Construction of Markov chane via algebraic approach for random generation of partition tables

设A=[symmetry]^c <i=0> Ai是域上的零维分次Gorenstein代数，×A→End（A）是正则表示.设z∈A是线性形式.设幂零矩阵×z∈End（V）可分解为大小为{f_1，.，f_s}的Jordan块.我们称模U_i=0：z^<f_i-1>+（z）/0：z^<f_i>+（z）为（A，z）的第i个中心单模。我们证明了以下结果。定理（1）每个U_i有一个对称的Hilbert函数。(2)如果每个U_i对所有i都有强Lefschetz性质，则A也有强Lefschetz性质.在这个定理中，如果我们去掉Gorenstein条件，而加上（1）U_i对所有i都有对称Hilbert函数，（2）A有对称Hilbert函数，那么我们就可以推出同样的结果.这有许多应用.例如证明了多项式环中由连续次数幂和生成的完全交理想具有强Lefschetz性质，在完全交A=K[x_1，x_2，.，x_n]/（x^d_1，..，x ^d_n）中设L=x_1+...+x_n.则（A，L）的中心单模模U_i是S_n-模。当d=2时，U_i由（i-1）次的Specht多项式生成.当n=2时，U_i是一维的。在任一情况下U_i都是不可约S_n-模。利用这一点以及L是强Lefschetz元的事实，可以将A分解为不可约S_k-模。

Let A=【symmetry】^c_<i=0>A_i be a zero-dimensional graded Gorenstein algebra over a field and let ×A→End(A) be the regular representation. Let z∈A be a linear form. Suppose that the nilpotent matirx ×z∈End(V) decomposes into Jordan blocks of sizes {f_1,…,f_s}. We call the module U_i=0:z^<f_i-1>+(z)/0:z^<f_i>+(z) ith central simple module of (A,z). We proved the following result. Theorem (1) Each U_i has a symmetric Hilbert fucntion. (2) If each U_i has the strong Lefschetz property, for all i, then A has the strong Lefschetz property.In this theorem if we drop the condition "Gorenstein", but add the conditions that (1) U_i has a symmetric Hilbert function, for all i and (2) A has a symmetric Hilbert function, then we may deduce the same result.This has many applications. For example it can be proved that a complete intersection ideal generated by power sums of consecutive degrees in a polynomial ring has the strong Lefschetz property.In the complete intersection A=K[x_1,x_2,…,x_n]/(x^d_1,…,x^d_n) put L=x_1+…+x_n. Then the central simple module module U_i of (A,L) is an S_n-module. When d=2, U_i is spanned by Specht polynomials of degree (i-1). When n=2, U_i is one-dimensional. In either case U_i is an irreducible S_n-module. Using this and the fact that L is a strong Lefschetz element, it is possible to decompose A into irreducible S_k-modules.