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Register a new userRegularity of Cohen-Macaulay Specht ideals
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For a partition $lambda$ of $n in {mathbb N}$, let $I^{rm Sp}_lambda$ be the ideal of $R=K[x_1,ldots,x_n]$ generated by all Specht polynomials of shape $lambda$. In the previous paper, the second author showed that if $R/I^{rm Sp}_lambda$ is Cohen-Macaulay, then $lambda$ is either $(n-d,1,ldots,1),(n-d,d)$, or $(d,d,1)$, and the converse is true if ${rm char}(K)=0$. In this paper, we compute the Hilbert series of $R/I^{rm Sp}_lambda$ for $lambda=(n-d,d)$ or $(d,d,1)$. Hence, we get the Castelnuovo-Mumford regularity of $R/I^{rm Sp}_lambda$, 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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For a partition $lambda$ of $n$, let $I^{rm Sp}_lambda$ be the ideal of $R=K[x_1, ldots, x_n]$ generated by all Specht polynomials of shape $lambda$. We show that if $R/I^{rm Sp}_lambda$ 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 if ${rm char}(K)=0$. To show the latter statement, the radicalness of these ideals and a result of Etingof et al. are crucial. We also remark that $R/I^{rm Sp}_{(n-3,3)}$ is NOT Cohen--Macaulay if and only if ${rm char}(K)=2$.
Scattered over the past few years have been several occurrences of simplicial complexes whose topological behavior characterize the Cohen-Macaulay property for quotients of polynomial rings by arbitrary (not necessarily squarefree) monomial ideals. The purpose of this survey is to gather the developments into one location, with self-contained proofs, including direct combinatorial topological connections between them.
We compute the minimal primary decomposition for completely squarefree lexsegment ideals. We show that critical squarefree monomial ideals are sequentially Cohen-Macaulay. As an application, we give a complete characterization of the completely squarefree lexsegment ideals which are sequentially Cohen-Macaulay and we also derive formulas for some homological invariants of this class of ideals.
We characterize unmixed and Cohen-Macaulay edge-weighted edge ideals of very well-covered graphs. We also provide examples of oriented graphs which have unmixed and non-Cohen-Macaulay vertex-weighted edge ideals, while the edge ideal of their underlying graph is Cohen-Macaulay. This disproves a conjecture posed by Pitones, Reyes and Toledo.
Let C be a uniform clutter and let I=I(C) be its edge ideal. We prove that if C satisfies the packing property (resp. max-flow min-cut property), then there is a uniform Cohen-Macaulay clutter C1 satisfying the packing property (resp. max-flow min-cut property) such that C is a minor of C1. For arbitrary edge ideals of clutters we prove that the normality property is closed under parallelizations. Then we show some applications to edge ideals and clutters which are related to a conjecture of Conforti and Cornuejols and to max-flow min-cut problems.
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