Symmetric Polynomials in Free Associative Algebras—II

Abstract

Let $K⟨X_d⟩$ be the free associative algebra of rank $d\ge 2$ over a field, K. In 1936, Wolf proved that the algebra of symmetric polynomials $K{⟨X_d⟩}^{Sym\left(d\right)}$ is infinitely generated. In 1984 Koryukin equipped the homogeneous component of degree n of $K⟨X_d⟩$ with the additional action of $Sym\left(n\right)$ 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 $K{⟨X_d⟩}^{Sym\left(d\right)}$ 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 $K{⟨X_d⟩}^{Sym\left(d\right)}$, taking into account that Koryukin’s action is infinitely generated, describe a minimal generating set.

1. Introduction

Noncommutative invariant theory started with the paper by Margarete Wolf [1] in 1936, where she studied the symmetric polynomials in the free associative algebra $K⟨X_d⟩$.

Here, $K⟨X_d⟩$ is the free d-generated associative algebra with 1, over a field K, i.e., the algebra of polynomials of $d\ge 2$ noncommuting variables. Wolf proved that the algebra $K{⟨X_d⟩}^{Sym\left(d\right)}$ of symmetric polynomials in $K⟨X_d⟩$ is not finitely generated. In the first part of this paper [2] we reviewed classical results of invariant theory of finite groups and compared them to similar ones in noncommutative invariant theory. We also briefly summarized the results of Wolf. We paid special attention to the result of Koryukin (1984) [3], who suggested an additional action $\circ$ of $Sym\left(n\right)$ of the homogeneous component of degree n of $K⟨X_d⟩$, namely permutation of the positions of the variables. We call the subalgebras of $(K⟨X_d⟩,\circ )$, which are invariant under this action, S-algebras. Koryukin established the finite generation of the S-algebra of invariants $\left(K{⟨X_d⟩}^G,\circ \right)$ of every reductive group, G. In [2], we showed that, over a field of characteristic 0 or of characteristic $p>d$, the S-algebra $(K{⟨X_d⟩}^{Sym\left(d\right)},\circ )$ is generated by the noncommutative generalizations of the elementary symmetric polynomials. We also conjectured that if the ground field, K, is of characteristic $p\le d$ then the S-algebra $\left(K{⟨X_d⟩}^{Sym\left(d\right)},\circ \right)$ is not finitely generated. In the present paper we give a positive answer to this conjecture.

2. Preliminaries

In this section we give some classical results from the theory of commutative invariants and compare them with their counterparts in the noncommutative case.

The classical invariant theory is often described over the complex field, $\mathbb{C}$; however, many of the results hold over any field, K, of characteristic 0. Let $K\left[X_d\right]=K[x_1,\dots ,x_d]$ be the algebra of polynomial functions on the d-dimensional vector space, $V_d$, with basis $\{v_1,\dots ,v_d\}$. The general linear group, $GL\left(V_d\right)={GL}_d\left(K\right)$, acts on $\mathrm{span}\{x_1,\dots ,x_d\}$ as on the dual $V_d^{*}$, i.e., one identifies each $x_i$ with the linear functional $V_d\to \mathbb{C}$ by

$$x_i({\xi }_1{v}_1+\cdots +{\xi }_d{v}_d)={\xi }_i,{\xi }_1,\dots ,{\xi }_d\in \mathbb{C}.$$

This induces an action of ${GL}_d\left(K\right)$ on $K\left[X_d\right]$ given by

$$g\left(f\right):v\to f\left(g^{-1}\left(v\right)\right),g\in {GL}_d\left(K\right),f\left(X_d\right)\in \mathbb{C}\left[X_d\right],v\in V_d.$$

The algebra of G-invariants for any subgroup, G, of ${GL}_d\left(K\right)$ is

$$\mathbb{C}{\left[X_d\right]}^G=\left\{f\in K\left[X_d\right]\mid g\left(f\right)=f\mathrm{for}\mathrm{all}g\in G\right\}.$$

The first result of classical commutative invariant theory is the fundamental theorem of symmetric polynomials (see [4,5] for the history of these results):

Symmetric polynomials can be written, in a unique way, as polynomials in the elementary symmetric polynomials.

In other words, for any field, K, of an arbitrary characteristic and for the action of the symmetric group, $Sym\left(d\right)$, of degree d on the vector space, $K{X}_d$, by

$$\sigma \left(x_i\right)=x_{\sigma \left(i\right)},\sigma \in Sym\left(d\right),i=1,\dots ,d,$$

the algebra $K{\left[X_d\right]}^{Sym\left(d\right)}$ of $Sym\left(d\right)$-invariants is generated by the elementary symmetric polynomials

$$e_m=\sum _{j_1<\cdots <j_m}{x}_{j_1}\cdots x_{j_m},m=1,2,\dots ,d,$$

moreover, the polynomials $e_1,e_2,\dots ,e_d$ are algebraically independent.

