Optimal line packings from finite group actions

# Optimal line packings from finite group actions

Joseph W. Iverson Department of Mathematics, University of Maryland, College Park, MD 20742 John Jasper Department of Mathematics and Statistics, South Dakota State University, Brookings, SD 57007  and  Dustin G. Mixon Department of Mathematics, The Ohio State University, Columbus, OH 43210
July 19, 2019
###### Abstract.

We provide a general program for finding nice arrangements of points in real or complex projective space from transitive actions of finite groups. In many cases, these arrangements are optimal in the sense of maximizing the minimum distance. We introduce our program in terms of general Schurian association schemes before focusing on the special case of Gelfand pairs. Notably, our program unifies a variety of existing packings with heretofore disparate constructions. In addition, we leverage our program to construct the first known infinite family of equiangular lines with Heisenberg symmetry.

###### Key words and phrases:
Equiangular tight frames, Schurian association schemes, Gelfand pairs, Heisenberg group
###### 2010 Mathematics Subject Classification:
Primary: 20B99, 42C15, 52C99 Secondary: 20C15, 94C30

## 1. Introduction

We consider the fundamental problem of packing points in real or complex projective space so that the minimum distance is maximized. A famous instance of this problem is the Tammes problem [38], which concerns the packing of points in . In this space, the optimal packing of 13 points was the subject of a celebrated argument between Newton and Gregory [35]. Recently, the general problem of packing in projective space has received renewed attention due to its applications in communication, coding, and quantum information theory [37, 45]. In fact, the last few years produced a multitude of disparate constructions of optimal packings [24, 9, 23, 19, 21, 10, 33, 20, 3, 18] (see [22] for a living survey), leaving one yearning for some sort of unified theory. This paper provides a modest step in that direction by identifying a fruitful correspondence with transitive actions of finite groups.

Points in projective space correspond to one-dimensional subspaces (lines) of some real or complex vector space, and for convenience, we represent each line with a spanning unit vector. Our packing problem then amounts to finding unit vectors that minimize coherence, defined by

 max1≤i≠j≤n|⟨ϕi,ϕj⟩|.

To minimize coherence, it suffices to achieve equality in some known lower bound, such as the Welch, orthoplex or Levenstein bounds [43, 14, 34]. For each of these bounds, there exist specific cases in which equality is achievable. Interestingly, a packing achieves equality in the Welch bound precisely when the Gram matrix is a scalar multiple of a projection with off-diagonal entries of constant modulus [37]. Such packings are known as equiangular tight frames (ETFs).

Conway, Hardin and Sloane [14] were perhaps the first to observe that highly symmetric arrangements of lines are frequently strong competitors in the packing problem. Packings exhibiting abelian symmetry are known as harmonic frames, and harmonic ETFs are constructed using combinatorial objects known as difference sets [37, 44]. Optimal packings of points in are conjectured to be ETFs with Heisenberg symmetry for every , which correspond to desirable measurement ensembles for quantum state estimation [45]. As a precursor to the present paper, the authors recently used group schemes to construct the first known infinite family of ETFs with nonabelian symmetry [33].

The main idea of this paper is illustrated in Figure 1. Every transitive action of a finite group determines a Schurian association scheme, which in turn produces a collection of distinguished projections through its spherical functions. Each projection inherits symmetries from the group action, and has a (small) number of distinct entries bounded by the dimension of the scheme’s adjacency algebra. Viewing each projection as a Gram matrix then produces a collection of vectors that will often generate an optimal line packing. In particular, the packing might be an ETF since it necessarily has a small number of angles.

While each of these individual relationships is known, the entire chain suggests a useful new discovery tool for researchers. For example, one may systematically search through finite group actions in GAP [25] to find worthy line packings. The authors used this program to find an ETF exhibiting Heisenberg symmetry, and then generalized it to the first known infinite family with such symmetry. While these ETFs are not exactly the packings desired in quantum information theory, we expect there to be some sort of relationship (as in [2]), and we leave this for future investigation.

