Graph homotopy, ideals of finite varieties and a surprising duality - - PowerPoint PPT Presentation

Ideals Discrete Homotopy Duality

Graph homotopy, ideals of finite varieties and a surprising duality

William J. Martin

Department of Mathematical Sciences and Department of Computer Science Worcester Polytechnic Institute

Modern Trends in Algebraic Graph Theory Villanova June 3, 2014

William J. Martin Homotopy and Ideals

Ideals Discrete Homotopy Duality

First Visit to Villanova

William J. Martin Homotopy and Ideals

Ideals Discrete Homotopy Duality

Configurations from characters

◮ Let A be a finite (abelian) group ◮ choose characters χ1, . . . , χm of A ◮ use these to plot the elements of A in m-dimensional complex

space

◮ Consider X = {[χ1(g), . . . , χm(g)] | g ∈ A} ⊂ Cm

William J. Martin Homotopy and Ideals

Ideals Discrete Homotopy Duality

Configurations from characters

◮ Let A be a finite (abelian) group ◮ choose characters χ1, . . . , χm of A ◮ use these to plot the elements of A in m-dimensional complex

space

◮ Consider X = {[χ1(g), . . . , χm(g)] | g ∈ A} ⊂ Cm ◮ We wish to study this configuration of points, especially when

we obtain a spherical code Problem: Find a simple set of polynomials Fi(Y ) = F(Y1, . . . , Ym) (1 ≤ i ≤ s) such that X is precisely the set of common zeros of the polynomials F1, . . . , Fs

William J. Martin Homotopy and Ideals

Ideals Discrete Homotopy Duality

Representations of distance-regular graphs

As our first simple example, consider the 3-cube. The second largest eigenvalue is θ = 1 and the characters χ100, χ010, χ001 form an orthogonal basis for the corresponding eigenspace.

William J. Martin Homotopy and Ideals

Ideals Discrete Homotopy Duality

Representations of distance-regular graphs

As our first simple example, consider the 3-cube. The second largest eigenvalue is θ = 1 and the characters χ100, χ010, χ001 form an orthogonal basis for the corresponding eigenspace. χa(b) = (−1)a·b a, b ∈ Z3

2

William J. Martin Homotopy and Ideals

Ideals Discrete Homotopy Duality

Representations of distance-regular graphs

As our first simple example, consider the 3-cube. The second largest eigenvalue is θ = 1 and the characters χ100, χ010, χ001 form an orthogonal basis for the corresponding eigenspace. χa(b) = (−1)a·b a, b ∈ Z3

2

Here 000 001 010 011 100 101 110 111 χ100 = [ 1 1 1 1

• 1
• 1
• 1
• 1]

χ010 = [ 1 1

• 1
• 1

1 1

• 1
• 1]

χ001 = [ 1

• 1

1

• 1

1

• 1

1

• 1]

William J. Martin Homotopy and Ideals

Ideals Discrete Homotopy Duality

3-cube, continued

We have three eigenvectors of the 3-cube, which are characters of Z3

2:

000 001 010 011 100 101 110 111 χ100 = [ 1 1 1 1

• 1
• 1
• 1
• 1]

χ010 = [ 1 1

• 1
• 1

1 1

• 1
• 1]

χ001 = [ 1

• 1

1

• 1

1

• 1

1

• 1]

This gives us a Euclidean configuration X = {(1, 1, 1), (1, 1, −1), (1, −1, 1), (1, −1, −1), (−1, 1, 1), (−1, 1, −1), (−1, −1, 1), (−1, −1, −1)} in R3.

William J. Martin Homotopy and Ideals

Ideals Discrete Homotopy Duality

3-cube, continued

We have three eigenvectors of the 3-cube, which are characters of Z3

2:

000 001 010 011 100 101 110 111 χ100 = [ 1 1 1 1

• 1
• 1
• 1
• 1]

χ010 = [ 1 1

• 1
• 1

1 1

• 1
• 1]

χ001 = [ 1

• 1

1

• 1

1

• 1

1

• 1]

This gives us a Euclidean configuration X = {(1, 1, 1), (1, 1, −1), (1, −1, 1), (1, −1, −1), (−1, 1, 1), (−1, 1, −1), (−1, −1, 1), (−1, −1, −1)} in R3. One easily checks that this is the zero set of the ideal I = Y 2

