New results on equiangular lines or How I caught a gold fish? - - PowerPoint PPT Presentation

New results on equiangular lines

• r ‘How I caught a gold fish?’

Ferenc Szöll˝

• si

szoferi@gmail.com

Department of Communications and Networking, Aalto University

Talk at CSD8 2017 Mons

Ferenc Szöll˝

• si (ComNet, Aalto University)

Equiangular lines CSD8, 2017/08/23 1 / 32

Overview

This talk covers some of my old and new results on equiangular lines. An overview, based on Equiangular lines in Euclidean spaces, arXiv:1402.6429 (jointly with Greaves, Koolen, and Munemasa) There are no 21 real equiangular lines in R12 (with α = 1/5), based on Enumeration of Seidel matrices, arXiv:1703.02943 (jointly with Östergård) There are at least 54 real equiangular lines in R18, based on A remark on a construction of D.S. Asche, arXiv:1703.04505 There are no 8 complex equiangular lines with angle

• 5/21,

based on All complex equiangular tight frames in dimension 3, arXiv:1402.6429

Ferenc Szöll˝

• si (ComNet, Aalto University)

Equiangular lines CSD8, 2017/08/23 2 / 32

Real equiangular lines

Definition

A set of n lines, spanned by the unit vectors v1, v2, . . . , vn ∈ Rd is called equiangular, if there is a common angle α such that |

• vi, vj
• | = α, i = j ∈ {1, . . . , n}.

Examples: The n vectors of an orthonormal basis in Rn (with α = 0); n = 3 vectors in R2 formed by rotation of 120 degrees (with α = 1/2), aka ‘the complex 3rd roots of unity’; n = 6 lines passing through the antipodal vertices of the icosahedron in R3 (with α = 1/ √ 5).

Problem

For d ≥ 2 fixed, what is the maximum number of real equiangular lines, N(d)?

Ferenc Szöll˝

• si (ComNet, Aalto University)

Equiangular lines CSD8, 2017/08/23 3 / 32

Historical remarks

Regarding the real case Research initiated by Haantjes 1948 Seidel and coauthors 1966–1995 Conway and Taylor 1972–1992 Makhnev 2002; and Bannai, Munemasa, Venkov 2005 Renewed interest in the complex case (quantum tomography) Godsil, Roy 2005 Appleby, Zauner 2000s Scott, Grassl 2010s This used to be a “sleepy field” (with very little activity) because it was believed that [at least in the real case] “things have already been completed in the ’70s (and what is left open is hopeless anyways), see reference [X], and/or ask Prof. Y about it”. The sleeper, however, has awakened! (Recent explosion in terms of the number of new papers).

Ferenc Szöll˝

• si (ComNet, Aalto University)

Equiangular lines CSD8, 2017/08/23 4 / 32

Representing the lines

We assume α > 0 (we exclude the case of the o.n.b) The unit vectors vi ∈ Rd can be represented by column vectors (w.r.t. a fixed basis). F = [v1, . . . , vn] is the frame operator. The Gram matrix G := F TF ([G]i,j :=

• vi, vj
• ).

The Seidel matrix S := (G −I)/α (|Si,j| = 1 for i = j and Si,i = 0). The adjacency matrix A = (J − S − I)/2 of the ambient graph. Note: If vi is replaced by −vi we obtain the same configuration. Permutation of the vectors within themselves yields the same configuration. Actually, we consider the equivalence classes of these objects.

Definition

Any two configurations {v1, . . . , vn} and {±vσ(1), . . . , ±vσ(n)} are called equivalent (σ ∈ Sn).

Ferenc Szöll˝

• si (ComNet, Aalto University)

Equiangular lines CSD8, 2017/08/23 5 / 32

n = 6 vectors in R4, α = 1/3

Assume that v1, v2, v3, and v4 are linearly independent. F = [v1, v2, v3, v4, v5, v6] :=               1

1 3 1 3 1 3 1 3 1 3 2 √ 2 3

−

√ 2 3

−

√ 2 3

−

√ 2 3

−

√ 2 3 √ 6 3

−

√ 6 3 √ 6 3

−

√ 6 3

              F is the frame operator, a short-fat matrix. In most cases it is inconvenient to work with F. Basis dependent representation. F is d × n. Column permutation and column negation does not change the line system what F represents. In this representation n, d, and α are visible.

