Symmetric Polynomials in Free Associative Algebras—II

Silvia Boumova (), Vesselin Drensky, Deyan Dzhundrekov and Martin Kassabov
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Silvia Boumova: Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, 1113 Sofia, Bulgaria
Vesselin Drensky: Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, 1113 Sofia, Bulgaria
Deyan Dzhundrekov: Faculty of Mathematics and Informatics, Sofia University, 1164 Sofia, Bulgaria
Martin Kassabov: Department of Mathematics, Cornell University, Ithaca, NY 14853, USA

Mathematics, 2023, vol. 11, issue 23, 1-10

Abstract: 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.

Keywords: free associative algebra; noncommutative invariant theory; symmetric polynomials; finite generation (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2023
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