \(n\) can be visualized using its Young diagram. The Young diagram of shape \(\lambda\) is a set of \(n\) empty boxes arranged to give the same information as \(\lambda.\) When these boxes are filled using all the integers from \(1\) to \(n\), such diagram takes the name of {\em Young tableau}. The symbol \(Tab(\lambda)\) denotes the set of all the Young tableaux of shape \(\lambda.\) Given a partition \(\lambda=(\lambda_1,\ldots, \lambda_t)\) of \(n\), and \(T=(T_{ij})\in Tab(\lambda)\), the {\em Specht polynomial} of \(T\) is an element in the polynomial ring \(R=K[x_1,\ldots,x_n]\), defined as a product of linear forms \[f_T=\prod_{i}\prod_{s<r}\left(x_{T(i,s)}-x_{T(i,r)}\right).\] Finally, the ideal of \(R\) generated by all the forms in \(\{f_T | T \in Tab(\lambda)\}\), is called {\em Specht ideal} of \(\lambda\) and denoted by \(I^{Sp}_{\lambda}\). Some results about the homological invariants of these ideals are known in literature. In particular, the Cohen-Macaulay property of the standard \(K\)-algebras \(R/I^{Sp}_{\lambda}\) was studied in the paper by \textit{K. Yanagawa} [``When is a Specht ideal Cohen-Macaulay?'', J. Commut. Algebra (accepted, in press) (2021)], where, see Proposition 2.8 and Corollary 4.4, it is shown that if \(R/I^{Sp}_{\lambda}\) is Cohen-Macaulay then \(\lambda\) must have one the following shapes \begin{itemize} \item[(1)] \(\lambda=(n-d, 1, \ldots, 1)\); \item[(2)] \(\lambda=(n-d, d)\); \item[(3)] \(\lambda=(d, d, 1)\). \end{itemize} And, if \(\mathrm{char}(K)=0\) then the converse is also true. The main result of this paper is Theorem 2.1, where the authors compute the Hilbert series of the Specht ideal of \(\lambda\) as in the cases (2) and (3) above. The Hilbert series for the case in (1), can be derived from the paper by \textit{J. Watanabe} and \textit{K. Yanagawa} [Math. Scand. 125, No. 2, 179--184 (2019; Zbl 1476.13016)], where the authors compute the graded Betti numbers of \(I^{Sp}_{(n-d, 1, \ldots, 1)}\).