Ferenc Szöll˝

• si (ComNet, Aalto University)

Equiangular lines CSD8, 2017/08/23 6 / 32

Gram matrices

G = F TF =               1

1 3 1 3 1 3 1 3 1 3 1 3

1 − 1

3

− 1

3

− 1

3

− 1

3 1 3

− 1

3

1

1 3

− 1

3 1 3 1 3

− 1

3 1 3

1

1 3

− 1

3 1 3

− 1

3

− 1

3 1 3

1

1 3 1 3

− 1

3 1 3

− 1

3 1 3

1               G is n × n, positive semidefinite. |Gi,j| = α, Gi,i = 1. rank(G) = d (the dimension is no longer visible). F is essentially the Cholesky-decomposition of G. Since the vectors of F live in R4, we have rank(G) = 4. In particular, there is some 4 × 4 nonvanishing principal minor. det(M4) = 32/81 > 0, where M4 is the leading 4 × 4 principal minor.

Ferenc Szöll˝

• si (ComNet, Aalto University)

Equiangular lines CSD8, 2017/08/23 7 / 32

Seidel matrices

S = (G − I)/α =         1 1 1 1 1 1 −1 −1 −1 −1 1 −1 1 −1 1 1 −1 1 1 −1 1 −1 −1 1 1 1 −1 1 −1 1         S is of n × n. S = ST. Sii = 0, Sij = ±1 for i = j. Neither the dimension d nor the common angle α is visible.

Lemma

The smallest eigenvalue of S, λmin(S) = −1/α of multiplicity n − d. Proof: Recall, that rank(G) = d, so G has exactly n − d eigenvalues 0 (and all other eigenvalues are positive). Then G − I has exactly n − d eigenvalues −1, and finally S = (G − I)/α has exactly n − d eigenvalues −1/α. This is its smallest eigenvalue. Example: Λ(S) = {−3, −3, 1, 1, 1, 3} = {[−3]2, [1]3, [3]1}.

Ferenc Szöll˝

• si (ComNet, Aalto University)

Equiangular lines CSD8, 2017/08/23 8 / 32

Graphs

Let J be the matrix with Jij = 1. A = (J − S − I)/2 =         1 1 1 1 1 1 1 1 1 1 1 1         A is of n × n A = AT Aii = 0, Aij ∈ {0, 1}. The matrix A is called the adjacency matrix of the ambient graph (or underlying graph) Γ(S) of the Seidel matrix S. Remark: Γ depends on the concrete representation of S. Remark: there is no (clear) correspondence between the eigenvalues

• f A, and the eigenvalues of S.

Example: Λ(A) = {[−1]2, [0]1, [1]1, [(1 ± √ 17)/2]1}.

Ferenc Szöll˝

• si (ComNet, Aalto University)

Equiangular lines CSD8, 2017/08/23 9 / 32

Upper bounds

Lemma[Gerzon’s bound, 1971]

Let d ≥ 2. Then N(d) ≤ d(d + 1)/2. Proof: counting argument. The right hand side is the dimension of the real vectorspace of real symmetric matrices. If there is equality, then d = 2, 3 or (d + 2) is the square of an odd integer. For d = 2, 3, 7, 23 we do have equality. For d = 47 we don’t have! (Makhnev, 2002)

Example[n = 28 lines in R7, α = 1/3]

Let v = [−3, −3, 1, 1, 1, 1, 1, 1] ∈ R8, and let V := {σ(v): σ ∈ S8}. |V| = 8

2

• = 28, and v, σ(v) = ±8. Normalize by

√ 24 to get unit

• vectors. Observe that σ(v), [1, 1, 1, 1, 1, 1, 1, 1] = 0, so we are in R7.

