Free functions with symmetry

In 1936, Margarete C. Wolf showed that the ring of symmetric free polynomials in two or more variables is isomorphic to the ring of free polynomials in infinitely many variables. We show that Wolf's theorem is a special case of a general theory of the ring of invariant free polynomials: every ring of invariant free polynomials is isomorphic to a free polynomial ring. Furthermore, we show that this isomorphism extends to the free functional calculus as a norm-preserving isomorphism of function spaces on a domain known as the row ball. We give explicit constructions of the ring of invariant free polynomials in terms of representation theory and develop a rudimentary theory of their structures. Specifically, we obtain a generating function for the number of basis elements of a given degree and explicit formulas for good bases in the abelian case.


Introduction
A symmetric free polynomial in the letters x 1 , . . . , x d is a free polynomial p which satisfies p(x 1 , . . . , x d ) = p(x σ(1) , . . . , x σ(d) ) for all permutations σ of {1, . . . , d}. For example, is a symmetric free polynomial in two variables. Classically, Wolf proved the following theorem about the structure of the ring of symmetric free polynomials [21]. Theorem 1.1 (Wolf). The symmetric free polynomials in d ≥ 2 variables is isomorphic to the ring of free polynomials in infinitely many variables.
More recently, free symmetric polynomials have been investigated by numerous authors [5,7,11], and some deep connections with representation theory are now known.
We are concerned with the ring of invariant free polynomials. Let G be a finite group. Let π : G → U d be a unitary representation. That is, π is a homomorphism from the group G to the group of d × d unitary matrices over C. An invariant free polynomial with respect to π is a free polynomial p ∈ C x 1 , . . . , x d which satisfies p(x 1 , . . . , x d ) = p(π(σ)(x 1 , . . . , x d )) for all σ in G. (Here, the d × d matrices π(σ) are acting on (x 1 , . . . , x d ) by matrix multiplication. That is, for example, if we define A(x 1 , x 2 ) = (x 1 + 2x 2 , 3x 1 + 4x 2 ).) Notably, the invariant free polynomials form an algebra, which is called the ring of invariant free polynomials. For example, the symmetric free polynomials in d variables are obtained in the special case where G = S d a symmetric group, and we define π : S d → U d be the representation satisfying π(σ)e i = e σ(i) , where e i is the i-th elementary basis vector for C d .
We now define the domains on which we intend to execute the function theory of invariant free polynomials. Fix an infinite dimensional separable Hilbert space H. There is only one infinite dimensional separable Hilbert space up to isomorphism [9], so what follows will be independent of the exact choice. Let Λ be an index set. Let C Λ be the set of (X λ ) λ∈Λ , sequences of elements in B(H) indexed by Λ, such that λ∈Λ X λ X * λ < 1, where A < B means that B − A is strictly positive definite and A ≤ B means that B − A is positive semidefinite. Here the sum λ∈Λ X λ X * λ is required to be absolutely convergent in the weak operator topology and have λ∈Λ X λ X * λ < 1, or equivalently, there is an ε > 0 such that for every finite Λ ′ ⊂ Λ, If Λ = d is a natural number, we identify d as a set with d elements.
That is, C d is the set of d-tuples of operators in B(H) such that Some authors refer to C Λ as the row ball, or the set of row contractions [14,18]. Given a free polynomial p in the letters (x λ ) λ∈Λ and a point X = (X λ ) λ∈Λ ∈ C Λ , we form p(X) via the formula which when Λ = d reduces to p(X) = p(X 1 , . . . , X d ).
Let R be a subalgebra of C x 1 , . . . , x d . We define a basis for R to be an indexed sequence (u λ ) λ∈Λ of free polynomials which generate R as an algebra, which is minimal in the sense that there is no Λ ′ Λ such that the sequence (u λ ) λ∈Λ ′ generate R as an algebra. We note that any basis for R is countable, since R has countable dimension as a vector space as it is a subspace of C x 1 , . . . , x d , and the minimality condition implies that the (u λ ) λ∈Λ are linearly independent.
We prove the following theorem.
Theorem 1.2. Let G be a finite group. Let π : G → U d be a unitary representation. There exists a basis for the ring of invariant free polynomials (u λ ) λ∈Λ such that the map Φ on C d defined by the formula Φ(X) = (u λ (X)) λ∈Λ satisfies the following properties: • Furthermore, for p in the ring of invariant free polynomials for π, there exists a unique free polynomialp such that p =p • Φ.
Namely, the map takingp to p is an isomorphism of rings from the free algebra C x λ λ∈Λ to the ring of invariant free polynomials. Theorem 1.2 follows from Theorems 4.2 and 6.4. We note that a similar result was obtained for free functions on the domain for symmetric free functions in two variables by Agler and Young [4].
In general, the basis in Theorem 1.2 is hard to compute. However, the number of elements of a certain degree is computed in Section 7.1. An explicit basis can be obtained if G is abelian, which we give in Section 7.2.
1.1. Some examples. We now give some concrete examples of what our main result says. First, we give an analogous theorem to that obtained in Agler and Young for symmetric free functions in two variables [4].
The map Φ satisfies the following properties: • For any free polynomial p such that there exists a unique free polynomialp such that p =p • Φ.
Another simple example concerns the ring of even functions in two variables, that is, the ring of free polynomials in two variables f satisfying the identity Here the group G in Theorem 1.2 is the cyclic group with two elements, Z 2 = {0, 1}, and the representation π is given by The map Φ satisfies the following properties: • Φ takes C 2 to C 4 .
• For any free polynomial p such that there exists a unique free polynomialp such that p =p • Φ. • Moreover, We discuss Proposition 1.4 in detail in Section 2. Here we see a concrete trade-off between the number of variables and degree in the optimization of a free polynomial: finding the maximum norm of a polynomial in 4 variables of degree d on C 4 is the same as finding the norm of an even polynomial in 2 variables of degree 2d on C 2 .
Both Proposition 1.3 and Proposition 1.4 follow from our basis construction in the abelian case given in Theorem 7.4.

