| In my signature work, we apply the knowledge of mathematics, statistics, and computer science to study the calculation of Zonal polynomials under the computational corollary of the Jack polynomials. We propose multiple algorithms to make the product of Zonal Polynomials as a linear combination form. In particular, we find out the cases of all neighboring values of linearization coefficients q[κ, λ] under different partitions κ, λ and finally use the definition and properties of symmetric polynomials to realize the rewriting of zonal polynomial productization. First, for the matrix Y possessing the corresponding dimensions, we use the recurrence relation to find the constant q[κ, λ] and the preparatory symmetric polynomial ℇ(Y), and successfully define the Zonal polynomial with corresponding linearization coefficient. Then, we encode the preparatory polynomial ℇ(Y) to realize the synthesis of its product under two different partitions. Finally, the product of two Zonal polynomials is successfully written in a new linear combinatorial form with the help of recurrence relations under our corollary. We also compare the speed and computational cost of different algorithms in achieving the linear combinatorial form and give the complete computational package to best save computational cost. In addition, in the search for the constant q[κ, λ], we use the matrix form to store the coefficients corresponding to each term in the zonal polynomial in order to facilitate the reader’s understanding. In addition, the occurrence of 0 values in the matrix is also explored to conform to the definition we give. |