Ferenc Szöll˝

• si (ComNet, Aalto University)

Equiangular lines CSD8, 2017/08/23 10 / 32

Upper bounds (ctd.)

Lemma[Relative bound]

If α ≤ 1/ √ d + 2 then n ≤ d(1 − α2)/(1 − dα2). When n > 2d then α is 1/(2k + 1) for some k ∈ N. The condition α ≤ 1/ √ d + 2 is essential. Equality iff the Seidel matrix has exactly two distinct eigenvalues. The interesting cases are usually those where α > 1/ √ d + 2. Message: when the angle is “too” small, then there are not so many lines. For maximizing n in Rd, the choice of α is crucial.

Example[n = 16 lines in R6 with α = 1/3]

Let d = 6, and assume that n > 2d = 12. Then 1/α ∈ {3, 5, . . . } and in particular α ≤ 1/ √ 6 + 2. Therefore n ≤ 6(1 − α2)(1 − 6α2) = 16. Take a symmetric Hadamard matrix H with constant diagonal −1 of order

• 16. It has spectrum Λ(H) = {[−4]10, [4]6}. Then consider S := H + I.

Ferenc Szöll˝

• si (ComNet, Aalto University)

Equiangular lines CSD8, 2017/08/23 11 / 32

Upper bounds (ctd.)

There are other ad-hoc bounds for (i) small d; and for (ii) specific values of α (Barg, King, Okuda with coauthors).

Lemma[Lemmens–Seidel, 1973]

Let α = 1/3, and d ≥ 15. Then n ≤ 2(d − 1). Proof: Lengthy, yet elementary argument.

Proposition[Neumaier, 1989]

Let α = 1/5, and d large. Then n ≤ ⌊3(d − 1)/2⌋. Proof: Graph theory (correspondence with Dynkin graphs).

Theorem[Bukh, 2015]

If α is fixed, then n ≤ c(α) · d. Proof: Clever probablistic argument. Message: when the angle is “too” large, then there are not so many lines either.

Ferenc Szöll˝

• si (ComNet, Aalto University)

Equiangular lines CSD8, 2017/08/23 12 / 32

Small cases

d N(d) Spectrum Remark 2 3 {[−2]1, [1]2} Gerzon bd. 3 6 {[− √ 5]3, [ √ 5]3} Gerzon bd. 4 6 {[−3]2, [1]3, [3]1} and the one above 5 10 {[−3]5, [3]5} Relative, Petersen 6 16 {[−3]10, [5]6} Hadamard mx. Rel. bd. 7 − 13 28 {[−3]21, [9]7} Gerzon bd. 14 28 − 29 {[−5]14, [3]7, [7]7} and the one above 15 36 {[−5]21, [7]15} Hadamard mx. Rel bd. 16 40 − 41 {[−5]24, [7]15, [15]1} From a SRG 17 48 − 50 {[−5]31, [7]8, [11]9} From an STS 18 54 − 60 {[−5]36, [7]6, [11]8,[13]2, [12 ± √ 37]1}, Sz. 2017 19 72 − 75 {[−5]53, [13]16, [19]3} D.S. Asche 1971 20 90 − 95 {[−5]70, [15]9, [19]10,[25]1} D.E. Taylor 1971 21 126 {[−5]105, [25]21}

• Rel. bd.

22 176 {[−5]154, [35]22}

• Rel. bd.

23 276 {[−5]253, [55]23} Golay code, Gerzon bd.

Ferenc Szöll˝

• si (ComNet, Aalto University)

Equiangular lines CSD8, 2017/08/23 13 / 32

Small open cases

d N(d) Spectrum Remark 14 28 − 29 {[−5]14, [3]7, [7]7} From a Hadamard mx. 16 40 − 41 {[−5]24, [7]15, [15]1} From a SRG 17 48 − 50 {[−5]31, [7]8, [11]9} From an STS 18 54 − 60 {[−5]36, [7]6, [11]8,[13]2, [12 ± √ 37]1}, Sz. 2017 19 72 − 75 {[−5]53, [13]16, [19]3} D.S. Asche, 1971 20 90 − 95 {[−5]70, [15]9, [19]10,[25]1} D.E. Taylor, 1971 42 276 − 288 Semidefinite Programming 44− ??? Weak bounds

Problem

Improve on this situation. Remarks and observations: It appears that N(d) is always even (cf. coding theory). In the optimal cases S has “few” distinct eigenvalues. The upper bounds shown here are nontrivial, and follow from elaborate case-by-case analysis (Azarija–Marc, Greaves et al).

