1. Macdonald polynomials Macdonald polynomials P λ (x; q, t) are orthogonal symmetric polynomials which are the natural multivariable generalisation of the continuous q-ultraspherical polyno-mials C n (x; β|q) [2] which, in their turn, constitute an important class of hyper-geometric orthogonal polynomials in one variable. Polynomials C n (x; β|q) can be obtained from the general Askey-Wilson p...

It is shown that two vectors with coordinates in the finite q-element field of characteristic p belong to same orbit under natural action symmetric group if each elementary polynomials degree pk,2pk,…,(q−1)pk, k=0,1,2,… has value on them. This separating set polynomial invariants for permutation representation not far from being minimal when q=p and dimension large compared p. A relatively smal...