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Symmetric polynomials in free associative algebras. (English) Zbl 1504.13008

Let us fix the following notation: \[ \begin{array}{rcl} K &:& \text{a field,}\\ X_d &:& \text{the set \(\{x_1,\dots, x_d\}\) consisting of \(d\) noncommuting variables},\\ K[X_d] &:& \text{a polynomial \(K\)-algebra generated by \(d\) commuting variables,} \\ K\langle X_d\rangle &:& \text{the noncommutative polynomial \(K\)-algebra generated by the elements of \(X_d\),} \\ K X_d &:& \text{the \(K\)-vector space spanned by \(X_d\)},\\ \mathrm{Sym}(d) &:& \text{the symmetric group on the set \(\{1,\dots, d\}\)}. \end{array} \] The algebra \(K\langle X_d\rangle\) is naturally graded by degree. Let \(\mathbb{Z}_{\geq 0}\) denote the set of nonnegative integers. For \(n\in \mathbb{Z}_{\geq 0}\), let \((K\langle X_d\rangle)^{(n)}\) denote the subspace consisting of homogeneous degree \(n\) polynomials from \(K\langle X_d\rangle\). On one hand, the general linear group \(GL_d(K)\) acts on the left on \(KX_d\) via its defining representation. This action lifts to an action on \(K\langle X_d\rangle\) as follows: \begin{align*} g\cdot f(x_1,\dots, x_d) := f(g(x_1),\dots, g(x_d)), \end{align*} where \(f(x_1,\dots, x_d)\in (K\langle X_d\rangle)^{(n)}\) and \(g\in GL_d(K)\). On the other hand, for every \(n\in \mathbb{Z}_{\geq 0}\), we have a natural right action of \(\mathrm{Sym}(n)\) on \((K\langle X_d\rangle)^{(n)}\) which is defined by \begin{align*} \sigma \cdot x_{i_1}\cdots x_{i_n} := x_{\sigma^{-1}(1)}\cdots x_{\sigma^{-1}(n)}, \end{align*} where \(\{i_1,\dots, i_n\}\subseteq \{1,\dots, d\}\) and \(\sigma \in \mathrm{Sym}(n)\). This action is called the \(S\)-action on \(K\langle X_d\rangle\). Notice that the \(S\)-action of \(\mathrm{Sym}(n)\) and the left action of \(GL_d(K)\) commute with each other. Now, let \(F\) be a graded subalgebra of \(K\langle X_d\rangle\). If \(F\) is stable under the \(S\)-action, then \(F\) is called an \(S\)-algebra, and denoted by \((F,\circ)\). For example, for every subgroup \(G\subset GL_d(K)\), the \(G\)-invariants in \(K\langle X_d\rangle\) form an \(S\)-algebra, which is denoted by \((K\langle X_d\rangle^G,\circ)\). The notion of an “\(S\)-subalgebra” is defined in the most straightforward way. An \(S\)-algebra \((F,\circ)\) is said to be finitely generated (as an \(S\)-algebra) if there exists a finite subset \(U\subset F\) consisting of homogeneous polynomials such that \((F,\circ)\) is the minimal \(S\)-subalgebra of \((K\langle X_d\rangle, \circ)\) containing \(U\). A well-known result of A. N. Koryukin [Algebra Logic 23, 290–296 (1984; Zbl 0587.20023); translation from Algebra Logika 23, No. 4, 419–429 (1984)] states that if \(G\) is a reductive subgroup of \(GL_d(K)\), then \((K\langle X_d\rangle^G,\circ)\) is finitely generated as an \(S\)-algebra. In this pleasure-to-read article, the authors consider the following problems:
Let \(G\) be a finite subgroup of \(GL_d(K)\).
(i)
Is there a bound on the cardinality of the minimal system of homogeneous generators of the \(S\)-algebra \((K\langle X_d\rangle^G,\circ)\)?
(ii)
Find a system of generators of \((K\langle X_d\rangle^G,\circ)\) for concrete groups.
(iii)
If the commutative algebra \(K[X_d]^G\) is generated by a homogeneous system \(\{f_1,\dots, f_m\}\), then is there a lift of this system to a system of generators for \((K\langle X_d\rangle^G,\circ)\)?
The authors offer solutions to these problems when \(G\) is the symmetric group \(\mathrm{Sym}(d)\). The main result of the article is the following result.
Theorem 4.5. Let \(\mathrm{char}(K) = 0\) or \(\mathrm{char}(K) = p > d\). Then the algebra \((K\langle X_d\rangle^{\mathrm{Sym}(d)},\circ)\) is generated as an \(S\) algebra by the (noncommutative analogs of the) elementary symmetric polynomials \(p_{1^i}\) for \(i=1,\dots, d\).
In addition to the several different proofs of their main result, the authors present new proofs of some classical results of M. C. Wolf [Duke Math. J. 2, 626–637 (1936; Zbl 0016.00501)].

MSC:

13A50 Actions of groups on commutative rings; invariant theory
16S50 Endomorphism rings; matrix rings
15A72 Vector and tensor algebra, theory of invariants
16W20 Automorphisms and endomorphisms
16W22 Actions of groups and semigroups; invariant theory (associative rings and algebras)
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References:

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