Ferenc Szöll˝

• si (ComNet, Aalto University)

Equiangular lines CSD8, 2017/08/23 14 / 32

A (computer) study of small Seidel matrices

Question: How many Seidel matrices are there up to equivalence? Recall that S1 ∼ S2 are equivalent, if S1 = MS2MT, where M is some monomial matrix (that is: M = PD for some permutation matrix P and some ±1 diagonal matrix D). The spectrum Λ(S) is invariant up to equivalence.

Example: Seidel matrices for n ≤ 3 (there are 2n(n−1)/2 of them)

S =

• ,

U = 1 1

• ∼
• −1

−1

• = V.

X =   1 1 1 1 1 1   ∼   0 −1 1 −1 0 −1 1 −1   ∼   1 −1 1 0 −1 −1 −1   ∼   0 −1 −1 −1 1 −1 1   , Y =   0 −1 −1 −1 0 −1 −1 −1   ∼   1 −1 1 1 −1 1   ∼   0 −1 1 −1 1 1 1   ∼   1 1 1 0 −1 1 −1   . Λ(X) = {[−1]2, [2]1}, Λ(Y) = {[−2]1, [1]2}.

Ferenc Szöll˝

• si (ComNet, Aalto University)

Equiangular lines CSD8, 2017/08/23 15 / 32

Historical and contemporary remarks

A: Quite a few... n # Generated by 1 1 2 1 3 2 4 3 5 7 6 16 7 54 Van Lint, Seidel (1966, by hand) 8 243 9 2.038 Bussemaker, Mathon, Seidel (1981) 10 33.120 Spence (early 1990s) 11 1.182.004 McKay (1990s) 12 87.723.296 Greaves, Koolen, Munemasa, Sz., 2014 13 12.886.193.064 Östergård and Sz., 2016 14 3.633.057.074.584 ?

Ferenc Szöll˝

• si (ComNet, Aalto University)

Equiangular lines CSD8, 2017/08/23 16 / 32

Enumeration of small Seidel matrices

Question: How many Seidel matrices are there up to equivalence?

Theorem[Mallows–Sloane, 1975]

The number of Seidel matrices up to equivalence equals to the number

• f Euler graphs up to graph isomorphism.

Euler graph: every vertex degree is even, but not necessarily connected. Explicit, computable formula is available (Robinson 1969). When n is odd, then every Seidel equivalence class contains a unique Euler graph. Therefore if Γi, i ∈ I are pairwise nonisomorphic Euler graphs, with adjacency matrices Ai, i ∈ I, then Si := J − 2Ai − I are pairwise inequivalent Seidel matrices. The previous correspondence fails to hold for n even (already for n = 4), nevertheless, the number of objects agree. We need a graph representation working for all n.

Ferenc Szöll˝

• si (ComNet, Aalto University)

Equiangular lines CSD8, 2017/08/23 17 / 32

(Another) graph representation of Seidel matrices

Recall that equivalent Seidel matrices can have nonisomorphic ambient graphs. The goal is to encode S as a graph X(S) such that if S1 ∼ S2 then X(S1) and X(S2) are isomorphic as graphs.

Graph representation of n × n Seidel matrices

Let S be n × n. We create a graph X(S) on 3n vertices in the following way: a row ri, i ∈ {1, . . . , n} of S is represented by a triplet of vertices in X(S) – a “cherry” – formed by a green vertex ui adjacent to two red vertices v(1)

i

and v(2)

i

. V = {u1, . . . , un} ∪ {v(1)

1 , v(2) 1 , . . . , v(1) n , v(2) n }.

