论文标题
$ \ mathfrak s_n $ -invariant单一理想的hochster公式
An equivariant Hochster's formula for $\mathfrak S_n$-invariant monomial ideals
论文作者
论文摘要
令$ r = \ bbbk [x_1，\ dots，x_n] $是field $ \ bbbk $上的多项式戒指，让$ i \ i \ subset r $是对称组$ \ mathfrak s_n $在$ r $上的对称组$ \ mathfrak s_n $的自然动作。我们提供了一种组合方法来确定$ \ mathrm {tor} _i（i，\ bbbk）$的$ \ mathfrak s_n $ - 模块结构。我们的公式表明，$ \ mathrm {tor} _i（i，\ bbbk）$是由与挂钩分区相关的SpecHT模块的诱导代表构建的，它们的多重性是由拓扑结构的某些简单复合物的拓扑结构来确定的。该结果可以看作是$ \ Mathfrak S_N $ - equivariant的Hochster公式的类似物，用于Betti数字的单一理想。我们将结果应用于确定$ \ Mathfrak S_N $ -INVARIANT单一理想的极端贝蒂数字，特别是为其Castelnuovo恢复公式 - 穆姆福德的规律性和投射性维度。我们还提供了一个具体的食谱，以了解贝蒂数字如何随着变量的增加而变化，并且在特征零（或$> n $）中，我们计算了$ \ mathrm {tor} _i（i，\ bbbk）$ of $ \ mathrm mathrm of $ \ i的$ \ iust y Mathrm of y Mathrm {tor} _i（i，\ bbbk）的$ \ mathrm {torrm {tor} $ rmmmat of tor tor tor tor tor的$ \ mathrm {tor} _i（i，\ bbbk）的一部分。
Let $R=\Bbbk[x_1,\dots,x_n]$ be a polynomial ring over a field $\Bbbk$ and let $I\subset R$ be a monomial ideal preserved by the natural action of the symmetric group $\mathfrak S_n$ on $R$. We give a combinatorial method to determine the $\mathfrak S_n$-module structure of $\mathrm{Tor}_i(I,\Bbbk)$. Our formula shows that $\mathrm{Tor}_i(I,\Bbbk)$ is built from induced representations of tensor products of Specht modules associated to hook partitions, and their multiplicities are determined by topological Betti numbers of certain simplicial complexes. This result can be viewed as an $\mathfrak S_n$-equivariant analogue of Hochster's formula for Betti numbers of monomial ideals. We apply our results to determine extremal Betti numbers of $\mathfrak S_n$-invariant monomial ideals, and in particular recover formulas for their Castelnuovo--Mumford regularity and projective dimension. We also give a concrete recipe for how the Betti numbers change as we increase the number of variables, and in characteristic zero (or $>n$) we compute the $\mathfrak S_n$-invariant part of $\mathrm{Tor}_i(I,\Bbbk)$ in terms of $\mathrm{Tor}$ groups of the unsymmetrization of $I$.
