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,
; however, many of the results hold over any field,
K, of characteristic 0. Let
be the algebra of polynomial functions on the
d-dimensional vector space,
, with basis
. The general linear group,
, acts on
as on the dual
, i.e., one identifies each
with the linear functional
by
This induces an action of
on
given by
The algebra of
G-invariants for any subgroup,
G, of
is
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, , of degree d on the vector space, , bythe algebra of -invariants is generated by the elementary symmetric polynomialsmoreover, the polynomials 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
for all subgroups,
G, of
.
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
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 is finitely generated for G, being a finite subgroup of , 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 is isomorphic to a polynomial algebra, i.e., has a system of algebraically independent generators if and only if is generated by pseudo-reflections (matrices of finite order with a conjugate that is a diagonal matrix of the form , where is a root of unity). The third important theorem is the Molien formula [
13] from 1897, which “counts” the invariants: the algebra
is graded for any group,
G, and decomposes as a direct sum
of its homogeneous components
of degree
n. The Hilbert (or Poincaré) series of
is
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
for some natural numbers,
, and a polynomial
. 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 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
be the free unitary associative algebra generated by the set of variables
. In the class of unitary associative algebras, the algebra
possesses a universal property as
in the class of commutative algebras: every mapping,
, of
to an algebra,
R, can be uniquely extended to a homomorphism,
. As in the commutative case, the general linear group,
, acts canonically on the vector space,
, with basis
. This action can be extended to
where each
g acts by an algebra homomorphism:
The algebra of G-invariants for a subgroup, G, of consists of all polynomials, , 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
with its natural action, by Margarete Wolf [
1] in 1936.
Theorem 4 (Wolf [
1])
. (i)
For any field, K, the algebra of symmetric polynomials , , is a free associative algebra.(ii) It has a homogeneous system of free generators with the property that there is at least one generator of degree n for any .
(iii) Any homogeneous system of generators of has the same number of polynomials of degree n.
(iv)
Ifthen the coefficients 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
and
behave in a completely different way with respect to the finite generation.
Theorem 5 ([
3])
. For an arbitrary subgroup, G, of over any field, K, of an arbitrary characteristic let be the minimal subspace of , such that . Then, the algebra is finitely generated if and only if G acts on by scalar multiplication. Corollary 1 ([
31,
32])
. When G is a finite subgroup of , the algebra of invariants 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 , i.e., if does not have nontrivial subspaces, W, such that , then is either trivial or not finitely generated. The analogue of the Chevalley–Shephard–Todd theorem also looks differently for the free associative algebra .
Theorem 6. (i) (Lane [
33]
and Kharchenko [
34]
) For any subgroup, G, of and for any field, K, the algebra is free. (ii) (Kharchenko [
34]
) For finite groups, G, there is a Galois correspondence between the free subalgebras of containing and the subgroups of G—a subalgebra, F, of with is free if and only if for a subgroup, H, of G. A variation of Molien’s formula also hold for :
Theorem 7 (Dicks and Formanek [
31])
. Let G be a finite group of and field K be with characteristic 0. Then The paper of Koryukin [
3] was our motivation for [
2]. Koryukin introduced an additional structure on
, which changes the notion of finite generation, and the algebra of invariants is “finitely generated” in this weak sense in some cases.
Let
be the homogeneous component of degree
n in
. Let us consider the action of the symmetric group
from the right on
by the rule
We name this action the S-action, and indicate by the algebra with the described action of on , . Given a graded subalgebra, F, of , which inherits the S-action, i.e., for any n it holds for its homogeneous component of degree n, we denote it by and label it as an S-algebra. The finite generation of means that there exists a finite subset, U, of homogeneous polynomials of F, such that U generates . Since the left action of on commutes with the right action of , the algebra of invariants is an S-algebra for any subgroup, G, of .