In 1900, at the International Congress of Mathematicians in Paris, Hilbert posed to the mathematical community and discussed 23 problems, the solutions of which he presented as a guide and challenge for the beginning of the century [6]. The 14th problem in this remarkable lecture, “Mathematische Probleme”, was inspired by the finite generation problem of $K{\left[X_d\right]}^G$ for all subgroups, G, of ${GL}_d\left(K\right)$.

Emmy Noether [7] gave an affirmative answer to the problem for finite groups, G, when the field K has characteristic 0, and extended this result to fields of arbitrary characteristic [8] in 1926. Hilbert’s earlier work [9] (1890–1893) contains nonconstructive proof of the finite presentability of the algebra $K{\left[X_d\right]}^G$ for reductive groups, G, over a field of zero characteristic. A counterexample to the problem was given by Nagata [10] in the 1950s in the general case.

The following three classical results on commutative invariant theory are considered to be cornerstones of the theory.

Theorem 1. 

(Endlichkeitssatz of Emmy Noether [7]) The algebra of invariants $K{\left[X_d\right]}^G$ is finitely generated for G, being a finite subgroup of ${GL}_d\left(K\right)$, and for a ground field, K, of characteristic 0. It has a system of homogeneous generators of degree bounded from above by the order of the group G.

Theorem 2. 

(Chevalley–Shephard–Todd [11,12]) For a finite group, G, over a field of characteristic 0, the algebra of invariants $K{\left[X_d\right]}^G$ is isomorphic to a polynomial algebra, i.e., $K{\left[X_d\right]}^G$ has a system of algebraically independent generators if and only if $G<{GL}_d\left(K\right)$ is generated by pseudo-reflections (matrices of finite order with a conjugate that is a diagonal matrix of the form $\mathrm{diag}(1,\dots ,1,\xi )$, where $\xi \ne 1$ is a root of unity).

The third important theorem is the Molien formula [13] from 1897, which “counts” the invariants: the algebra $K{\left[X_d\right]}^G$ is graded for any group, G, and decomposes as a direct sum

$$K{\left[X_d\right]}^G=K\oplus {\left(K{\left[X_d\right]}^G\right)}^{\left(1\right)}\oplus {\left(K{\left[X_d\right]}^G\right)}^{\left(2\right)}\oplus \cdots ,$$

of its homogeneous components ${\left(K{\left[X_d\right]}^G\right)}^{\left(n\right)}$ of degree n. The Hilbert (or Poincaré) series of $K{\left[X_d\right]}^G$ is

$$H\left(K{\left[X_d\right]}^G,t\right)=\sum _{n\ge 0}dim{\left(K{\left[X_d\right]}^G\right)}^{\left(n\right)}·t^n$$

The Hilbert–Serre theorem gives that the Hilbert series of a finitely generated graded commutative algebra is rational, which implies for a finite group, G, that

$$H\left(K{\left[X_d\right]}^G,t\right)=p\left(t\right)\prod _{i=1}^m\frac{1}{1-t^{a_i}},$$

for some natural numbers, $a_i$, and a polynomial $p\left(t\right)\in \mathbb{Z}\left[t\right]$. In the case of algebras of invariants, the Molien formula makes more precise:

Theorem 3. 

Let the field K be of characteristic 0 and G be a finite group. Then

$$H\left(K{\left[X_d\right]}^G,t\right)=\frac{1}{\left|G\right|}\sum _{g\in G}\frac{1}{det\left(1-gt\right)}.$$

We juxtapose the results in the invariant theory of finite groups in the commutative case and the noncommutative one.

Let K be an arbitrary field and $K⟨X_d⟩=K⟨x_1,\cdots ,x_d⟩$ be the free unitary associative algebra generated by the set of variables $X_d=\{x_1,\dots ,x_d\}$. In the class of unitary associative algebras, the algebra $K⟨X_d⟩$ possesses a universal property as $K\left[X_d\right]$ in the class of commutative algebras: every mapping, $X_d\to R$, of $X_d$ to an algebra, R, can be uniquely extended to a homomorphism, $K⟨X_d⟩\to R$. As in the commutative case, the general linear group, ${GL}_d={GL}_d\left(K\right)=GL\left(K{X}_d\right)$, acts canonically on the vector space, $K{X}_d$, with basis $X_d$. This action can be extended to $K⟨X_d⟩$ where each g acts by an algebra homomorphism:

$$g\left(f(x_1,\cdots ,x_d)\right)=f(g\left(x_1\right),\cdots ,g\left(x_d\right)),g\in G{L}_d,f(x_1,\cdots ,x_d)\in K⟨X_d⟩.$$

The algebra of G-invariants $K{⟨X_d⟩}^G$ for a subgroup, G, of ${GL}_d\left(K\right)$ consists of all polynomials, $f(x_1,\dots ,x_d)\in K⟨X_d⟩$, that remain unchanged under the action of G.