The following section covers preliminary information about Schurian association schemes and in particular, the commutative instances corresponding to Gelfand pairs. In the theory of Lie groups, Gelfand pairs are used widely for the reproducing properties of their spherical functions on homogeneous spaces [27]. However, Gelfand pairs of finite groups appear to have received comparatively little attention from the frame theory community. As far as we are aware, the current article represents the first systematic attempt to mine Gelfand pairs as sources of finite frames. Section 3 then discusses the packings that arise from Schurian schemes, known as homogeneous frames. We illustrate the theory with examples in Section 4. Sections 5 and 6 then explain how to leverage the chain of relationships illustrated in Figure 1 to produce an infinite family of ETFs with Heisenberg symmetry. A nontrivial consequence of Zauner’s conjecture [45] is that an infinite family like this exists, and so our construction gives theoretical evidence in favor of that conjecture.

## 2. Preliminaries

We begin by recalling the basic theory of frames, Schurian association schemes, and Gelfand pairs. See [13, 11, 5, 12] for more details.

### 2.1. Frames

Let be a -dimensional Hilbert space (), either real or complex. A sequence of vectors in is called a frame if there are constants such that

 A∥ψ∥2≤n∑j=1|⟨ψ,ϕj⟩|2≤B∥ψ∥2

for all . We call and the frame bounds. When , the frame is called tight, and when , it is called Parseval. In abuse of notation, we sometimes think of as a short, fat matrix whose columns describe the frame vectors . With this in mind, the Gram matrix for is the matrix

 Φ∗Φ=[⟨ϕj,ϕi⟩]ni,j=1,

which is clearly positive semidefinite. To say that is a tight frame for its span means precisely that is a constant times a projection. Moreover, it is possible to recover from its Gram matrix, up to a unitary equivalence. (For example, when is Parseval we can just take the columns of in their span.) We say that Parseval frames and are Naimark complements if , meaning their Gram matrices project onto orthogonally complementary spaces.

In this paper, we want to find equiangular tight frames (ETFs), which are tight frames with two more properties: (1) All of the frame vectors have the same (nonzero) length, and (2) The inner product is constant across all pairs of distinct frame vectors . When an ETF occurs, it can be rescaled so that its vectors are unit norm with coherence matching the Welch bound

 μ(Φ):=max1≤i≠j≤n|⟨ϕi,ϕj⟩|≥√n−dd(n−1).

In particular, an ETF is an optimal line packing.

Our strategy is to hunt for ETFs via their Gram matrices, which are recognizable from three features: (1) They are constant multiples of projections, (2) They have constant diagonal, and (3) They have constant modulus off the diagonal. We are going to look in particular for Gram matrices that lie in the adjacency algebras of Schurian association schemes, described below.

### 2.2. Schurian schemes

Let be a finite group acting transitively on a finite set , from the left. This action determines a matrix algebra in the following way. Let act on by setting for and , and let be the orbits of this action, indexed in such a way that . We can express each orbit as an matrix whose entries are given by

 (2.1) (Ai)x,y={1,if (x,y)∈Ri;0,otherwise.

Then , the matrix of all ones, and it turns out that is a complex -algebra under the usual matrix multiplication. In other words, form a (possibly noncommutative) association scheme. Schemes of this type are called Schurian, or group-case. We call the adjacency matrices of the scheme, and the adjacency algebra; it consists of all complex matrices with the property that for all and . In other words, is the -algebra of all -stable matrices.

For Schurian schemes, the adjacency algebra has another description in terms of representation theory. Let be the space of complex-valued functions on , with the inner product

 ⟨f,g⟩L2(X)=∑x∈Xf(x)¯¯¯¯¯¯¯¯¯¯g(x)(f,g∈L2(X)).

Using the canonical basis of point masses in , we can think of our adjacency matrices as linear operators . The action of on produces a unitary representation ,

 [λ(g)f](x)=f(g−1⋅x)(g∈G,f∈L2(X),x∈X),

and the adjacency algebra coincides with its commutant

 C(λ)={T∈B(L2(X)):Tλ(g)=λ(g)T for all % g∈G}.

For a third realization of the adjacency algebra, fix a point and let be its stabilizer in . As a -set, is isomorphic to with the usual action on the left. For each , there is a unique double coset

 HaiH={haih′:h,h′∈H}

for in such that

 (2.2) (Ai)g⋅x0,h⋅x0={1,if h−1g∈HaiH;0,otherwise.