1.2.
Geometry. Although the image of the map Φ in Theorem 1.2 may have high codimension, in the sense that it is far from being literally surjective, the functionf is completely determined by f and has the same norm. We view this as an analogue of the celebrated work of Agler and M c Carthy on norm preserving extensions of functions on varieties to whole domains [2,1]. Additionally, since f totally determines f , the dimension of the Zariski closure of the image of a free polynomial map can apparently go up, in contrast with the commutative case [12]. Exploiting the aforementioned phenomenon is a critical step in the theory of change of variables for free polynomials and their generalizations, the free functions, which had been thought to be extremely rigid [13].
To prove Theorem 1.2, we develop a geometric theory of the ring of invariant free functions. As a consequence of the geometric structure, the ring of free invariant polynomials is always free (but perhaps infinitely generated), in contrast with the Chevalley-Shepard-Todd Theorem in the commutative case, in which freeness depends on the structure of G [8, 20].

Free analysis.
A free function on C Λ is a function from C Λ to B(H) which is the uniform limit of free polynomials on the sets rC Λ . We let H(C Λ ) denote the algebra of free functions on C Λ . The Banach algebra of bounded free functions We note that there are many equivalent characterizations of a free function, such as [3,14,15,18].
We can reinterpret Theorem 1.2 as an isomorphism of function algebras analogous to Wolf's theorem. Let H ∞ π (C d ) be the Banach algebra of bounded invariant free functions for π on C d .
is such an isomorphism. The map is injective and surjective since it is already since it is an isomorphism on the level of formal power series, which uniquely define a free function [15].

Example: An even free function in two variables
To begin, we discuss a simple nontrivial example of a free ring of invariants. We will now explain what our main result, Theorem 1.2, says about a specific even free polynomial in two variables.
We say a free polynomial p ∈ C x 1 , x 2 is even if We note that the even free functions form an algebra. Consider the even free polynomial as a map on the domain C 2 .
We first note that it is clearly not a coincidence that p has no odd degree terms. Furthermore, if we let we get that . We are interested in the analytical properties ofp and Φ.
We will show the remarkable fact that Let X ∈ C 2 . That is, We will now show a curious equality: Since p is a free polynomial and is thus continuous on the closure Define operators X 1 and X 2 with the following block structure First, That is, (X 1 , X 2 ) ∈ C 2 . Note whereJ denotes some matrix tuple which is irrelevant to our aims. So,

