Let be the usual inner product on and the Euclidean norm on . For a lattice of full rank (that is a discrete co-compact subgroup of ) the minimal norm of is
and its set of minimal or shortest vectors is
The automorphism group of the lattice , , is the group of all real orthogonal matrices that map to itself. A particularly interesting class of lattices are eutactic lattices: a lattice is called eutactic if its set of minimal vectors satisfies a eutaxy condition, i.e. there exist positive real numbers , (called eutaxy coefficients) such that
for all . If , is said to be strongly eutactic. Eutactic and strongly eutactic lattices are central objects of lattice theory due to their importance in connection with well studied optimization problems. A theorem of Voronoi (1908) asserts that is a local maximum of the packing density function on the space of lattices in if and only if it is eutactic and perfect ( is perfect if the set spans the space of real symmetric matrices) [Vor08]. More details on eutactic, strongly eutactic and perfect lattices can be found in J. Martinet’s book [Mar03].
Two lattices and are called similar, written , if for a nonzero scalar and an orthogonal transformation . Similarity is an equivalence relation on lattices that preserves inner products between vectors and, as a result, lattice’s automorphism group; it also gives a bijection between sets of minimal vectors. Consequently, all the geometric properties that we discuss here, such as eutaxy, strong eutaxy and perfection are preserved on similarity classes.
In the previous papers [BFG16] and [BF17] of the first two authors lattices generated by equiangular tight frames (ETFs) were studied and examples of strongly eutactic such lattices were constructed. Here we aim to take this discussion further. Let and let be a set of vectors, not necessarily distinct, such that . Such a set is called an -frame, the name originating in a 1952 paper of Duffin and Schaeffer in connection with their study of nonharmonic Fourier series [DS52]. A frame is called uniform if all of its vectors have the same norm, and it is called tight if there exists a real constant such that for every
and the tight frame is called Parseval if : clearly, any tight frame can be rescaled to a Parseval frame. Notice the similarity between this equation and the equation (1) above. Although the tightness condition (2) above is well studied in several contemporary branches of mathematics, the closely related eutaxy condition precedes it by half a century. Voronoi’s study [Vor08] of quadratic forms in 1908 gave rise to the introduction of eutaxy condition (1). Nonetheless, we can say that a lattice is strongly eutactic whenever its set of minimal vectors forms a uniform tight frame. Another way to view uniform tight frames is as spherical -designs, a subclass of more general spherical -designs introduced by Delsarte, Goethals, and Seidel in their groundbreaking 1977 paper [DGS91]. A special class of tight frames are examples of optimal packings of lines in projective space. These uniform tight frames are called equiangular (abbreviated ETF) if is the same for all . Tight frames in general and ETFs in particular are extensively studied objects in harmonic analysis; see S. Waldron’s book [Wal18] for detailed information on this subject.
Given a real -frame , define
If we write for the matrix with vectors as columns, then
The norm-form associated with is the quadratic form
We call the frame rational if is (a constant multiple of) a rational quadratic form, i.e. the Gram matrix is (a constant multiple of) a rational matrix. This is equivalent to saying that the inner products are (up to a constant multiple) rational numbers for all . In [BF17], it was proved that if is rational, then is a lattice. Further, in the case that is an ETF, is a lattice if and only if is rational (the converse was previously proved in [BFG16]). More generally, it was shown in [BF17] that when the dimension or and is a tight -frame for any so that is a lattice, then must be rational. Our first result is an extension of this observation to any dimension.
Suppose that is a uniform tight -frame so that is a lattice. Then must be rational.
We give two different proofs of Theorem 1 in Section 2, one of them as a consequence of a stronger result about a larger class of matrices than just the tight frames. We can now use this result to pick out lattices generated by tight frames. We are especially interested in frames that give rise to lattices with nice geometric properties. For this we need some more notation. Let the automorphism group of a frame be
where is the group of real orthogonal matrices. As usual, we write to indicate that is a subgroup of the group .
We now discuss group frames; see Chapter 10 of [Wal18] for a detailed exposition. Let be a vector and let a finite group of orthogonal matrices. Define to be the orbit of under the action of by left multiplication, i.e.
then all the vectors in have the same norm. If spans , then is a uniform frame, which we refer to as a -frame. is said to act irreducibly on the space if there is no nonzero proper subspace of that is closed under the action of , that is, for any . A -frame with such an irreducible action corresponding to on is similarly called irreducible. All irreducible group frames are tight. In fact, if is a group with an irreducible action on , then the orbit of under , , is an irreducible tight -frame for any nonzero vector (see Sections 10.5 - 10.9 of [Wal18] for details).
