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

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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.