The edge set in addition contains edges based on the elements Sij. E = {{ui, v(k)

i

}} ∪ {{v(k)

i

, v(k)

j

}: Sij = 1} ∪ {{v(k)

i

, v(3−k)

j

}: Sij = −1}.

Example for n = 2

The Seidel matrix S = 1 1

• is represented by the hexagon with a

pair of antipodal green vertices.

Ferenc Szöll˝

• si (ComNet, Aalto University)

Equiangular lines CSD8, 2017/08/23 18 / 32

Example of graph representation

S =   −1 1 −1 −1 1 −1   The rows r1, r2, and r3 correspond to the encircled “cherries”.

r1 r2 r3

X(S)

S is represented by X(S), a 2-colored graph on 9 vertices. Equivalence is now reduced to deciding graph isomorphism. X(S) should be designed carefully.

Ferenc Szöll˝

• si (ComNet, Aalto University)

Equiangular lines CSD8, 2017/08/23 19 / 32

Generation of Seidel matrices

Assume that Xn is a complete set of representatives of Seidel matrices

• f order n. For example, X1 = {
• }. Starting from Xn, we generate

Xn+1 by the method of canonical augmentation (McKay, 1998): For every S ∈ Xn. Set up a container C ← ∅. For every possible row v ∈ {±1}n, append v to S forming its last row and column (S S, the dependence on v is not shown). Discard S if S is not its canonical parent. Add S to C if this equiv. class has not yet been found. End for. Output C. End for. (Some irrelevant, minor details are being skipped here, such as why anything like this would work, and how one should choose a canonical parent in the first place...) The main point is that the equivalence class S can be obtained from multiple starting point matrices, say from S0 := S, S1, . . . , Sk ∈ Xn such that S0 ∼ S1 ∼ · · · ∼

• Sk. We avoid duplicates by declaring a

canonical parent, say Sj, to the equivalence class of

• S. Then

Si is kept

• nly if i = j.

Ferenc Szöll˝

• si (ComNet, Aalto University)

Equiangular lines CSD8, 2017/08/23 20 / 32

Interlacing eigenvalues

Q: How do we get large Seidel matrices (corresponding to large set of equiangular lines)? We have yet to use the spectrum of S.

Theorem[Interlacing:Basic version]

Assume that S is a Seidel matrix, and λ ∈ Λ(S) of multiplicity m ≥ 2. Let T be any principal submatrix of S. Then λ ∈ Λ(T) of multiplicity at least m − 1. Moreover, if λ is the smallest eigenvalue of S, then it is the smallest eigenvalue of T. If m is “large”, then we can use this result iteratively to conclude that a “small” principal submatrix of S has a prescribed eigenvalue. This structural information can be exploited during the matrix generation.

Lemma

Let S be a Seidel matrix of order n, and let λ ∈ Λ(S) of multiplicity m ≥ 1. Then m = n − rank(S − λI). Moreover if λ / ∈ Λ(S) then rank(S − λI) = n.

Ferenc Szöll˝

• si (ComNet, Aalto University)

Equiangular lines CSD8, 2017/08/23 21 / 32

Exploiting interlacing: a test case

Example[n = 28 lines in R7 with common angle α = 1/3]

Consider a Seidel matrix S corresponding to n = 28 equiangular lines in R7 (the common angle is 1/3). S is of order 28 with λmin(S) = −3 of multiplicity exactly m = 21. So any 27 × 27 principal minor of S should have λ as an eigenvalue of multiplicity at least 20, . . . , any 8 × 8 principal minor of S should have λ = −3 as (the smallest) an eigenvalue! The number of Seidel matrices (up to equivalence) with λmin = −3 of multiplicity at least n − 7: n 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 # 23 37 54 70 90 101 103 101 90 70 54 37 23 16 10 5 3 2 1 1 1 0 Recall that |X8| = 243, and |X13| ≈ 1.2 × 1010. Note that there are 2n ways to augment an n × n Seidel matrix with a new row/column (actually, enough to check half of these). For n = 28 this is about 268 × 106, a manageable number of cases.