Our next result demonstrates a certain two-way correspondence between irreducible group frames and strongly eutactic lattices.
Let be a group of real orthogonal matrices and be a vector so that is an irreducible rational group frame in . Then the lattice is strongly eutactic. Conversely, suppose is a strongly eutactic lattice of rank . Then its set of minimal vectors forms a uniform tight frame in . If in addition acts transitively on , then is an -frame, and if the action of on is irreducible then is an irreducible group frame.
We prove Theorem 2 in Section 3. This theorem raises a question: which irreducible group frames are rational? One steady source of such frames comes from vertex transitive graphs, as detailed in Section 10.7 of [Wal18].
Let be a vertex transitive graph on vertices and its automorphism group. Let be the adjacency matrix of and an eigenvalue of multiplicity -dimensional eigenspace to eigenvalue
an eigenvalue of multiplicity. Assume is rational and let be the corresponding
-dimensional eigenspace to eigenvalue. Let be a rational orthogonal projection matrix of onto . Then the lattice in is strongly eutactic, has rank and its automorphism group contains .
We review all the necessary notation and prove Theorem 3 in Section 4. There are plenty of examples of vertex transitive graphs with rational eigenvalues. In fact, for any , there exist such lattices on vertices having eigenvalues of multiplicity being an increasing function of (for instance complete graphs, Kneser graphs, Johnson graphs, various product and line graphs, etc.), so that this construction yields strongly eutactic lattices in arbitrarily high dimensions. We demonstrate several examples of this construction in Section 4, some of which are summarized in Table 1. A separate collection of lattices coming from several Johnson graphs is given in Table 2 in Section 4. Furthermore, in Theorem 13
we give a characterization of lattices coming from product graphs in terms of tensor products and orthogonal direct sums of component lattices.
For the purposes of all of our examples and constructions, the lattices are viewed up to similarity and eigenspaces of graphs are identified with real Euclidean spaces for the appropriate dimension equal to the multiplicity of the corresponding eigenvalue. Our examples have been computed in Maple [map] using online catalog [Bai] of distance regular graphs and online catalog [Mar] of strongly eutactic lattices. It can be seen from these examples that a graph and its complement produce the same lattices. This is true in general, as is shown in Proposition 14 in Section 4. At the end of Section 4 we also demonstrate an interesting correspondence between contact polytopes of lattices , and and our construction of lattices from their skeleton graphs.
|Graph||# of vertices||Eig.||Mult. of||Lattice|
|Disconnected graph||()||()||Integer lattice|
|Complete graph||()||()||Root lattice|
|Petersen graph||()||()||, dual of|
|Petersen graph||()||()||Coxeter lattice|
|Petersen line graph||()||()||, dual of|
|Petersen line graph||()||()||Coxeter lattice|
|Clebsch graph||()||()||, dual of|
|Clebsch graph complement||()||()||, dual of|
|Shrikhande graph complement||()||()|
|Schläfli graph||()||()||, dual of|
|Schläfli graph complement||()||()||, dual of|
|Gosset graph||()||()||, dual of|
It is also interesting to consider Theorem 3 in view of the properties of eutactic configurations, i.e. finite sets of vectors satisfying the eutaxy condition (1). The famous theorem of Hadwiger ([Mar03], Theorem 3.6.12) asserts that a set of cardinality in -dimensional space , , is eutactic if and only if it is an orthogonal projection onto of an orthonormal basis in an -dimensional space containing . In fact, our construction considers precisely such a projection, namely the set of vectors where is the standard basis in . This set is therefore eutactic by Hadwiger. Our result, however, implies more, specifically that in our setting these vectors generate a lattice whose set of minimal vectors is strongly eutactic.
Finally, in Section 5 we discuss a possible relation between coherence of a lattice and its sphere packing density, as well as potential applications of tight frames coming from sets of minimal vectors of lattices in compressed sensing.
2. Rationality of lattice-generating frames
We start with a simple proof of Theorem 1.
Proof of Theorem 1.
With notation as in the statement of the theorem, let be a real matrix whose columns are vectors of the tight frame and is a lattice. Let be a basis matrix for . Then, there exists a integer matrix so that . Thus
for some . Since is invertible,
so that . Therefore
Since has rational entries, we have that is a multiple of a rational matrix. Therefore is a rational tight frame. ∎
The above argument implies that if as in (3) is a quadratic form corresponding to an irrational tight frame then the corresponding integer span is not a lattice (i.e. is not discrete) because cannot be bounded away from on integer points. This argument, however, relies heavily on the norm-form coming from a tight frame. On the other hand, it is not difficult to construct other irrational quadratic forms (not corresponding to tight frames) which are bounded away from on integer points. For instance, take to be rational linear forms in variables and any positive real numbers. Let
This is a positive semidefinite quadratic form. Suppose for some integer vector , then there must exist such that . Since has rational coefficients, , where is the least common multiple of the denominators of these coefficients. Let and , then we have
for all for which . In particular, if some of the ’s are irrational, is a form with irrational coefficients.