Thus, sup
To see that sup

The space of free polynomials as an inner product space
We now seek to understand the geometry of free polynomials as a Hilbert space.
Definition 3.1. We define H 2 d to be the Hilbert space of free formal power series in d variables whose coefficients are in ℓ 2 . We define [H 2 d ] n to be the subspace of homogeneous free polynomials of degree n. For a free polynomial f in d variables, we define M f to be the operator on The following lemma describes a grading structure on H 2 d . Lemma 3.2 (Grading lemma). Let p, q be homogenous free polynomials of degree n and r, s be homogenous free polynomials of degree m. Then, pr, qs = p, q r, s .
Proof. Write where I k is the set of free multi-indicies of degree k. Observe that p, q = I∈In p I q I and r, s = J∈Im r J s J , which gives p, q r, s = I∈In J∈Im Observe that which gives pr, qs = I∈In J∈Im Let u 1 , . . . , u k be an orthonormal basis for V and v 1 , . . . , v k be an orthonormal basis for V.
Proof. Let m X = (X I ) |I|=n be the row vector of free monomials of degree n. Note that, for any So we get that Note that u i u * i is the orthogonal projection onto V, and thus did not depend on the choice of basis, so we are done.

Superorthogonality
We can now define superorthogonality. We will later see that the ring of invariant free polynomials is itself superorthogonal.  • The map Φ takes C d to C Λ .
• Furthermore, for p in the ring of invariant free polynomials for π, there exists a unique free polynomialp such that p =p • Φ. • Moreover, Namely, the map takingp to p is an isomorphism of rings from the free algebra C x λ λ∈Λ to A.  Proof. Note that it is enough to show that for any two distinct products n i=1 u λ i , m j=1 u κ j of the same degree as free polynomials in A ⊂ C x 1 , . . . , x d that since then the words in u λ are linearly independent. Since the words are distinct, there is a p such that λ i = κ i for all i < p and λ p = κ p . So, Note that n i=p u λ i ∈ u λp H 2 d and m j=p u κ j ∈ u κp H 2 d which implies that the two products are orthogonal since u λp and u κp are superorthogonal and thus the desired inner product is 0.
We now show that the superothogonal basis maps C d into C Λ .
Let I be the left ideal generated by A \ {1} in C x 1 , . . . , x d , that is, the span of the elements of the form ab where a ∈ A \ {1} and b ∈ C x 1 , . . . , x d .
To show that λ∈Λ u λ (X)u λ (X) * < 1, we will show by induction that where w i,n form an orthonormal basis for (I ∩ [H 2 d ] n ) ⊥ . Note that for n = 1, Equation (4.1) becomes

Now suppose that Equation (4.1) holds for n. That is,
We will now show that it holds for n + 1. Since Note that any u λ of degree n+1 must be in the subspace To show that sup we show the equivalent inequality Similarly to our example for even functions (Section 2), given a U ∈ C Λ we would like to find an X ∈ C d such that there is some projection P so that Pp(Φ(X))P =p(U) (4.2) and thus that p(X) ≥ p(U) .
Since the algebra A is a joint invariant subspace for all the M u λ , we get that with respect to the decomposition, where J λ and K λ are some operators which will be irrelevant to this discussion. The Frazho-Popescu dilation theorem [10,17] states that for any U ∈ C Λ , there is a projectionP such that for any free polynomial q, P q((M x λ ⊗ I))P = q(U).
Thus, there is indeed an X and a projection as desired in (4.2), namely, taking X = (M x 1 ⊗ I, . . . , M x d ⊗ I) and the projection as constructed above.
Thus, for every U ∈ C Λ there exists an X ∈ C d such that and so sup To see that sup we note that Φ(X) ∈ C Λ by Lemma 4.4. Now Theorem 4.2 follows immediately.