The adjacency matrices are in one-to-one correspondence with the double cosets through the mapping . Moreover, for any , the convolution kernel given by

 (2.3) ψA(g)=1|H|Ag⋅x0,x0(g∈G)

belongs to the space

 L2(H∖G/H)={ψ:G→C:ψ(gh)=ψ(hg)=ψ(g) for all g∈G,h∈H}

of bi--invariant functions on . This has the structure of a -algebra with the usual convolution and involution,

 (ψ1∗ψ2)(g)=∑h∈Gψ1(h)ψ2(h−1g)% andψ∗(g)=¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯ψ(g−1)(g∈G),

and if we set , the mapping is a -algebra isomorphism from to .

###### Definition 2.1.

We call a Gelfand pair when is commutative.

This notion, too, has a representation-theoretic interpretation. Let be the trace character of , namely

 χλ(g)=|{x∈X:g⋅x=x}|(g∈G),

and let

 χλ=n0χ0+⋯+nrχr

be its decomposition into irreducible characters , with for all , and for . Then is commutative if and only if for all . In other words, is a Gelfand pair if and only if is multiplicity free.

### 2.3. Spherical functions

Whether or not is commutative, many of its projections can be constructed explicitly from the character table of , as follows. Each of the constituent characters above determines a spherical function given by

 (2.4) ωj(g)=1|H|∑h∈Hχj(g−1h)=1|HgH|∑h∈HgH¯¯¯¯¯¯¯¯¯¯¯¯¯χj(h)(g∈G).

Writing for the degree of , the matrix

 (2.5) Ej:=mj|X|c∑i=0ωj(ai)Ai

is orthogonal projection onto the isotypical component corresponding to . In other words, is the unique -invariant subspace of on which the restriction of has trace character . Since is -invariant, is an orthogonal projection in with rank . Moreover, , and whenever . In particular, every sum for is an orthogonal projection in . If it happens that is commutative, then form the basis of primitive idempotents promised by the spectral theorem, and every projection in takes the form just described. We summarize these results below.

###### Proposition 2.2.

Every subset produces an orthogonal projection in with entries

 (2.6) (GD)g⋅x0,h⋅x0=1|X|∑j∈Dmjωj(h−1g)(g,h∈G).

When is a Gelfand pair, every orthogonal projection in takes this form.

We have written code for the computer program GAP [25] to compute the spherical functions associated with any transitive group action, and to produce the corresponding projections [32].

The spherical functions have another description in terms of invariant vectors. If is any unitary representation of affording as its trace character, then the space

 HHj={v∈Hj:πj(h)v=v for all h∈H}

of -stable vectors in has dimension . If is an orthonormal basis for , then

 (2.7) ωj(g)=nj∑i=1⟨ui,πj(g)ui⟩(g∈G).

### 2.4. Examples

###### Example 2.3.

acts trivially on the subspace of constant functions in , so one of the constituents of , say , is the trivial character . The corresponding spherical function is constantly equal to . Hence,

 E0=1|X|c∑i=0Ai=1|X|J,

which is indeed orthogonal projection onto the subspace of constant functions.

###### Example 2.4.

Let be any finite group, acting on by left translation. The adjacency algebra consists of all -circulant matrices. Since the stabilizer of any point in is the trivial group , is isomorphic to the -algebra . Thus, we have a Gelfand pair if and only if is abelian. In that case, the spherical functions are given by the Pontryagin dual group , which consists of all homomorphisms , under the operation of pointwise multiplication. Indeed, the permutation representation is the left regular representation of , and the Peter-Weyl Theorem tells us that every character appears as a constituent of . From (2.4), we see that the spherical function corresponding to is . Any choice of subset prescribes, via Proposition 2.2, a orthogonal projection with entries

 (2.8) (GD)g,h=1|G|∑α∈Dα(h)¯¯¯¯¯¯¯¯¯¯α(g)(g,h∈G).
###### Example 2.5.

Let be any finite group (possibly nonabelian), and let , acting on by the formula for . The orbits of the corresponding action on are indexed by the conjugacy classes of , and take the form

 Ri={(gh,h):g∈Ci,h∈K}.

If is the matrix representation for left translation by on , it follows that

 Ai=∑g∈CiLg.

The adjacency algebra is the center of the group algebra of -circulant matrices described in Example 2.4. The stabilizer of the point is the diagonal

 Δ(K)={(g,g):g∈K}.

Since is commutative, is a Gelfand pair.