The first results of invariant theory (as in the commutative case) were in the case of the symmetric group $Sym\left(d\right)$ with its natural action, by Margarete Wolf [1] in 1936.

Theorem 4 

(Wolf [1]). (i) For any field, K, the algebra of symmetric polynomials $K{⟨X_d⟩}^{Sym\left(d\right)}$, $d\ge 2$, is a free associative algebra.

(ii) It has a homogeneous system of free generators $\{f_j\mid j\in J\}$ with the property that there is at least one generator of degree n for any $n\ge 1$.

(iii) Any homogeneous system of generators of $K{⟨X_d⟩}^{Sym\left(d\right)}$ has the same number of polynomials of degree n.

(iv) If

$$f=\sum _{j=(j_1,\dots ,j_m)}{\alpha }_j{f}_{j_1}\cdots f_{j_m}\in K{⟨X_d⟩}^{Sym\left(d\right)},{\alpha }_j\in K,$$

then the coefficients ${\alpha }_j$ are linear combinations with integer coefficients of the coefficients of f.

Thirty years later, Bergman and Cohn [14] generalized Wolf’s [1] main result. Different aspects in the theory have been considered by many authors, see for example [15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30].

The following theorem was proved independently by Dicks and Formanek [31] and Kharchenko [32] for finite groups, and by Koryukin [3] in the general case. It demonstrates that the algebras $K{\left[X_d\right]}^G$ and $K{⟨X_d⟩}^G$ behave in a completely different way with respect to the finite generation.

Theorem 5 

([3]). For an arbitrary subgroup, G, of ${GL}_d\left(K\right)$ over any field, K, of an arbitrary characteristic let $K{Y}_m$ be the minimal subspace of $K{X}_d$, such that $K{⟨X_d⟩}^G\subseteq K⟨Y_m⟩$. Then, the algebra $K{⟨X_d⟩}^G$ is finitely generated if and only if G acts on $K{Y}_d$ by scalar multiplication.

Corollary 1 

([31,32]). When G is a finite subgroup of ${GL}_d\left(K\right)$, the algebra of invariants $K{⟨X_d⟩}^G$ is finitely generated if and only if G is a finite cyclic group consisting of scalar matrices.

Corollary 2 

([3]). If G acts irreducibly on $K{X}_d$, i.e., if $K{X}_d$ does not have nontrivial subspaces, W, such that $G\left(W\right)=W$, then $K{⟨X_d⟩}^G$ is either trivial or not finitely generated.

The analogue of the Chevalley–Shephard–Todd theorem also looks differently for the free associative algebra $K⟨X_d⟩$.

Theorem 6. 

(i) (Lane [33] and Kharchenko [34]) For any subgroup, G, of ${GL}_d\left(K\right)$ and for any field, K, the algebra $K{⟨X_d⟩}^G$ is free.

(ii) (Kharchenko [34]) For finite groups, G, there is a Galois correspondence between the free subalgebras of $K⟨X_d⟩$ containing $K{⟨X_d⟩}^G$ and the subgroups of G—a subalgebra, F, of $K⟨X_d⟩$ with $K{⟨X_d⟩}^G\subseteq F$ is free if and only if $F=K{⟨X_d⟩}^H$ for a subgroup, H, of G.

A variation of Molien’s formula also hold for $K⟨X_d⟩$:

Theorem 7 

(Dicks and Formanek [31]). Let G be a finite group of ${GL}_d\left(K\right)$ and field K be with characteristic 0. Then

$$H\left(K{⟨X_d⟩}^G,t\right)=\frac{1}{\left|G\right|}\sum _{g\in G}\frac{1}{1-tr\left(g\right)t}.$$

The paper of Koryukin [3] was our motivation for [2]. Koryukin introduced an additional structure on $K{⟨X_d⟩}^G$, which changes the notion of finite generation, and the algebra of invariants is “finitely generated” in this weak sense in some cases.