Ferenc Szöll˝

• si (ComNet, Aalto University)

Equiangular lines CSD8, 2017/08/23 22 / 32

Exploiting interlacing: towards catching a BIG fish

Theorem[Östergård and Sz., 2015]

The maximum number of equiangular lines in R12 with common angle α = 1/5 is 20. There are exactly 32 distinct configurations. Proof: along the same lines as previous example. Assume that there exist 21 equiangular lines in R12. Then it is represented by a 21 × 21 Seidel matrix with smallest eigenvalue −5 of multiplicity exactly 9. Then any 13 × 13 principal minor T has λmin(T) = −5. There are at most 26.030.960 such Seidel matrices. Augment these with a new row/column and increase the multiplicity of −5. There are no 21 × 21 examples. The number of Seidel matrices with λ = −5 ∈ Λ(S) of multiplicity at least n − 12: n 13 14 15 16 17 18 19 20 21 # 26030960 8897086 2931650 851892 155223 16385 852 32 0

Ferenc Szöll˝

• si (ComNet, Aalto University)

Equiangular lines CSD8, 2017/08/23 23 / 32

The BIG FISH

Problem[aka the BIG FISH]

Are there n = 29 equiangular lines in R14 (with α = 1/5)? The framework developed previously could be – in principle – applied for this long standing open problem. There is a tiny caveat, however. First steps of a (potential) proof: Assume that there exist 29 equiangular lines in R14. Then it is represented by a 29 × 29 Seidel matrix with smallest eigenvalue −5 of multiplicity exactly 15. Then any 15 × 15 principal minor T has λmin(T) = −5. We estimate, that there are about 3 × 1010 such Seidel matrices... Remark: one may circumvent the problem of 15 × 15 matrices by considering 14 linearly independent equiangular vectors in R14. This corresponds to 14 × 14 Seidel matrices without the eigenvalue −5. Experiments show that there are about the same number of such matrices as above (ie. in the range of 1010). We should do 1000 times better (theory+implementation+CPUs time)

Ferenc Szöll˝

• si (ComNet, Aalto University)

Equiangular lines CSD8, 2017/08/23 24 / 32

Equiangular lines in Rd, d ∈ {18, 19, 20, 21, 22, 23}

We have this magic construction, the “Golay code”. From the Golay code one – in a relatively simple way – construct 276 equiangular lines in R23. This achieves Gerzon’s bound hence it is maximal. Idea: consider a hyperplane, and look at those vectors which are

• rthogonal to it. Which hyperplane to consider?

Ansatz: consider “the most obvious one” (obvious to group theorists) which maintains the highest amount of symmetry. In this way, you can find 176 lines in R22, 126 lines in R21, 90 lines in R20, 72 lines in R19, and 48 lines in R18. This has been known since the 1980s. New idea: throw away some of the symmetry, and choose a “non

• bvious” hyperplane.

Theorem[Sz., 2017]

We have N(18) ≥ 54.

Ferenc Szöll˝

• si (ComNet, Aalto University)

Equiangular lines CSD8, 2017/08/23 25 / 32

Complex ETFs

What is an (n, d) ETF?

A collection of n equiangular unit vectors {vi}n

i=1 in Cd at angle

α = |

• vi, vj
• | =
• n − d

d(n − 1). An (n, n) ETF is an orthonormal basis. An (n + 1, n) ETF is the simplex An (n2, n) ETF is a SIC-POVM. Remark: n ≤ d2. Problem: For what pair of values (n, d) exist an ETF? Conjecture: (n2, n) ETFs exist for all n ≥ 2.

Ferenc Szöll˝

• si (ComNet, Aalto University)

Equiangular lines CSD8, 2017/08/23 26 / 32

Gram matrices of ETFs

Let {vi}n

i=1 be an (n, d) ETF.

The Gram matrix G associated to an ETF is [G]i,j =

• vi, vj
• Properties:

G = G∗ Gii = 1, |Gi,j|2 =

n−d d(n−1)

dG2 = nG Remark: From G one can get back the set {vi}n

i=1.

Theorem[F .Sz., 2014]

There is no (8, 3) complex ETF. Proof: Gröbner basis computation.

