elementary_inv.sagews

e = SymmetricFunctions(QQ).elementary()
h = SymmetricFunctions(QQ).homogeneous()
s = SymmetricFunctions(QQ).schur()
m = SymmetricFunctions(QQ).monomial()

#let n=4
#we want to check that the involution that maps e1 to e3 sends s_k to s_kkk if we assume e4=1 and e5=e6=...=0
#say, k=5
e(s([5]))#expand s5 in e's
e(s([5,5,5])) #expand s555 in e's

e[1, 1, 1, 1, 1] - 4*e[2, 1, 1, 1] + 3*e[2, 2, 1] + 3*e[3, 1, 1] - 2*e[3, 2] - 2*e[4, 1] + e[5] e[3, 3, 3, 3, 3] - 4*e[4, 3, 3, 3, 2] + 3*e[4, 4, 3, 2, 2] + 3*e[4, 4, 3, 3, 1] - 2*e[4, 4, 4, 2, 1] - 2*e[4, 4, 4, 3] + 3*e[5, 3, 3, 2, 2] - 3*e[5, 3, 3, 3, 1] - 2*e[5, 4, 2, 2, 2] - 2*e[5, 4, 3, 2, 1] + 4*e[5, 4, 3, 3] + e[5, 4, 4, 1, 1] + 2*e[5, 4, 4, 2] + e[5, 5, 2, 2, 1] + 2*e[5, 5, 3, 1, 1] - 4*e[5, 5, 3, 2] - 2*e[5, 5, 4, 1] + e[5, 5, 5] - 2*e[6, 3, 2, 2, 2] + 4*e[6, 3, 3, 2, 1] - 2*e[6, 3, 3, 3] + 2*e[6, 4, 2, 2, 1] - 4*e[6, 4, 3, 1, 1] + 2*e[6, 4, 4, 1] - 2*e[6, 5, 2, 1, 1] + 2*e[6, 5, 2, 2] + 2*e[6, 5, 3, 1] - 2*e[6, 5, 4] + e[6, 6, 1, 1, 1] - 2*e[6, 6, 2, 1] + e[6, 6, 3] + e[7, 2, 2, 2, 2] - 3*e[7, 3, 2, 2, 1] + e[7, 3, 3, 1, 1] + 2*e[7, 3, 3, 2] + 2*e[7, 4, 2, 1, 1] - 2*e[7, 4, 2, 2] - 2*e[7, 4, 3, 1] + e[7, 4, 4] - e[7, 5, 1, 1, 1] + 2*e[7, 5, 2, 1] - e[7, 5, 3]

#for example, you can see that the second term - 4*e[2, 1, 1, 1] is mapped to - 4*e[4, 3, 3, 3, 2]=-4*e4*e3*e3*e3*e2=-4*e3*e3*e2 because e4=1