Straightening Law and Powers of Determinantal Ideals of Hankel Matrices
Straightening Law and Powers of Determinantal Ideals of Hankel Matrices
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Hankel矩阵行列式理想的矫直律和幂
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发表时间：
1998
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通讯作者：
A. Conca
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作者：
A. Conca
本文建立了一般汉克尔矩阵的标准单项式理论。我们所说的一般汉克尔矩阵是指矩阵Y=(yij)，其中yij=x i+ j&1，其中x i是域k上的不定式。我们利用这种结构确定了Y的行列式理想It的符号幂和幂的初等分解。进一步证明了它的符号代数和普通Rees代数是Cohen Macaulay正规域。第一个标准单项式理论是由Hodge [H]提出的，用于研究格拉斯曼变种的齐次坐标环。后来，Doubilet、Rota和Stein [DRS]为一般矩阵建立了标准单项式理论，De Concini和Procesi [DP]为一般对称和一般偏对称矩阵建立了标准单项式理论。这些都是在偏集或剂量集上具有矫直定律（简称ASL）的代数的例子。ASL的抽象概念是由Eisenbud [E1]和De Concini、Eisenbud和Procesi [DEP2]提出并发展起来的，参见[BV]。这些结构被证明是研究由上述泛型矩阵引起的行列式环和理想的一个极其强大的工具。现在设x1，…， xn在任意域k上不确定，对于j=1，…， n我们用Xj表示j_（n+1& j）汉克尔矩阵，其元素为x1，…， xn，即，
In this paper we establish a standard monomial theory for generic Hankel matrices. By a generic Hankel matrix we mean a matrix Y=( yij) with yij=x i+ j&1 where the x i are indeterminates over a field K. We use this structure to determine the symbolic powers and the primary decomposition of the powers of the determinantal ideals It of Y. Further we prove that the symbolic and ordinary Rees algebras of It are Cohen Macaulay normal domains. The first standard monomial theory was developed by Hodge [H] to study the homogeneous coordinate ring of the Grassmannian variety. Later standard monomial theories were established for generic matrices by Doubilet, Rota and Stein [DRS], and for generic symmetric and generic skew symmetric matrices by De Concini and Procesi [DP]. These are all examples of algebras with straightening law (ASL for short) over a poset or over a doset. The abstract notion of ASL was introduced and developed by Eisenbud [E1], and by De Concini, Eisenbud and Procesi [DEP2], see also [BV]. These structures turned out to be an extremely powerful tool in studying determinantal rings and ideals arising from the above mentioned generic matrices. Let now x1 , ..., xn be indeterminates over an arbitrary field K. For j=1, ..., n we denote by Xj the j_(n+1& j) Hankel matrix with entries x1 , ..., xn , that is,