In view of this observation, it is interesting to understand what are the necessary and sufficient conditions on a real matrix so that is a lattice to imply that must be rational? In the rest of this section we prove a sufficient condition that is weaker than being a tight frame. Write for the elements of a frame (a sequence of vectors spanning ), written as column vectors of a matrix , where . Let the first columns in be denoted in matrix form by and the remaining column vectors by , so that , , .
Suppose that is such that and is discrete. Then .
If is discrete, it is a full-rank lattice in , and so has a basis matrix such that . Hence there exist some integer matrices such that , and . Since is full rank and invertible, is invertible and . ∎
Let be an orthogonal real matrix, then multiplication by preserves inner products of vectors in . Hence a collection of vectors generates a lattice over if and only if does. Let be orthogonally equivalent to , that is for some ( denotes the set of real orthogonal matrices). , the identity matrix, so that the matrix of outer products for is . Having information about the entries of this matrix for certain (arising in this case from the -decomposition of a matrix) allows for an easy way to check rationality of inner products. When is a tight frame given in matrix form, (as above) for some , and so collapses to the same matrix as . In general, however the relationship between and can get “muddled” by transformation so that determining lattice properties of integer combinations of vectors in a tight frame is easier than the general case.
Given , the factorization of , so that and is upper-triangular with positive entries along the diagonal, it will be useful to work with the alternative representation of : . Additionally, the entries of may be given as
where are taken to be positive (possible since has positive diagonal entries). From now on, let denote a matrix of the form when not specified otherwise.
Suppose a collection of vectors , , is given as column vectors of a matrix of the form (as in the preceding remark). Suppose these column vectors have the following properties:
the row-vectors of satisfy for some , for , that is, for , and
for all .
Then the inner products must all be rational, i.e. .
For each column vector from , Lemma 4 implies there exists a vector such that . We now use the rational numbers and to show that the inner products, which are the entries of must be rational.
Recall that has rows and columns. From now on, denote the last column vectors of by , . The condition gives that , so for all , i.e., the numbers are rational. In the same manner, we obtain equations:
which imply that for all , as well as
which implies . Now, these equations can be written together in a matrix equation:
The matrix formed above on the left is invertible as all the rows of index greater than one are orthogonal to the first (this may be checked using the condition ) and the lower right block being the negative identity shows the last rows (arising from the first set of equalities above) are linearly independent amongst themselves. Thus by applying the inverse of the matrix on the left to each side we can express the coordinates of the vector on the left as rational numbers.
Proceeding, the idea now is to induct on “levels” (each level is determined by the smallest index in the variables appearing in the matrix equations of the type above) supposing that all the variables appearing in the previous level (with the exception of variables of the form which must be treated separately later) have been demonstrated to be rational. At the -th such level the arising matrix equation analogous to the one above is of the form:
As all entries in the matrix on the left appear in the left or right hand side vector of some matrix equation from a previous level, the inductive hypothesis implies that they are rational. A few observations are in order. The first rows in the matrix above are linearly independent by the fact the first column sub-matrix is upper-triangular with ones along the diagonal. Second, the remaining row vectors have inner products with the first row vectors which are zero as the expressions resulting in computing these inner products come exactly as the equations . Lastly, note that the last row vectors are linearly independent amongst themselves by the lower right block being minus the identity in . Together, these observations justify the claim that the above matrix is invertible, so that the variables may be expressed as rationals. This completes the inductive portion of the argument.
Reflect on what is known about the variables which have appeared in this process so far. For each , the variables have been shown to be rational along with the variables . There is one set of equations which have not appeared yet, along with a set of variables which have yet to play a role (the variables ). Treating these will be the last step of this argument.
The diagonal elements of give rise to the equations
where the convention is that a sum with starting index larger than the ending index is zero. For , the corresponding equation is
Since all of the variables are rational, so is . An analogous argument establishes that is rational as in those equations, and are rational (by the previous inductive argument). All that remains now is to compute the inner products. These are of the form
which are all rational. ∎
Suppose that is a matrix with column vectors given by , a Parseval tight frame. Then is discrete if and only if are rational.
Second proof of Theorem 1.