As in the abelian case, primitive idempotents in are indexed by the set of irreducible characters of , with a character corresponding to the matrix

 Eχ:=χ(1)|K|[χ(g−1h)]g,h∈K.

See, for instance, [26, Thm. 10.6.1]. Once again, projections in are uniquely determined by subsets , through the formula

 (2.9) (GD)g,h=1|K|∑χ∈Dχ(1)χ(g−1h)(g,h∈G).

This completes our review of background material.

## 3. Homogeneous frames and Schurian schemes

In general, we are interested in association schemes primarily as sources of finite frames, represented by their Gram matrices in the corresponding adjacency algebra. Any positive semidefinite matrix can be viewed as the Gram matrix of some frame, and the resulting frame is Parseval if and only if the Gram matrix is a projection. By the spectral theorem, any commutative -algebra of square matrices therefore determines a finite set of Parseval frames through its projections. In the case of association schemes, the resulting frames have few inner products—no more than the number of adjacency matrices. In this sense, association schemes may be well suited for the construction of low-coherence tight frames.

In fact, the adjacency algebra of any association scheme of matrices contains the Gram matrices of three trivial ETFs: an orthonormal basis for (represented by the identity matrix ), identical vectors in (represented by ), and the -vector simplex in (represented by ). Note that these last two examples are Naimark complements of each other, and in general, adjacency algebras are closed under such complementation.

Among association schemes, the Schurian schemes are particularly attractive for two reasons. First, they provide a channel from the discrete world of finite group actions to the continuous setting of finite frames in . Second, the spherical functions make it easy to compute projections from a character table. Here, the identity matrix corresponds to in Proposition 2.2, whereas projection onto the all-ones vector comes from the trivial action of on constant functions in , as in Example 2.3. In addition, the Naimark complement corresponds to the set complement of . Now that we have established how trivial ETFs naturally arise from Schurian schemes, we are ready to pursue nontrivial constructions.

This section is devoted to the general theory of frames whose Gram matrices have Schurian structure. We begin by relating symmetry in the Gram matrix to symmetry in the frame vectors themselves. Subsection 3.2 continues with a series of techniques to identify group structure in a given frame. In subsection 3.3, we explain how this machinery distinguishes a small class of frames associated with a regular subgroup. Finally, subsection 3.4 introduces an important technical tool for squeezing additional line packings out of a given scheme. Illustrative examples are sprinkled throughout this section; more substantial examples appear in Sections 4 and 6.

### 3.1. Homogeneous frames

###### Definition 3.1.

Let be a finite group, and let be a unitary representation. Any frame of the form , with , is called a group frame, or more specifically, a -frame. If there is a subgroup such that for all , then we can reduce to form a new frame, . We call a homogeneous frame, or if we wish to emphasize the particular groups involved, a -frame.

###### Theorem 3.2.

Let be a finite group acting transitively on a set , and let be the adjacency algebra of all -stable, matrices. After changing indices through any -set isomorphism for a subgroup , the positive semidefinite matrices in are precisely the Gram matrices for -frames.

###### Proof.

First, let be a -frame. Then its Gram matrix is positive semidefinite, with entries given by

 GgH,hH=⟨ρ(h)v,ρ(g)v⟩

for . For any , we have

 GkgH,khH=⟨ρ(kh)v,ρ(kg)v⟩=⟨ρ(k)ρ(y),ρ(k)ρ(g)v⟩=GgH,hH,

since is unitary. Hence, .

In the other direction, if is any positive semidefinite matrix in , then its convolution kernel is a positive element of the finite-dimensional -algebra , so it has a unique positive square root . For each , let be the translation operator given by . We define and

 H=span{Lgv:g∈G}⊆L2(G),

and let describe left translation on . As a finite spanning set, is a frame for . Since is -stable, we obtain a -frame .

It remains to show that is the Gram matrix for . For any , we have

 ⟨λ(h)v,λ(g)v⟩=|H|∑k∈Gψ1/2G(k)ψ1/2G(k−1h−1g)=|H|(ψ1/2G∗ψ1/2G)(h−1g)=|H|ψG(h−1g).