1 − 1,

Y 2

2 − 1,

Y 2

3 − 1

William J. Martin Homotopy and Ideals

Ideals Discrete Homotopy Duality

ETFs from Difference Sets

Our second example comes from the area of “compressive sensing”. It has been shown that every (v, k, λ) difference set in an abelian group gives rise to an “equiangular tight frame” (ETF):

William J. Martin Homotopy and Ideals

Ideals Discrete Homotopy Duality

ETFs from Difference Sets

Our second example comes from the area of “compressive sensing”. It has been shown that every (v, k, λ) difference set in an abelian group gives rise to an “equiangular tight frame” (ETF): {a1, . . . , av} ⊂ Cm ai = 1 ∀ i |ai, aj| = c ∀ i = j (c const) m v

• i

b, ai ai = b ∀ b ∈ Cm

William J. Martin Homotopy and Ideals

Ideals Discrete Homotopy Duality

ETFs from Difference Sets

Every (v, k, λ) difference set in an abelian group gives rise to an “equiangular tight frame” (ETF). Ex: The quadratic residues in Z7 form a (7, 3, 1) difference set S = {1, 2, 4} satisfies 1−2 = 6, 1−4 = 4, 2−1 = 1, 2−4−5, 4−1 = 3, 4−2 = 2. The corresponding characters χ1, χ2, χ4 give us 7 vectors We have three eigenvectors of the 3-cube, which are characters of Z3

2:

With ω = e2πi/7 1 2 3 4 5 6 χ1 = [ 1 ω ω2 ω3 ω4 ω5 ω6] χ2 = [ 1 ω2 ω4 ω6 ω ω3 ω5] χ4 = [ 1 ω4 ω ω5 ω2 ω6 ω3]

William J. Martin Homotopy and Ideals

Ideals Discrete Homotopy Duality

ETFs from Difference Sets

With ω = e2πi/7 1 2 3 4 5 6 χ1 = [ 1 ω ω2 ω3 ω4 ω5 ω6] χ2 = [ 1 ω2 ω4 ω6 ω ω3 ω5] χ4 = [ 1 ω4 ω ω5 ω2 ω6 ω3] This gives us a configuration in C3 X =

• (1, 1, 1), (ω, ω2, ω4), (ω2, ω4, ω), (ω3, ω6, ω5),

(ω4, ω, ω2), (ω5, ω3, ω6), (ω6, ω5, ω3)

• We’d like to describe the ideal I(X) of polynomials in

C[Y1, Y2, Y3] that vanish on X.

William J. Martin Homotopy and Ideals

Ideals Discrete Homotopy Duality

Ideal of a finite set

Let X be a finite subset of Cm. For a ∈ X, write a = (a1, . . . , am). Now consider polynomials in m variables F(Y ) = F(Y1, . . . , Ym) from the polynomial ring R = C[Y1, . . . , Ym]. We wish to study the ideal I(X) = {F ∈ R | F(a1, . . . , am) = 0 ∀ a ∈ X}

• f all polynomials in m variables that vanish at every point of X.

William J. Martin Homotopy and Ideals

Ideals Discrete Homotopy Duality

Ideal of a finite set

Let X be a finite subset of Cm. For a ∈ X, write a = (a1, . . . , am). Now consider polynomials in m variables F(Y ) = F(Y1, . . . , Ym) from the polynomial ring R = C[Y1, . . . , Ym]. We wish to study the ideal I(X) = {F ∈ R | F(a1, . . . , am) = 0 ∀ a ∈ X}

• f all polynomials in m variables that vanish at every point of X.

William J. Martin Homotopy and Ideals

Ideals Discrete Homotopy Duality

Ideal of a finite set

Let X be a finite subset of Cm. For a ∈ X, write a = (a1, . . . , am). Now consider polynomials in m variables F(Y ) = F(Y1, . . . , Ym) from the polynomial ring R = C[Y1, . . . , Ym]. We wish to study the ideal I(X) = {F ∈ R | F(a1, . . . , am) = 0 ∀ a ∈ X}

• f all polynomials in m variables that vanish at every point of X.