Let ${\left(K⟨X_d⟩\right)}^{\left(n\right)}$ be the homogeneous component of degree n in $K⟨X_d⟩$. Let us consider the action of the symmetric group $Sym\left(n\right)$ from the right on ${\left(K⟨X_d⟩\right)}^{\left(n\right)}$ by the rule

$$(x_{i_1}\cdots x_{i_n})\circ \sigma =x_{i_{{\sigma }^{-1}\left(1\right)}}\cdots x_{i_{{\sigma }^{-1}\left(n\right)}},\sigma \in Sym\left(n\right).$$

We name this action the S-action, and indicate by $(K⟨X_d⟩,\circ )$ the algebra $K⟨X_d⟩$ with the described action of $Sym\left(n\right)$ on ${\left(K⟨X_d⟩\right)}^{\left(n\right)}$, $n=0,1,2,\dots$. Given a graded subalgebra, F, of $K⟨X_d⟩$, which inherits the S-action, i.e., for any n it holds $F^{\left(n\right)}\circ Sym\left(n\right)=F^{\left(n\right)}$ for its homogeneous component $F^{\left(n\right)}$ of degree n, we denote it by $(F,\circ )$ and label it as an S-algebra. The finite generation of $(F,\circ )$ means that there exists a finite subset, U, of homogeneous polynomials of F, such that U generates $(F,\circ )$. Since the left action of ${GL}_d\left(K\right)$ on ${\left(K⟨X_d⟩\right)}^{\left(n\right)}$ commutes with the right action of $Sym\left(n\right)$, the algebra of invariants $\left(K{⟨X_d⟩}^G,\circ \right)$ is an S-algebra for any subgroup, G, of ${GL}_d\left(K\right)$.

Theorem 8 

(Koryukin [3]). For any field, K, and any reductive subgroup, G, of ${GL}_d\left(K\right)$ the S-algebra $\left(K{⟨X_d⟩}^G,\circ \right)$ is finitely generated.

By the Maschke theorem, if the field K has characteristic 0 or characteristic $p>0$, and p does not divide the order of G, then the finite dimensional representations of G are completely reducible. Hence, Theorem 8 inspires the following problem.

Problem 1. 

Let G be a finite subgroup of ${GL}_d\left(K\right)$ and let $char\left(K\right)=0$ or $char\left(K\right)=p>0$ and p does not divide the order of G.

(i) For a minimal homogeneous generating system of the S-algebra $\left(K{⟨X_d⟩}^G,\circ \right)$ is there a bound of the degree of the generators in terms of the order $\left|G\right|$ of G, the rank d of $K⟨X_d⟩$ and the characteristic of K?

(ii) Find a finite system of generators of $\left(K{⟨X_d⟩}^G,\circ \right)$ for concrete groups, G.

(iii) If the commutative algebra $K{\left[X_d\right]}^G$ is generated by a homogeneous system $\{f_1,\dots ,f_m\}$, can this system be lifted to a system of generators of $\left(K{⟨X_d⟩}^G,\circ \right)$?

Remark 1. 

By the Endlichkeitsatz of Emmy Noether [7], if $char\left(K\right)=0$, then $K{\left[X_d\right]}^G$ has a set of generators of degree $\le \left|G\right|$ for any finite group, G. Fleischmann [35] and Fogarty [36] proved that the same upper bound holds if $char\left(K\right)=p>0$ does not divide the order $\left|G\right|$ of G. Hence, in Problem 1 (i), it is reasonably to restrict our attention to the order of G and the rank, d, of $K⟨X_d⟩$.

In [2], we gave the answer to the questions in Problem 1 if G is the symmetric group of degree, d, and showed that $\left(K{⟨X_d⟩}^G,\circ \right)$ is generated (as S-algebra) by analogs of the elementary symmetric functions.

3. Infinite Generation in the Case $\mathit{p}\le \mathit{d}$

Koryukin’s result (Theorem 8) does not extend to the non-reductive case. Our main result shows that when $d\ge p$ the S-algebra $\left(K{⟨X_d⟩}^{Sym\left(d\right)},\circ \right)$ is not finitely generated.

Remark 2. 

For $d^{\prime }>d$, we have a projection from $K⟨X_{d^{\prime }}⟩$ to $K⟨X_d⟩$, which sends the extra generators to 0. It is easy to see that this projection induces a surjective map between the S-algebras of symmetric polynomials. Thus, it is enough to establish that the S-algebra $\left(K{⟨X_d⟩}^{Sym\left(d\right)},\circ \right)$ in not finitely generated in the case $char\left(K\right)=p=d$. In the sequel, we shall assume that $d=p$.

