Symmetric Polynomials in Free Associative Algebras—II

Let K ⟨ X d ⟩ be the free associative algebra of rank d ≥ 2 over a field, K . In 1936, Wolf proved that the algebra of symmetric polynomials K ⟨ X d ⟩ Sym ( d ) is infinitely generated. In 1984 Koryukin equipped the homogeneous component of degree n of K ⟨ X d ⟩ with the additional action of Sym ( n ) by permuting the positions of the variables. He proved finite generation with respect to this additional action for the algebra of invariants K ⟨ X d ⟩ G of every reductive group, G . In the first part of the present paper, we established that, over a field of characteristic 0 or of characteristic p > d , the algebra K ⟨ X d ⟩ Sym ( d ) with the action of Koryukin is generated by (noncommutative version of) the elementary symmetric polynomials. Now we prove that if the field, K , is of positive characteristic at most d then the algebra K ⟨ X d ⟩ Sym ( d ) , taking into account that Koryukin’s action is infinitely generated, describe a minimal generating set.