W-Operators and Permutation Groups

Hao Sun

Published 2016-10-20Version 1

W-operators are differential operators on the polynomial ring. A special example of W-operators is the cut-and-join operator. We study the relation between W-operators and permutation groups. We find the W-operator $W([d])$ can be written as the sum of $d!$ summations. Each summation can be determined by a unique permutation in the permutation group $S_d$.