By (2.3), , as desired. ∎

###### Example 3.3.

We now explain how to produce homogeneous frames with Gram matrices as in Proposition 2.2. Following the notation of Section 2, fix a subset , and let , where denotes the direct sum of copies of . For each , choose an orthonormal basis for the space of -stable vectors in . Then let

 v=1√|X|{√mju(j)i}j∈D,1≤i≤nj∈⨁j∈DH(nj)j.

It is clearly stabilized by . Comparing (2.6) and (2.7), we see that is a -frame with Gram matrix .

When (for instance, when is a Gelfand pair), there is an easy way to find a spanning vector for . Indeed, we can start with any , and then the vector

 ~v:=∑h∈Hπj(h)v

will be stabilized by . As long as is not orthogonal to , will be nonzero. In particular, if we apply this procedure to an entire orthonormal basis for , then we are guaranteed to find at least one nonzero -stable vector, which we can rescale and use as .

###### Example 3.4.

Let be a finite group acting on itself by left translation, as in Example 2.4. In this case, Theorem 3.2 tells us that positive semidefinite -circulant matrices are precisely the Gram matrices of -frames. For projections in the adjacency algebra and tight -frames, this is a theorem of Vale and Waldron [39].

If is abelian, the projections in are determined by subsets as in (2.8). As the reader can verify, is the Gram matrix of the harmonic frame whose synthesis matrix is made by extracting the rows indexed by from the discrete Fourier transform (DFT) matrix

 F=1√|G|[α(g)]α∈^G,g∈G.

Harmonic frames were an early and abundant source of ETFs [37, 44, 15]. Conditions for equiangularity are completely understood in terms of combinatorics. Namely, represents an ETF if and only if is a difference set in , meaning there is a constant such that

 ∣∣{(α,β)∈D×D:αβ−1=γ}∣∣=λ

for all in .

###### Example 3.5.

Let be any finite group, and let be the adjacency algebra of the conjugacy class scheme described in Example 2.5. Tight frames with Gram matrices in are called central -frames in [40]. In this case, Theorem 3.2 tells us that central -frames are equivalent to tight -frames.

The authors investigated conditions for equiangularity in [33]. Briefly, has the structure of a hypergroup, which is a probabilistic generalization of a group, and represents an ETF if and only if is a hyperdifference set, which is a corresponding generalization of a difference set. An infinite family of ETFs with this form, with nonabelian, appears in [33].

Any ETF made by one of the Gelfand pairs in Examples 2.4 and 2.5 is bound by an integrality constraint: if it consists of vectors in or , then must divide . For the harmonic frames in Example 2.4, this is an easy consequence of the difference set condition. For the central group frames in Example 2.5, this was proved in [33]. As the following example demonstrates, ETFs from more general Gelfand pairs enjoy greater latitude.

###### Example 3.6.

Let be the affine linear group over , acting transitively on the set of lines in in the natural way. Using the computer program GAP [25] with package FinInG [4], we check that this action is multiplicity free. One of its primitive idempotents is the matrix in Figure 2, which describes an ETF of 28 vectors in . The exact code used to produce this example is available online [32].

### 3.2. More conditions for homogeneity

Let be any finite frame, regardless of its origin. In practice, we often find nice frames by accident and then work very hard to “reverse engineer” them, in the hope of finding a larger family. If we think that symmetry might play a role in our accident, then it is natural to wonder if might be homogeneous, and if so, what the groups involved might be. We now address this problem.

For each permutation , let be the corresponding permutation matrix. Writing for the Gram matrix of , we define

 G(Φ)={σ∈S(X):PσG=GPσ},

which is clearly a subgroup of . Following the proof of [41, Lemma 3.5], it is easy to show that if and only if there is a unitary , necessarily unique, such that for all . When is tight, it follows by [41, Example 1] that is precisely the symmetry group of introduced by Vale and Waldron [41].

When restricted to tight frames, the first part of the following theorem recasts [39, Theorem 4.8] in the language of Schurian association schemes.

###### Theorem 3.7.

Fix any point , and let be the stabilizer of in . Then the following hold:

1. is a homogeneous frame if and only if acts transitively on . In that case, is a -frame.

2. If is a -frame for any Gelfand pair , then is also a Gelfand pair.

###### Proof.

First, suppose that is a transitive permutation group, and let

 A={M∈MatX×X(C):PσM=MPσ for all σ∈G(Φ)}

be the corresponding adjacency algebra. We have by definition, so is a -frame by Theorem 3.2.

