Pensieve Header: Verifying that the Nil-Hecke algebra acts on polynomials in n variables.

del[i_][f_] := Simplify[
(f-(f /. {x[i] -> x[i+1], x[i+1] -> x[i]}))/(x[i]-x[i+1])
]

del[1][f[x[1], x[2], x[3]]]

del[2][del[1][f[x[1], x[2], x[3]]]]

lhs=del[1][del[2][del[1][f[x[1], x[2], x[3]]]]]

rhs=del[2][del[1][del[2][f[x[1], x[2], x[3]]]]]

lhs==rhs