Ferenc Szöll˝

• si (ComNet, Aalto University)

Equiangular lines CSD8, 2017/08/23 27 / 32

Nonexistence of (8, 3) ETFs

Assume that such a frame exist, and consider its Gram matrix. G = 1 α             α 1 1 1 1 1 1 1 1 α g23 g24 g25 g26 g27 g28 1 1/g23 α g34 g35 g36 g37 g38 1 1/g24 1/g34 α g45 g46 g47 g48 1 1/g25 1/g35 1/g45 α g56 g57 g58 1 1/g26 1/g36 1/g46 1/g56 α g67 g68 1 1/g27 1/g37 1/g47 1/g57 1/g67 α g78 1 1/g28 1/g38 1/g48 1/g58 1/g68 1/g78 α             21α2 − 5 = 0, G2 − 8/3G = 0, u

• 1<i<j<9

gij − 1 = 0 Polynomial equations in 21 + 1 + 1 variables (just clear the denominators). Attempt to compute a Groebner basis... and fail.

Ferenc Szöll˝

• si (ComNet, Aalto University)

Equiangular lines CSD8, 2017/08/23 28 / 32

Eliminate the variables from the last two rows/columns

There is a (series of) “magic” polynomial identity (due to Haagerup) Φ relating the entries within the principal 6 × 6 minor. It is a consequence

• f the Gram equations. Consider

H = 1 α         α 1 1 1 1 1 1 α g23 g24 g25 g26 1 1/g23 α g34 g35 g36 1 1/g24 1/g34 α g45 g46 1 1/g25 1/g35 1/g45 α g56 1 1/g26 1/g36 1/g46 1/g56 α         21α2−5 = 0, Φ(H) = 0, rank(H) ≤ 3, rank(3H2−8H) ≤ 2, u

• 1<i<j<7

gij − 1 = 0 Polynomial equations in 10 + 1 + 1 variables. If there is G, then there is H, however H does not exist.

Ferenc Szöll˝

• si (ComNet, Aalto University)

Equiangular lines CSD8, 2017/08/23 29 / 32

So what is Φ?

A toy example

Imagine, that you have two equations (assume that all variables are unimodular, that is, |z|2 = 1, that is, z = 1/z): x1 + y1 = a, x2 + y2 = a, ux1x2y1y2a − 1 = 0. Now this is equivalent to x1 + y1 = a, 1/x2 + 1/y2 = a, ux1x2y1y2a − 1 = 0. This is a system of three polynomial equations in 5 + 1 variables. From this, we can eliminate a and consequently have x2y2(x1 + y1) = x2 + y2, ux1x2y1y2 − 1 = 0. This is a reduced system of polynomial equations in 4 + 1 variables. In real life, Φ connects the variables in three distinct rows of G.

Ferenc Szöll˝

• si (ComNet, Aalto University)

Equiangular lines CSD8, 2017/08/23 30 / 32

But seriously, what is Φ??

Consider three distinct rows (i, j, k) of the Gram matrix G, and look at the corresponding Gram equations [G2]x,y = [n/mG]x,y, (x, y) ∈ {(i, j), (j, k), (k, j)}. You have three equations like this: ngx,y/m −

n−2

• ℓ=1

gx,ℓgy,ℓ = gx,n−1gy,n−1 + gx,ngy,n On the left hand side variables of H show up, while on the right hand side variables of the last two columns of G show up, so we have L1 = R1, L2 = R2, and L3 = R3. Now multiply up these three equations: L1L2L3 = R1R2R3 ≡ |R1|2 + |R2|2 + |R3|2 − 4 = |L1|2 + |L2|2 + |L3|2 − 4. Do this for all triplets (i, j, k); Φ is the collection of all these equations.

Ferenc Szöll˝

• si (ComNet, Aalto University)

Equiangular lines CSD8, 2017/08/23 31 / 32

Thank you!

Ferenc Szöll˝

• si (ComNet, Aalto University)

Equiangular lines CSD8, 2017/08/23 32 / 32