Let us consider the augmentation ideal ${\left(K{⟨X_d⟩}^{Sym\left(d\right)},\circ \right)}^{+}$ of $\left(K{⟨X_d⟩}^{Sym\left(d\right)},\circ \right)$, i.e., the ideal of polynomials without a constant term in $\left(K{⟨X_d⟩}^{Sym\left(d\right)},\circ \right)$. Let $M_d$ be the quotient of ${\left(K{⟨X_d⟩}^{Sym\left(d\right)},\circ \right)}^{+}$ by its square, i.e.,

$$M_d:={\left(K{⟨X_d⟩}^{Sym\left(d\right)},\circ \right)}^{+}/\circ \left({\left({\left(K{⟨X_d⟩}^{Sym\left(d\right)},\circ \right)}^{+}\right)}^2\right),$$

where $\circ \left(V\right)$ denotes the submodule of $K⟨X_d⟩$ generated by V under the action $\circ$. ${\displaystyle M_d=\underset{n\in \mathbb{N}}{⨁}{M}_d^{\left(n\right)}}$ is naturally graded and each homogeneous component, $M_d^{\left(n\right)}$, is an $Sym\left(n\right)$-module, i.e., there is a natural $\circ$-action on M.

Finding the minimal generating set $\left(K{⟨X_d⟩}^{Sym\left(d\right)},\circ \right)$ is essentially equivalent to identifying $M_d$ and its graded components. The first step of achieving that is constructing a small generating set of $M_d$ as a vector space over K.

Lemma 1. 

The vector space, $M_d$, is generated as a $\circ$-module and as a vector space by the images of the power sums

$$p_n=x_1^n+\cdots +x_d^n,n=1,2,\dots .$$

Proof. 

It was shown in [2] (Lemma 4.1) that, over any field, K, of arbitrary characteristic, the S-algebra $\left(K{⟨X_d⟩}^{Sym\left(d\right)},\circ \right)$ is generated by the power sums $p_n$, $n=1,2\dots$. This implies that $M_d$ is generated as a $\circ$-module by the images of $p_n$. Since, for each n, the power sum, $p_n$, is invariant under the action of $Sym\left(d\right)$, these images also generate $M_d$ as a vector space.

This does not imply that $\left(K{⟨X_d⟩}^{Sym\left(d\right)},\circ \right)$ is infinitely generated, since some of elements $p_n$ might become trivial when projected to $M_d$. The main result in [2] shows that this indeed happens when $char\left(K\right)=0$ or $char\left(K\right)>d$ and the image of $p_n$ in $M_d$ is trivial for $n>d$. □

The observation that the power sums in $K⟨X_d⟩$ are fixed under the $\circ$-action suggests the idea that we can obtain useful information by passing to a suitable quotient of $K{⟨X_d⟩}^{Sym\left(d\right)}$ where the $\circ$-action becomes trivial. Consider the abelianization map $\pi :K⟨X_d⟩\to K\left[X_d\right]$ and the map induced by it on the subalgebras of symmetric polynomials. The map, $\pi$, is clearly an algebra homomorphism.

Lemma 2. 

The map, π, sends a generating set of the S-algebra $\left(K{⟨X_d⟩}^{Sym\left(d\right)},\circ \right)$ to a generating set of the commutative algebra $\pi \left(\left(K{⟨X_d⟩}^{Sym\left(d\right)},\circ \right)\right)\subset K{\left[X_d\right]}^{Sym\left(d\right)}$.

Proof. 

The statement follows immediately from the observation that the extra action, $\circ$, disappears (becomes trivial) after the map, $\pi$. □

The above lemma allows us to prove that the S-algebra $\left(K{⟨X_d⟩}^{Sym\left(d\right)},\circ \right)$ is not finitely generated if we can show that its image under $\pi$ is not finitely generated.

Remark 3. 

Although the map $\pi :K⟨X_d⟩\to K\left[X_d\right]$ is surjective it does not induce a surjective map between $K{⟨X_d⟩}^{Sym\left(d\right)}$ and $K{\left[X_d\right]}^{Sym\left(d\right)}$. For example, if $char\left(K\right)=2$, then $\pi (x_1{x}_2+x_2{x}_1)=0\in K{\left[X_2\right]}^{\mathrm{Sym}\left(2\right)}$ and the elementary symmetric function $e_2=x_1{x}_2$ is not in the image of π.

In order to compute the image for of the map $\pi$ we will introduce a notation which can be slightly misleading. Let u be a monomial (either in $K⟨X_d⟩$ or in $K\left[X_d\right]$). Since the action of $Sym\left(d\right)$ preserves the set of monomials, one can construct invariants by summing over the orbits of $Sym\left(d\right)$ acting on the set of monomials, i.e.,

$$\sum u=\sum _{g\in Sym\left(d\right)/H_u}g\left(u\right)$$

is in the algebra of invariants, where $H_u$ is the stabilizer of the monomial u under the action of the symmetric group $Sym\left(d\right)$.