Example: Free functions in three variables invariant under the natural action of the cyclic group with three elements
We now show how to construct a superorthonormal basis for the ring of free polynomials in three variables which are invariant under the natural action of the free group. That is, we want to understand free functions which satisfy the identity and show that they form a superorthogonal algebra.
Let σ denote a generator of the cyclic group with three elements. Define the action of σ on free functions in three variables by (σ · f )(X 1 , X 2 , X 3 ) = f (X 2 , X 3 , X 1 ).
With this notation we are trying to understand functions such that Let ω be a nontrivial third root of unity. Consider the following three linear polynomials: Clearly, the function u 0 is fixed by the natural action of the cyclic group with three elements. However, σ · u 1 = ωu 1 , and σ · u −1 = ωu 1 .
That is, they are eigenfunctions of the action σ on free polynomials in three variables. In fact, any product of the u i will be an eigenfunction of the action of σ. For example, Thus, it can be observed that j u i j is in the ring of invariant free polynomials under the action of the cyclic group with three elements if and only if j i j ≡ 3 0, and furthermore that products satisfying this condition span the algebra of free polynomials in three variables which are fixed by the natural action of the cyclic group with three elements.
So, if we choose products N j=1 u i j such that j i j ≡ 3 0, which are primitive in that no partial product n j=1 u i j is in the ring of invariant free polynomials, we will obtain a basis for our algebra. To show that this basis is superorthogonal, it is enough to show that any two distinct products N j=1 u i j , N j=1 u k j are orthogonal. However, by the grading lemma, Since the two products were assumed to be not equal, the orthogonality of the u i implies that at least one of the u i j , u k j = 0, so we are done.
For the action of a general finite group, an explicit construction of a superorthogonal basis for the ring of invariants is more difficult. However, the existence of such a basis can be established using some basic representation theory which we do in the next section. Later, we will return to the issue of an explicit construction for specific classes of groups for which the problem is tractible.