William J. Martin Homotopy and Ideals

Ideals Discrete Homotopy Duality

Two “dual girth” parameters

For interesting structures represented by subsets X of complex space, we are interested in two measures of complexity:

◮ γ1(X): smallest degree of a “non-trivial” polynomial in I(X)

(not divisible by eqn of sphere)

◮ γ2(X): the smallest k for which I(X) admits a generating set

• f polynomials of degree k or less

William J. Martin Homotopy and Ideals

Ideals Discrete Homotopy Duality

The Icosahedron and Famous Lattices

Fact: If X is a spherical t-design, γ1(X) ≥ t/2. Shortest vectors of some famous lattices gives examples: Name Dim strength γ1(X) γ2(X) icos. 3 5 3 3 E6 6 5 3 3 E7 7 5 3 3 E8 8 7 4 4 Leech 24 11 6 6

for lattices, the shortest vectors often form an association scheme

(joint with Corre Love Steele arXiv:1310.6626)

William J. Martin Homotopy and Ideals

Ideals Discrete Homotopy Duality

Pause

good idea! William J. Martin Homotopy and Ideals

Ideals Discrete Homotopy Duality Lewis’s Homotopy

A New Take on Graph Homotopy

The following idea appears in the thesis work of Heather Lewis (Discrete Math. (2000)) under the supervision of Paul Terwilliger. Consider equivalence classes of closed walks in G starting and ending at basepoint a.

William J. Martin Homotopy and Ideals

Ideals Discrete Homotopy Duality Lewis’s Homotopy

A New Take on Graph Homotopy

The following idea appears in the thesis work of Heather Lewis (Discrete Math. (2000)) under the supervision of Paul Terwilliger. Consider equivalence classes of closed walks in G starting and ending at basepoint a.

William J. Martin Homotopy and Ideals

Ideals Discrete Homotopy Duality Lewis’s Homotopy

Discrete Homotopy on a Graph

Closed walk atwa is in the same equivalence class as atwswa. These both have “essential length” 3.

William J. Martin Homotopy and Ideals

Ideals Discrete Homotopy Duality Lewis’s Homotopy

Discrete Homotopy on a Graph

Our group operation is concatenation of walks. In this case, the concatenation of these two walks is represented by another cycle.

William J. Martin Homotopy and Ideals

Ideals Discrete Homotopy Duality Lewis’s Homotopy

Homotopy: the group operation

atwa ⋆ awsva = atwsva

William J. Martin Homotopy and Ideals

Ideals Discrete Homotopy Duality Lewis’s Homotopy

Homotopy: the group operation

In this way, larger cycles are built from smaller ones. For example, take our first walk atwa

William J. Martin Homotopy and Ideals

Ideals Discrete Homotopy Duality Lewis’s Homotopy

Homotopy: the group operation

. . . and concatenate with the walk awtuwa

William J. Martin Homotopy and Ideals

Ideals Discrete Homotopy Duality Lewis’s Homotopy

Homotopy: the group operation

. . . and concatenate with the walk awtuwa which also has essential length 3 as it has form pqp−1 for a walk q = wtuw of length three and a path p.

William J. Martin Homotopy and Ideals

Ideals Discrete Homotopy Duality Lewis’s Homotopy

Homotopy: the group operation

In our fundamental group, we have atwa ⋆ awtuwa = atuwa a walk of essential length 4.

William J. Martin Homotopy and Ideals

Ideals Discrete Homotopy Duality Lewis’s Homotopy

Subgroups of the Fundamental Group

Let π(G, a) be the homotopy group, as just defined, w basepoint a.

William J. Martin Homotopy and Ideals

Ideals Discrete Homotopy Duality Lewis’s Homotopy

Subgroups of the Fundamental Group

Let π(G, a) be the homotopy group, as just defined, w basepoint a. For each k, let πk(G, a) be the subgroup generated by walks of essential length k.

William J. Martin Homotopy and Ideals

Ideals Discrete Homotopy Duality Lewis’s Homotopy

Subgroups of the Fundamental Group

Let π(G, a) be the homotopy group, as just defined, w basepoint a. For each k, let πk(G, a) be the subgroup generated by walks of essential length k. For example, if G is a simple graph, πk(G, a) = 1 for k = 0, 1, 2.