Theorem 8 (Koryukin [
3])
. For any field, K, and any reductive subgroup, G, of the S-algebra is finitely generated. By the Maschke theorem, if the field K has characteristic 0 or characteristic , 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 and let or and p does not divide the order of G.
(i) For a minimal homogeneous generating system of the S-algebra is there a bound of the degree of the generators in terms of the order of G, the rank d of and the characteristic of K?
(ii) Find a finite system of generators of for concrete groups, G.
(iii) If the commutative algebra is generated by a homogeneous system , can this system be lifted to a system of generators of ?
Remark 1. By the Endlichkeitsatz of Emmy Noether [
7]
, if , then has a set of generators of degree for any finite group, G. Fleischmann [
35]
and Fogarty [
36]
proved that the same upper bound holds if does not divide the order of G. Hence, in Problem 1 (i), it is reasonably to restrict our attention to the order of G and the rank, d, of . In [
2], we gave the answer to the questions in Problem 1 if
G is the symmetric group of degree,
d, and showed that
is generated (as
S-algebra) by analogs of the elementary symmetric functions.
3. Infinite Generation in the Case
Koryukin’s result (Theorem 8) does not extend to the non-reductive case. Our main result shows that when the S-algebra is not finitely generated.
Remark 2. For , we have a projection from to , 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 in not finitely generated in the case . In the sequel, we shall assume that .
Let us consider the augmentation ideal
of
, i.e., the ideal of polynomials without a constant term in
. Let
be the quotient of
by its square, i.e.,
where
denotes the submodule of
generated by
V under the action
.
is naturally graded and each homogeneous component,
, is an
-module, i.e., there is a natural
-action on
M.
Finding the minimal generating set is essentially equivalent to identifying and its graded components. The first step of achieving that is constructing a small generating set of as a vector space over K.
Lemma 1. The vector space, , is generated as a -module and as a vector space by the images of the power sums Proof. It was shown in [
2] (Lemma 4.1) that, over any field,
K, of arbitrary characteristic, the
S-algebra
is generated by the power sums
,
. This implies that
is generated as a
-module by the images of
. Since, for each
n, the power sum,
, is invariant under the action of
, these images also generate
as a vector space.
This does not imply that
is infinitely generated, since some of elements
might become trivial when projected to
. The main result in [
2] shows that this indeed happens when
or
and the image of
in
is trivial for
. □
The observation that the power sums in are fixed under the -action suggests the idea that we can obtain useful information by passing to a suitable quotient of where the -action becomes trivial. Consider the abelianization map and the map induced by it on the subalgebras of symmetric polynomials. The map, , is clearly an algebra homomorphism.
Lemma 2. The map, π, sends a generating set of the S-algebra to a generating set of the commutative algebra .
Proof. The statement follows immediately from the observation that the extra action, , disappears (becomes trivial) after the map, . □
The above lemma allows us to prove that the S-algebra is not finitely generated if we can show that its image under is not finitely generated.
Remark 3. Although the map is surjective it does not induce a surjective map between and . For example, if , then and the elementary symmetric function is not in the image of π.
In order to compute the image for of the map
we will introduce a notation which can be slightly misleading. Let
u be a monomial (either in
or in
). Since the action of
preserves the set of monomials, one can construct invariants by summing over the orbits of
acting on the set of monomials, i.e.,
is in the algebra of invariants, where
is the stabilizer of the monomial
u under the action of the symmetric group
.
There are several important comments to make. First, the notation is defined only for monomials and not for arbitrary elements in the algebra (technically we can use the above formula to define 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 or in because the stabilizer depends on that. Finally, we can observe that since both or have bases consisting of monomials, one has that the algebras of invariants and are spanned by . Therefore the image of under is generated as a K-vector space by .
The key observation is the following lemma, which informally says that almost commutes with ∑, even though ∑ is not an algebraic operation.
Lemma 3. For any monomial , there exists an integer constant such that Moreover, in the case the constant is 0 in K if and only if for some .