The ring of invariant free polynomials is superorthogonal
In order to prove Theorem 1.2 by Theorem 4.2, it is sufficient to show that the ring of invariant free polynomials is superorthogonal. Definition 6.1. Let π be a unitary representation of a finite group G. Let π n denote the action of π on [H 2 d ] n . We make the identification that π 1 = π.
Note that π n+m · pq = (π n · p)(π m · q), which translates to the following formal observation by the grading lemma.
Observation 6.2. Let π be a unitary representation of a finite group G. Then, π n ⊗ π m = π n+m with the identification made in the grading lemma.
Observation 6.2 implies that the action on homogeneous polynomials of degree n is determined by the action on homogeneous polynomials of degree 1. Corollary 6.3. Let π be a unitary representation of a finite group G. Then, π n = π ⊗n with the identification made in the grading lemma.
We can now prove that the ring of invariant free polynomials is superorthogonal by constructing a basis for it. Theorem 6.4. Let G be a finite group. Let π : G → U d be a group representation. The ring of invariant free polynomials for π is superorthogonal.
Proof. The action of the group on homogeneous polynomials of degree i is given by π ⊗i (g) = π i (g). To show that the ring of invariant free polynomials is superorthogonal, we construct a superorthogonal basis recursively. Let whereṼ i is the space which is fixed by the action ofπ i . Let (u i,k ) be an orthonormal basis of V i . We will show that for all p, q in the ring of invariant free polynomials, u i,k p ⊥ u j,l q for j ≤ i, i.e. that u i,k and u j,l are superorthogonal.
When i = j, for l = k, by the grading lemma, u i,k p, u j,l q = u i,k , u j,l p, q = 0.
When j < i, by the recursive construction, for some a, b, and therefore u i,k p, u j,l q = 0.
To show that our recursively generated sequence (u i,k ) is a basis, consider the following. If p ∈ [H 2 d ] n is in the ring of invariant free polynomials, note that the projection of p onto any u i,k [H 2 d ] n−i is in the ring of invariant free polynomials. So for any p in the ring of invariant free polynomials, where q i,k is in the ring of invariant free polynomials, and r ∈Ṽ n ⊥ and in the ring of invariants. So, by construction, r = 0.
7. Structure of the superorthogonal basis for the ring of invariants 7.1. Counting. We now calculate the number of elements in any superorthonormal basis for the ring of invariant free polynomials of a given degree in terms of generating functions. The main result of this section is as follows.
Theorem 7.1. Let π be a unitary representation of a group G and χ = tr π be the character corresponding to π. Let C G be a set of representatives for the conjugacy classes of G. Let g n be the number of free polynomials of degree n in some superorthogonal basis for the ring of invariant free polynomials of π. Then, where #C τ denotes the number of elements in the conjugacy class of τ.
Namely, the number of generators of a given degree is independent of the choice of superorthogonal basis.
We prove Theorem 7.1 in Section 7.1.1. For example, consider symmetric functions in three variables. That is, take the group S 3 acting on C 3 via the representation π(σ)e i = e σ(i) . According to Theorem 7.1, the necessary information can be conveniently compiled in the following table.
Thus, in general, with the help of computer algebra software, it is a simple exercise to calculate the number of free polynomials of degree n in some superorthogonal basis for the ring of invariant free polynomials of π.
Proof. It can be shown using enumerative combinatorics that Thus, Calculating g from f gives that In order to calculate f n , we use character theory (see Serre [19, Chapter 2]).
Let ρ be a representation for G. For each σ ∈ G, put χ ρ (g) = tr(ρ(σ)). χ ρ is the character of ρ. Let φ 1 , φ 2 be functions from G into C. The scalar product ·, · is defined by Let τ : G → U 1 be the trivial representation. The dimension of the space of fixed vectors of the action of ρ is given by χ ρ , τ . (See Serre [19,Chapter 2]) We make the identification χ = χ π . Lemma 7.3.
Proof. By Lemma 6.3, the action of π on homogenous polynomials is given by π ⊗n .
So, any invariant free polynomial is in the span of the v J ∈ B, which are themselves invariant homogeneous free polynomials, since otherwise χ J , τ = 0 by the orthogonality of characters.
To show thatB is superorthogonal, note for any two distinct words, v K , v L = 0 by the grading lemma, which implies the claim.
So, the calculation of a superorthogonal basis for the ring of invariants of a finite abelian group is tractable in general.
The superorthogonal basis given in Theorem 7.4 also implies that off of some variety, the ring of invariant functions is finitely generated. So, there are finitely many invariant free rational functions such that any invariant free polynomial can be written in terms of them, in spite of the fact that the ring of invariant free polynomials is not itself finitely generated. Similar phenomena occur in real algebraic geometry, such as the fact that positive polynomials cannot be written as sums of squares of polynomials [16], but can be written as sums of squares of rational functions, i.e. Artin's resolution of Hilbert's seventeenth problem [6]. Proof. We note that by Pontryigan duality theorem [22,Theorem 1.7.2], the characters of an abelian group G form a groupĜ under multiplication which is noncanonically isomorphic to G. Let F d be the free group with d generators. Let H = {I ∈ F d |χ I = 0}. Consider the short exact sequence 0 → H → F d →Ĝ → 0. So, H is of finite index in F d and is thus finitely generated, which then implies that the ring is finitely generated. 7.3. Example: Ad hoc methods for symmetric functions in three variables. We now turn our attention to symmetric functions in three variables.
Let u 0 , u 1 , u −1 be as in Section 5 and let (b n ) n∈N be the constructed superorthogonal basis for cyclic free polynomials in 3 variables, where we fix b 0 = u 0 .
We recall that each b n is of the form b n = k u i k such that k i k ≡ 3 0. So, τ · b n = bñ for someñ, since τ · k u i k = k u −i k , and so k −i k ≡ 3 0. For n > 0 define For n = 0, let c 0 0 = u 0 . Note that each c 0 n is a symmetric free polynomial and that the product of an even number of c 1 n is also symmetric. In fact, τ · c 0 n = c 0 n , and τ · c 1 n = −c 1 n . Set Now, B is a superorthogonal basis for the symmetric free polynomials in 3 variables. The elements of B of degree 4 and less are given in the following table. degree basis elements 1 u 0 2 Note the table agrees with the generating function obtained earlier. As there are 13 elements of the basis of degree 5, we will stop here. We remark that the method above of iteratively constructing can be applied, in principal, for any solvable group, since we exploited the fact that Z 2 ∼ = S 3 /Z 3 acts on the ring of free polynomials invariant under Z 3 . Since the ring of free polynomials invariant under Z 3 is itself isomorphic to a free algebra in infinitely many variables, we are essentially repeating the construction done for any abelian group.