There are several important comments to make. First, the notation $\sum u$ is defined only for monomials and not for arbitrary elements in the algebra (technically we can use the above formula to define $\sum u$ for arbitrary algebra elements but this operation will NOT be linear). Second, one needs to be careful weather u is considered as an element in $K⟨X_d⟩$ or in $K\left[X_d\right]$ because the stabilizer $H_u$ depends on that. Finally, we can observe that since both $K⟨X_d⟩$ or $K\left[X_d\right]$ have bases consisting of monomials, one has that the algebras of invariants $K{⟨X_d⟩}^{Sym\left(d\right)}$ and $K{\left[X_d\right]}^{Sym}$ are spanned by $\sum u$. Therefore the image of $K{⟨X_d⟩}^{Sym\left(d\right)}$ under $\pi$ is generated as a K-vector space by $\pi \left(\sum u\right)$.

The key observation is the following lemma, which informally says that $\pi$ almost commutes with ∑, even though ∑ is not an algebraic operation.

Lemma 3. 

For any monomial $u\in K⟨X_d⟩$, there exists an integer constant $c_u\in \mathbb{N}$ such that

$$\pi \left(\sum u\right)=c_u\left(\sum \pi \left(u\right)\right).$$

Moreover, in the case $p=d$ the constant $c_u$ is 0 in K if and only if $\pi \left(u\right)=x_1^s{x}_2^s\cdots x_p^s$ for some $s\ge 1$.

Proof. 

As mentioned above, ∑ has two different meanings in $K{⟨X_d⟩}^{Sym\left(d\right)}$ and in $K\left[X_d\right]$, but in both cases

$$\sum u=\sum _{g\in Sym\left(d\right)/H_u}g\left(u\right),$$

where $H_u$ is the stabilizer of the monomial, u, under the action of the symmetric group, $Sym\left(d\right)$. The difference arises since $H_u$ is not equal to $H_{\pi \left(u\right)}$.

Since $\pi$ is a homomorphism from $K⟨X_d⟩$ to $K\left[X_d\right]$, which is compatible with the action of $Sym\left(d\right)$, we have that $H_u\subset H_{\pi \left(u\right)}$. Thus, the constant, $c_u$, is equal to the index of $H_u$ in $H_{\pi \left(u\right)}$. It is not hard to see that $H_u$ is the symmetric group on the variables, which does not appear in u, and $H_{\pi \left(u\right)}$ is the product of symmetric groups on the variables, which appears in u with the same degree. Therefore, ${\displaystyle c_u=\prod _{i\ge 1}{k}_i!}$, where $k_i$ is the number of variables in u, which appear exactly i times. In the case $d=charK$, $c_u=0$ (as an element in K, i.e., that $p|c_u$) if and only if $k_s=p=d$ for some $s\ge 1$ (since $p/|k_i!$ unless $k_i=p$). □

Lemma 4. 

In the case $d=p=charK$, the commutative algebra

$$\pi \left(K{⟨X_d⟩}^{Sym\left(d\right)}\right)\subset K[e_1,\dots ,e_d]=K{\left[X_d\right]}^{Sym\left(d\right)}$$

is spanned by all products, $e_1^{m_1}\cdots e_d^{m_d}$, of the elementary symmetric polynomials except the powers, $e_p^m$, of $e_p$.

Proof. 

The image of $K{⟨X_d⟩}^{Sym\left(d\right)}$ in $K{\left[X_d\right]}^{Sym\left(d\right)}$ under $\pi$ is a subalgebra of $K{\left[X_d\right]}^{Sym\left(d\right)}$ and contains the elements $e_i{e}_p^m$ for all $1\le i<p$ and all $m\ge 0$—to see this it is enough to construct a monomial, u, such that $\pi \left(u\right)$ is a non-zero multiple of $e_i{e}_p^m$, which follow by the previous Lemma 3 and the observation $e_i{e}_p^m$ is equal to $\sum (x_1\dots x_i){(x_1\dots x_p)}^m$ in $K\left[X_d\right]$. By the same lemma, $\pi \left(K{⟨X_d⟩}^{Sym\left(d\right)}\right)$ does not contain the m-th power of $e_p$, otherwise $\pi$ will induce a surjective map in degree $mp$. □

Theorem 9. 

When $d\ge char\left(K\right)=p>0$, the S-algebra $\left(K{⟨X_d⟩}^{Sym\left(d\right)},\circ \right)$ is not finitely generated.

Proof. 

