On a conjecture of Hivert and Thiéry about Steenrod operators

We prove some results related to a conjecture of Hivert and Thiéry about the dimension of the space of q-harmonics (F. Hivert and N. Thiéry, 2004 [HT]). In the process we compute the actions of the involved operators on symmetric and alternating functions, which have some independent interest. We then use these computations to prove other results related to the same conjecture.