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 χ 1 , . . . , χ m of A ◮ use these to plot the elements of A in m -dimensional complex space ◮ Consider X = { [ χ 1 ( g ) , . . . , χ m ( g )] | g ∈ A } ⊂ C m William J. Martin Homotopy and Ideals

Ideals Discrete Homotopy Duality Configurations from characters ◮ Let A be a finite (abelian) group ◮ choose characters χ 1 , . . . , χ m of A ◮ use these to plot the elements of A in m -dimensional complex space ◮ Consider X = { [ χ 1 ( g ) , . . . , χ m ( g )] | g ∈ A } ⊂ C m ◮ We wish to study this configuration of points, especially when we obtain a spherical code Problem: Find a simple set of polynomials F i ( Y ) = F ( Y 1 , . . . , Y m ) (1 ≤ i ≤ s ) such that X is precisely the set of common zeros of the polynomials F 1 , . . . , F s 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 χ 100 , χ 010 , χ 001 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 χ 100 , χ 010 , χ 001 form an orthogonal basis for the corresponding eigenspace. χ a ( b ) = ( − 1) a · b a , b ∈ Z 3 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 χ 100 , χ 010 , χ 001 form an orthogonal basis for the corresponding eigenspace. χ a ( b ) = ( − 1) a · b a , b ∈ Z 3 2 Here 000 001 010 011 100 101 110 111 χ 100 = [ 1 1 1 1 -1 -1 -1 -1] χ 010 = [ 1 1 -1 -1 1 1 -1 -1] χ 001 = [ 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 Z 3 2 : 000 001 010 011 100 101 110 111 χ 100 = [ 1 1 1 1 -1 -1 -1 -1] χ 010 = [ 1 1 -1 -1 1 1 -1 -1] χ 001 = [ 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 R 3 . 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 Z 3 2 : 000 001 010 011 100 101 110 111 χ 100 = [ 1 1 1 1 -1 -1 -1 -1] χ 010 = [ 1 1 -1 -1 1 1 -1 -1] χ 001 = [ 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 R 3 . One easily checks that this is the zero set of the ideal I = � Y 2 Y 2 Y 2 1 − 1 , 2 − 1 , 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): C m { a 1 , . . . , a v } ⊂ � a i � = 1 ∀ i |� a i , a j �| = ∀ i � = j ( c const) c m � ∀ b ∈ C m � b , a i � a i = b v i 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 Z 7 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 χ 1 , χ 2 , χ 4 give us 7 vectors We have three eigenvectors of the 3-cube, which are characters of Z 3 2 : With ω = e 2 π i / 7 0 1 2 3 4 5 6 ω 2 ω 3 ω 4 ω 5 ω 6 ] χ 1 = [ 1 ω ω 2 ω 4 ω 6 ω 3 ω 5 ] χ 2 = [ 1 ω ω 4 ω 5 ω 2 ω 6 ω 3 ] χ 4 = [ 1 ω William J. Martin Homotopy and Ideals

Ideals Discrete Homotopy Duality ETFs from Difference Sets With ω = e 2 π i / 7 0 1 2 3 4 5 6 ω 2 ω 3 ω 4 ω 5 ω 6 ] χ 1 = [ 1 ω ω 2 ω 4 ω 6 ω 3 ω 5 ] χ 2 = [ 1 ω ω 4 ω 5 ω 2 ω 6 ω 3 ] χ 4 = [ 1 ω This gives us a configuration in C 3 (1 , 1 , 1) , ( ω, ω 2 , ω 4 ) , ( ω 2 , ω 4 , ω ) , ( ω 3 , ω 6 , ω 5 ) , � X = ( ω 4 , ω, ω 2 ) , ( ω 5 , ω 3 , ω 6 ) , ( ω 6 , ω 5 , ω 3 ) � We’d like to describe the ideal I ( X ) of polynomials in C [ Y 1 , Y 2 , Y 3 ] 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 C m . For a ∈ X , write a = ( a 1 , . . . , a m ) . Now consider polynomials in m variables F ( Y ) = F ( Y 1 , . . . , Y m ) from the polynomial ring R = C [ Y 1 , . . . , Y m ]. We wish to study the ideal I ( X ) = { F ∈ R | F ( a 1 , . . . , a m ) = 0 ∀ a ∈ X } of 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 C m . For a ∈ X , write a = ( a 1 , . . . , a m ) . Now consider polynomials in m variables F ( Y ) = F ( Y 1 , . . . , Y m ) from the polynomial ring R = C [ Y 1 , . . . , Y m ]. We wish to study the ideal I ( X ) = { F ∈ R | F ( a 1 , . . . , a m ) = 0 ∀ a ∈ X } of 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 C m . For a ∈ X , write a = ( a 1 , . . . , a m ) . Now consider polynomials in m variables F ( Y ) = F ( Y 1 , . . . , Y m ) from the polynomial ring R = C [ Y 1 , . . . , Y m ]. We wish to study the ideal I ( X ) = { F ∈ R | F ( a 1 , . . . , a m ) = 0 ∀ a ∈ X } of 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: ◮ γ 1 ( X ): smallest degree of a “non-trivial” polynomial in I ( X ) (not divisible by eqn of sphere) ◮ γ 2 ( X ): the smallest k for which I ( X ) admits a generating set of 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, γ 1 ( X ) ≥ t / 2. Shortest vectors of some famous lattices gives examples: Name Dim strength γ 1 ( X ) γ 2 ( X ) icos. 3 5 3 3 6 5 3 3 E 6 E 7 7 5 3 3 8 7 4 4 E 8 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 Lewis’s Homotopy Duality 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 Lewis’s Homotopy Duality 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 Lewis’s Homotopy Duality 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 Lewis’s Homotopy Duality 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 Lewis’s Homotopy Duality Homotopy: the group operation atwa ⋆ awsva = atwsva William J. Martin Homotopy and Ideals