By Remark 2, it is sufficient to show that the image $\pi \left(K{⟨X_d⟩}^{Sym\left(d\right)}\right)$ of $K{⟨X_d⟩}^{Sym\left(d\right)}$ in $K{\left[X_d\right]}^{Sym\left(d\right)}$ is not finitely generated for $d=p$ only. Let B denote $\pi \left(K{⟨X_d⟩}^{Sym\left(d\right)}\right)$ and let $B^{+}$ be the augmentation ideal of B. Clearly, $B=K\oplus B^{+}$, and therefore B is finitely generated if and only if ${\left(B^{+}\right)}^2$ is of finite co-dimension in $B^{+}$. However, the description of B as a subalgebra of $K[e_1,\dots ,e_d]$ in Lemma 4 allows us to see that the quotient space $B^{+}/{\left(B^{+}\right)}^2$ has a basis consisting of the images of $e_i{e}_p^m$ for all $i<p$ and all $m\ge 0$, and thus it is infinite dimensional. Hence, the algebra $\pi \left(K{⟨X_d⟩}^{Sym\left(d\right)}\right)$ is not finitely generated. □

This argument shows that any generating set of the S-algebra $\left(K{⟨X_d⟩}^{Sym\left(d\right)},\circ \right)$ needs to contain a generator of degree $k=mp+i$ for any $i=1,\dots ,p-1$. Therefore, the image of $p_n$ in $M_d$ is nontrivial when $p/|n$.

4. Generating Set

As remarked above, the proof of Theorem 9 gives that any homogeneous generating set of the S-algebra $\left(K{⟨X_d⟩}^{Sym\left(d\right)},\circ \right)$ contains generators of degree $mp+i$ for any $m\ge 0$ and $1\le i<p$; however, it says nothing about degrees $mp$ and it does not give a minimal system of generators. In this section we shall show that any generating set contains also generators of degree $mp$. More precisely, we shall prove that $\{p_n\mid n=1,2,\dots \}$ is a minimal generating set of the S-algebra $\left(K{⟨X_d⟩}^{Sym\left(d\right)},\circ \right)$.

Example 1. 

We shall illustrate the idea of the proof that the power sums $p_n$, $n=1,2,\dots$, form a minimal generating set of the S-algebra $\left(K{⟨X_d⟩}^{Sym\left(d\right)},\circ \right)$ in the special case $p=2$ and $n=3$. Note that this case is covered in the proof of Theorem 9; however, the case $n=p=2$ is too easy and the case $n=4$ leads to an unnecessary large system of linear equations.

We want to show that $p_3$ does not belong to the S-subalgebra F of $\left(K{⟨X_2⟩}^{Sym\left(2\right)},\circ \right)$ generated by $p_1$ and $p_2$. The following polynomials of degree 3 span the homogeneous component of degree 3 of the S-algebra F:

$$p_1^3=(x_1^3+x_1^2{x}_2+x_1{x}_2{x}_1+x_1{x}_2^2)+(x_2^3+x_2^2{x}_1+x_2{x}_1{x}_2+x_2{x}_1^2),$$

$$p_1{p}_2=(x_1^3+x_1{x}_2^2)+(x_2^3+x_2{x}_1^2),$$

$$\left(p_1{p}_2\right)\circ \left(12\right)=(x_1^3+x_1{x}_2{x}_1)+(x_2^3+x_2{x}_1{x}_2),$$

$$\left(p_1{p}_2\right)\circ \left(13\right)=(x_1^3+x_1^2{x}_2)+(x_2^3+x_2^2{x}_1).$$

(Here we use that $p_1^s$ is invariant under the action of $Sym\left(3\right)$, and the orbit of $p_1{p}_2$ under the operation $\circ$ is spanned by $p_1{p}_2$, $\left(p_1{p}_2\right)\circ \left(12\right)$ and $\left(p_1{p}_2\right)\circ \left(13\right)$ since $\left(23\right)$ stabilizes $p_1{p}_2$. The orbit of $p_1{p}_2$ is the same as the one for $p_2{p}_1$ because $p_2{p}_1=\left(p_1{p}_2\right)\circ \left(13\right)$.)

We want to see if it is possible to express $p_3$ as a linear combination of these four expressions:

$$p_3=x_1^3+x_2^3={\alpha }_1{p}_1^3+{\alpha }_2{p}_1{p}_2+{\alpha }_3\left(p_1{p}_2\right)\circ \left(12\right)+{\alpha }_4\left(p_1{p}_2\right)\circ \left(13\right),$$

where the αs are unknown coefficients. Comparing the coefficients of the monomials of degree 3, which start with $x_1$, we obtain the following linear system

$$\begin{array}{cccccc}\hfill x_1^3:& {\alpha }_1\hfill & +{\alpha }_2\hfill & +{\alpha }_3\hfill & +{\alpha }_4\hfill & =1\hfill \\ \hfill x_1^2{x}_2:& {\alpha }_1\hfill & & & +{\alpha }_4\hfill & =0\hfill \\ \hfill x_1{x}_2{x}_1:& {\alpha }_1\hfill & & +{\alpha }_3\hfill & & =0\hfill \\ \hfill x_1{x}_2^2:& {\alpha }_1\hfill & +{\alpha }_2\hfill & & & =0.\hfill \end{array}$$