Conversely, if is a -frame for some group and subgroup , then there is a transitive group action with point stabilizer for which lies in the adjacency algebra

 A′={M∈MatX×X(C):PσM=MPσ for all σ∈α(G)},

by Theorem 3.2. Consequently, . It follows that is also transitive. Moreover, , so is commutative whenever is. When is a Gelfand pair, this means that is, too. ∎

###### Corollary 3.8.

A finite frame is homogeneous if and only if, for every , there is a unitary such that is a permutation of with .

###### Remark 3.9.

We mention two more tricks that might help a researcher find a group that is “responsible” for a given frame . First, arguing just as in the proof of Theorem 3.7, one can easily see that is -stable for any transitive subgroup . Thus, if it happens that acts transitively and multiplicity-freely on , one can search the subgroup lattice of to find more groups with that description. Both of those properties are preserved by inclusion into a larger permutation group, so it could be that some subgroup provides a starting point for a simpler description of , and that inherited those properties only by default.

Second, if one can establish that lies in the adjacency algebra of some association scheme , then one can use graph-theoretic algorithms to determine whether or not is a Schurian scheme, and if so, what the corresponding permutation group is. See [28, Theorem 2.5] and [47, Theorem 6.3.1]. We have written a GAP function that does precisely this [32], returning the relevant permutation group when it exists. When this procedure succeeds, we necessarily have . Just as above, one can then dig deeper into the subgroup lattice of to look for alternative “explanations” for .

### 3.3. Regular subgroups

Now consider the reverse problem: instead of starting with a frame and trying to find a group responsible for it, we start with a group (like the Heisenberg group) and try to isolate a nice set of -frames. As we now explain, this is sometimes possible using an action of a larger group .

###### Definition 3.10.

Let be a finite group acting transitively on a set . A subgroup is called regular for this action if:

1. The action of on is transitive; and

2. The stabilizer of each in is trivial.

For the remainder of this subsection, we fix a finite group acting transitively on a set , and we let denote the stabilizer of a fixed base point . It is easy to prove that a subgroup is regular if and only if and , if and only if is a complete and irredundant set of left coset representatives for in . This leads us to the following simple, but crucial, observation.

###### Proposition 3.11.

If is a regular subgroup for the action of on , then every -frame is a -frame.

###### Proof.

The expression for a -frame does not depend on the choice of coset representatives , so . ∎

###### Remark 3.12.

It may very well happen that is a Gelfand pair, while a regular subgroup is nonabelian. In that case, the spherical functions for the Gelfand pair distinguish a finite subset of the uncountably infinite collection of tight -frames. (See [40] for the cardinality of -frames.)

An important class of examples occurs when and forms a Gelfand pair. For the researcher exploring the class of -frames, it may be worthwhile to traverse the lattice of subgroups and examine the spherical functions formed by any Gelfand pairs . In fact, that is exactly how the authors discovered the class of Heisenberg ETFs described in Section 6.

We can say more by involving the adjacency algebra. When is regular for the action of on , the mapping determines a bijection , which turns into a -set with the action

 g⋅k=k′⟺gk⋅x0=k′⋅x0(g∈G; k,k′∈K),

or equivalently,

 (3.1) g⋅k=k′⟺gkh=k′ for some h∈H(g∈G; k,k′∈K).

If it happens that normalizes (for instance, if ), then the action of on is simply conjugation; however, we do not assume this in general. The theorem below says that we can identify with the space

 L2(K)H:={φ∈L2(K):φ(h⋅k)=φ(k) for all% k∈K and h∈H}.

This appears to be a folk theorem. The closest reference we could locate is [5, Theorem 6.1]. We include a proof here for completeness.

###### Theorem 3.13.

When is a regular subgroup, is a -subalgebra of . It is isomorphic to the adjacency algebra of -stable matrices by mapping to the function

 ~φA(k)=Ax0,k⋅x0(k∈K).
###### Proof.

The main idea is to identify with . We will write for the -orbit of , and for the set of -orbits in , with a similar notation in . From (3.1), we see that

 [k]=HkH∩K(k∈K).

The mapping , , is an isomorphism of -sets, so by the bijection . On the other hand, -orbits in are precisely described by double cosets through the correspondence . Consequently, through the mapping . It follows that there is a linear isomorphism with