Proof. As mentioned above, ∑ has two different meanings in
and in
, but in both cases
where
is the stabilizer of the monomial,
u, under the action of the symmetric group,
. The difference arises since
is not equal to
.
Since is a homomorphism from to , which is compatible with the action of , we have that . Thus, the constant, , is equal to the index of in . It is not hard to see that is the symmetric group on the variables, which does not appear in u, and is the product of symmetric groups on the variables, which appears in u with the same degree. Therefore, , where is the number of variables in u, which appear exactly i times. In the case , (as an element in K, i.e., that ) if and only if for some (since unless ). □
Lemma 4. In the case , the commutative algebrais spanned by all products, , of the elementary symmetric polynomials except the powers, , of . Proof. The image of in under is a subalgebra of and contains the elements for all and all —to see this it is enough to construct a monomial, u, such that is a non-zero multiple of , which follow by the previous Lemma 3 and the observation is equal to in . By the same lemma, does not contain the m-th power of , otherwise will induce a surjective map in degree . □
Theorem 9. When , the S-algebra is not finitely generated.
Proof. By Remark 2, it is sufficient to show that the image of in is not finitely generated for only. Let B denote and let be the augmentation ideal of B. Clearly, , and therefore B is finitely generated if and only if is of finite co-dimension in . However, the description of B as a subalgebra of in Lemma 4 allows us to see that the quotient space has a basis consisting of the images of for all and all , and thus it is infinite dimensional. Hence, the algebra is not finitely generated. □
This argument shows that any generating set of the S-algebra needs to contain a generator of degree for any . Therefore, the image of in is nontrivial when .
4. Generating Set
As remarked above, the proof of Theorem 9 gives that any homogeneous generating set of the S-algebra contains generators of degree for any and ; however, it says nothing about degrees 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 . More precisely, we shall prove that is a minimal generating set of the S-algebra .
Example 1. We shall illustrate the idea of the proof that the power sums , , form a minimal generating set of the S-algebra in the special case and . Note that this case is covered in the proof of Theorem 9; however, the case is too easy and the case leads to an unnecessary large system of linear equations.
We want to show that does not belong to the S-subalgebra F of generated by and . The following polynomials of degree 3 span the homogeneous component of degree 3 of the S-algebra F:(Here we use that is invariant under the action of , and the orbit of under the operation is spanned by , and since stabilizes . The orbit of is the same as the one for because .) We want to see if it is possible to express as a linear combination of these four expressions:where the αs are unknown coefficients. Comparing the coefficients of the monomials of degree 3, which start with , we obtain the following linear system The matrix of the system is 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 rowwhich means that the system does not have a solution. This proves that does not belong to the S-subalgebra of generated by and . Theorem 10. If , then the set is a minimal generating set of the S-algebra .
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.,
. As in Example 1 we shall show that
does not belong to the
S-subalgebra of
generated by
. Let
be the set of all pairwise different polynomials of degree
n depending on the power sums
. Hence,
in all products. Since each
is a sum of
p monomials
, we demonstrate that each product
is a sum of
pairwise different monomials
. In particular,
of these monomials start with
. Let us assume that
Comparing the coefficients of the monomials
, which start with
in the above sum, we obtain a linear system with unknowns
. Since all products
contain
as a summand, the same holds for
. Hence, the equation corresponding to
is
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
, i.e., it consists of 1s only. The other rows of the matrix are of the form
, where
(actually it is an integer multiple of
, so it is an element in
).
The number of monomials starting with 1 in is equal to . Since , 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 is 0, since when viewed as integers these elements sum to , which represents 0 in K. Thus, if we add all rows of the matrix to the first row we shall obtain the row . This means that the system does not have a solution, i.e., does not belong to the S-subalgebra of generated by .
This, combined with Lemma 1, gives that is a minimal generating set of when . □
The findings and their implications should be discussed in the broadest context possible. Future research directions may also be highlighted.