William J. Martin Homotopy and Ideals

Ideals Discrete Homotopy Duality Lewis’s Homotopy

Discrete Homotopy on a Graph

In this example, π(G, a) = π3(G, a)

William J. Martin Homotopy and Ideals

Ideals Discrete Homotopy Duality Lewis’s Homotopy

Discrete Homotopy on a Graph

In this example, π3(G, a) = π4(G, a) = π5(G, a) = π(G, a)

William J. Martin Homotopy and Ideals

Ideals Discrete Homotopy Duality Lewis’s Homotopy

Discrete Homotopy on a Graph

In this example, π3(G, a) = π4(G, a) = π(G, a)

William J. Martin Homotopy and Ideals

Ideals Discrete Homotopy Duality Lewis’s Homotopy

Some results of Heather Lewis

If simple graph G has diameter d, then

◮ π0(G, x) = π1(G, x) = π2(G, x) ⊆ π2d+1(G, x) = π(G, x)

William J. Martin Homotopy and Ideals

Ideals Discrete Homotopy Duality Lewis’s Homotopy

Some results of Heather Lewis

If simple graph G has diameter d, then

◮ π0(G, x) = π1(G, x) = π2(G, x) ⊆ π2d+1(G, x) = π(G, x) ◮ a Q-polynomial distance-regular graph has girth at most 6 ◮ For any Q-polynomial distance-regular graph, π6(G, x) = {e} ◮ . . . and either π6(G, x) = π(G, x) or [something strange

happens] This suggests a second “girth” parameter for graphs: g2(G) := min{k | πk(G, x) = π(G, x)}

What about Moore graphs? cages? coset graphs of additive codes? William J. Martin Homotopy and Ideals

Ideals Discrete Homotopy Duality

Translation Graphs

Let G be a Cayley graph for abelian group A: G = C(A, S) for some “connection set” S ⊆ A. Assume G is a connected graph (i.e., S generates group A). For convenience, assume S = −S so that G is undirected.

William J. Martin Homotopy and Ideals

Ideals Discrete Homotopy Duality

Characters are Eigenvectors

It is well-known that the (linear) characters of group A provide an eigenbasis for the adjacency matrix of graph G. Natural isomorphism from A to its group of irreducible characters.

William J. Martin Homotopy and Ideals

Ideals Discrete Homotopy Duality

Characters are Eigenvectors

It is well-known that the (linear) characters of group A provide an eigenbasis for the adjacency matrix of graph G. Natural isomorphism from A to its group of irreducible characters. Now take only those characters corresponding to elements of S S = {g1, . . . , gm} → {χ1, . . . , χm} (χi = χgi). This gives us a map A → Cm mapping g ∈ A to [χ1(g), . . . , χm(g)] Denote by X this set of |A| vectors in Cm.

mutually unbiased bases, equiangular tight frames William J. Martin Homotopy and Ideals

Ideals Discrete Homotopy Duality

Main Theorem

Claim: If πk(G, e) = π(G, e), then I(X) is generated by some set

• f polynomials each of degree ⌈k/2⌉ or less.

William J. Martin Homotopy and Ideals

Ideals Discrete Homotopy Duality

Polynomials from Closed Walks

Proof: First observe that each closed walk w in graph G gives a polynomial Fw(Y ) in I(X). If w = h0 = e ∼ h1 ∼ h2 ∼ . . . ∼ hℓ−1 ∼ hℓ = e with hi = hi−1 + gji (gji ∈ S) then gj1 + gj2 + · · · + gjℓ = e in A. So, the entrywise product of characters χj1 ◦ χj2 ◦ · · · · · · χjℓ = 1 in CA. So Yj1Yj2 · · · Yjℓ − 1 ∈ I(X) and we can halve the degree using S = −S.

William J. Martin Homotopy and Ideals

Ideals Discrete Homotopy Duality

Subideal Generated by Closed Walks

Let J denote the ideal generated by all polynomials arising from closed walks in G in the above fashion. By hypothesis, J is generated by some set of polynomials each of degree at most ⌈k/2⌉. We have J ⊆ I. To prove equality, we show Z(J ) ⊆ Z(I).

William J. Martin Homotopy and Ideals

Ideals Discrete Homotopy Duality

Phantom Zeros

Now let a ∈ Cm be any zero of the polynomials in ideal J . We have ℓ

i=1 aji = 1 whenever gj1 + · · · + gjℓ = e in A.

William J. Martin Homotopy and Ideals

Ideals Discrete Homotopy Duality

Phantom Zeros

Now let a ∈ Cm be any zero of the polynomials in ideal J . We have ℓ

i=1 aji = 1 whenever gj1 + · · · + gjℓ = e in A.

Now view a as a function S → C and extend this to a function α : A → C using α(g + h) = α(g)α(h). Since a satisfies all equations coming from J , this is consistent. and defines a character on group A! So a ∈ X and we’re done.

William J. Martin Homotopy and Ideals

Ideals Discrete Homotopy Duality

Thank You

Villanova University Art Gallery Upstairs in the Connelly Center

William J. Martin Homotopy and Ideals