Symmetric Polynomials in Free Associative Algebras—II

Abstract

Let K⟨Xd⟩ be the free associative algebra of rank d≥2 over a field, K. In 1936, Wolf proved that the algebra of symmetric polynomials K⟨Xd⟩Sym(d) is infinitely generated. In 1984 Koryukin equipped the homogeneous component of degree n of K⟨Xd⟩ 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⟨Xd⟩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⟨Xd⟩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⟨Xd⟩Sym(d), taking into account that Koryukin’s action is infinitely generated, describe a minimal generating set.

Keywords

• free associative algebra
• noncommutative invariant theory
• symmetric polynomials
• finite generation