Suppose that is a uniform tight -frame so that is a lattice. Then for all ,
for an appropriate constant . Hence is a Parseval tight frame and is again a lattice. Then Corollary 6 implies that inner products of vectors in are rational, and so inner products of vectors in are rational multiples of . ∎
3. Lattices from irreducible group frames
In this section we focus on group frames and lattices generated by them. As in Section 1, let be a vector and let a finite group of orthogonal matrices. Assume that
spans , that is, it is a -frame. If is a cyclic group, is called a cyclic frame. An example of a cyclic frame is the -ETF discussed, for instance, in Section 5 of [BFG16]:
If is an abelian group, is a harmonic frame (see Section 11.3 of [Wal18], Theorem 11.1). Notice that for any -frame , . We also make a simple observation about the size of the -frame .
Let be a -frame in , then . Further, if and only if if an eigenvector for some non-identity matrix
if an eigenvector for some non-identity matrixwith the corresponding eigenvalue equal to 1.
The fact that is clear from the definition. Now assume . This means that for some two distinct , , hence , meaning that is an eigenvector for with the corresponding eigenvalue equal to 1, and since is a group. On the other hand, if if an eigenvector for some non-identity matrix with the corresponding eigenvalue equal to 1, then , and so there are at most distinct vectors in . ∎
We now prove Theorem 2, splitting it into two lemmas.
Suppose that is an irreducible -frame and is a lattice. Then is strongly eutactic.
The automorphism group of , , is the group of all orthogonal matrices that permute the lattice. Then we have
and the action of on is irreducible. Let be the set of minimal vectors of and let . Since the automorphisms of permute the minimal vectors, it must be true that is closed under the action of . Thus we must have , and so acts irreducibly on , the space spanned by the minimal vectors of . Then Theorem 3.6.6 of [Mar03] guarantees that is a strongly eutactic configuration, and hence is a strongly eutactic lattice. ∎
Let be a strongly eutactic lattice of full rank. Then its set of minimal vectors forms a uniform tight frame in . In addition, if acts transitively on , then is a -frame, and if the action of on is irreducible then is an irreducible group frame.
By Corollary 16.1.3 of [Mar03], is strongly eutactic if and only if its set of minimal vectors is a spherical 2-design, which is a condition equivalent to the tightness condition (2). Since all minimal vectors have the same norm, is a uniform tight frame. Now suppose some acts transitively on . Let , then for any there exists a such that . Hence
and so is an -frame. ∎
4. Vertex transitive graphs
Constructions of irreducible group frames from vertex transitive graphs are described in Section 10.7 of [Wal18]. We briefly review this subject here, proving Theorem 3 and providing some applications.
Let be a graph on vertices labeled by integers with automorphism group . is called vertex transitive if for each pair of vertices there exists such that . From here on graphs considered will always be vertex transitive. Let denote the standard basis vectors in . Then acts on by
for every and vector . Let be the adjacency matrix of , so that if vertices and are connected by an edge and otherwise. The condition that is vertex transitive translates to for all . The matrix is symmetric, with real eigenvalues , each of multiplicity , so that . From now on, we call these the eigenvalues of graph . For each let be the corresponding -dimensional eigenspace. The group acts irreducibly on each eigenspace , hence for any nonzero vector the orbit of under the action of is an irreducible group frame in . Further, if is the orthogonal projection onto , then for any and ,
As indicated in Section 10.7 of [Wal18], this identity is true since the action of and the action of the adjacency matrix on a vector commute, i.e.
Proof of Theorem 3.
Suppose now that an eigenvalue is an integer and the group consists of integer matrices. Pick a nonzero integer vector . Then and the frame is rational, and hence generates a strongly eutactic lattice so that . We will refer to lattices obtained in this manner as lattices generated by the graph . Let us consider some examples of such lattices. For instance, let , then the corresponding frame
consists of column vectors of (possibly with repetitions), since is some for every , and every is representable as for some , since the graph is vertex transitive. We will refer to the corresponding lattice as just , or as when the graph need be specified. These observations complete the proof of Theorem 3. ∎
For the rest of this section, we consider examples of this lattice construction when applied to various graphs and their products. One class of lattices that will figure prominently in our examples are root lattices, that is, integral lattices generated by vectors of norm , which are called its roots (recall that a lattice is integral if the inner product between any two vectors is always an integer). Also recall that the dual lattice of a full rank lattice is
If is integral, then .
Let be a completely disconnected graph on vertices, then generates the integer lattice .
The adjacency matrix for is the -matrix, and so it has one eigenvalue with multiplicity with the corresponding eigenspace being the entire . The automorphism group of