The matrix of the system is

$$\left(\begin{array}{ccccc}1& 1& 1& 1& 1\\ 1& 0& 0& 1& 0\\ 1& 0& 1& 0& 0\\ 1& 1& 0& 0& 0\end{array}\right).$$

Each column of the matrix contains an even number of 1s. Since we work over a field of characteristic 2, if we add all rows to the first row we shall obtain the row

$$\left(\begin{array}{ccccc}0& 0& 0& 0& 1\end{array}\right)$$

which means that the system does not have a solution. This proves that $p_3$ does not belong to the S-subalgebra of $\left(K{⟨X_2⟩}^{Sym\left(2\right)},\circ \right)$ generated by $p_1$ and $p_2$.

Theorem 10. 

If $d\ge char\left(K\right)=p>0$, then the set $\{p_n\mid n=1,2,\dots \}$ is a minimal generating set of the S-algebra $\left(K{⟨X_d⟩}^{Sym\left(d\right)},\circ \right)$.

Proof. 

As in the proof of Theorem 9 and in virtue of Lemma 1, it is sufficient to prove the theorem when the characteristic of the ground field, K, is equal to the number of the variables, i.e., $charK=p=d\ge 2$. As in Example 1 we shall show that $p_n$ does not belong to the S-subalgebra of $\left(K{⟨X_d⟩}^{Sym\left(d\right)},\circ \right)$ generated by $p_1,\dots ,p_{n-1}$. Let $\{(p_{n_1}\cdots p_{n_k})\circ \sigma \}$ be the set of all pairwise different polynomials of degree n depending on the power sums $p_1,\dots ,p_{n-1}$. Hence, $k\ge 2$ in all products. Since each $p_{n_i}$ is a sum of p monomials $x_1^{n_i},\dots ,x_p^{n_i}$, we demonstrate that each product $\{(p_{n_1}\cdots p_{n_k})\circ \sigma \}$ is a sum of $p^k$ pairwise different monomials $x_{j_1}\cdots x_{j_n}$. In particular, $p^{k-1}$ of these monomials start with $x_1$. Let us assume that

$$p_n=\sum {\alpha }_{(n_1,\dots ,n_k)\circ \sigma }(p_{n_1}\cdots p_{n_k})\circ \sigma ,{\alpha }_{(n_1,\dots ,n_k)\circ \sigma }\in K.$$

Comparing the coefficients of the monomials $x_1{x}_{j_2}\cdots x_{j_n}$, which start with $x_1$ in the above sum, we obtain a linear system with unknowns ${\alpha }_{(n_1,\dots ,n_k)\circ \sigma }$. Since all products $p_{n_1}\cdots p_{n_k}$ contain $x_1^n$ as a summand, the same holds for $(p_{n_1}\cdots p_{n_k})\circ \sigma$. Hence, the equation corresponding to $x_1^n$ is

$$\sum {\alpha }_{(n_1,\dots ,n_k)\circ \sigma }=1$$

and the right hand side of all other equations are equal to 0. Let us consider the matrix of the linear system. Its first row is $(1,\dots ,1|1)$, i.e., it consists of 1s only. The other rows of the matrix are of the form $(\ast ,\dots ,\ast |0)$, where $\ast \in K$ (actually it is an integer multiple of $1\in K$, so it is an element in ${\mathbb{F}}_p$).

The number of monomials starting with 1 in $(p_{n_1}\cdots p_{n_k})\circ \sigma$ is equal to $p^{k-1}$. Since $k\ge 2$, we demonstrate that this number of monomials is divisible by p. As in Example 1, the sum of the elements in any column of the matrix of the system corresponding to the product $(p_{n_1}\cdots p_{n_k})\circ \sigma$ is 0, since when viewed as integers these elements sum to $p^{k-1}$, which represents 0 in K. Thus, if we add all rows of the matrix to the first row we shall obtain the row $(0,\dots ,0|1)$. This means that the system does not have a solution, i.e., $p_n$ does not belong to the S-subalgebra of $\left(K{⟨X_d⟩}^{Sym\left(d\right)},\circ \right)$ generated by $p_1,\dots ,p_{n-1}$.

This, combined with Lemma 1, gives that $\left\{p_n\right|n=1,\dots \}$ is a minimal generating set of $\left(K{⟨X_d⟩}^{Sym\left(d\right)},\circ \right)$ when $d\ge charK>0$. □

The findings and their implications should be discussed in the broadest context possible. Future research directions may also be highlighted.