An ideal associated to any cometric association scheme William J. - PowerPoint PPT Presentation

Prelude Definitions from a Simple Example Some Theory The Ideal An ideal associated to any cometric association scheme William J. Martin Department of Mathematical Sciences and Department of Computer Science Worcester Polytechnic Institute IPM20 May 19, 2009 William J. Martin The Ideal of E 1

Prelude Definitions from a Simple Example Some Theory The Ideal Outline Prelude Definitions from a Simple Example The 6-cycle Some Theory Main Parameters Main Results and Conjectures The known examples Dismantlability The Ideal Small degree Conjecture William J. Martin The Ideal of E 1

Prelude Definitions from a Simple Example Some Theory The Ideal Why Association Schemes? ◮ Coding Theory ◮ Design Theory William J. Martin The Ideal of E 1

Prelude Definitions from a Simple Example Some Theory The Ideal Why Association Schemes? ◮ Coding Theory ◮ Design Theory ◮ “Distinguishability” ◮ “Approximation” William J. Martin The Ideal of E 1

Prelude Definitions from a Simple Example Some Theory The Ideal Why Association Schemes? ◮ Coding Theory ◮ Design Theory ◮ “Distinguishability” ◮ “Approximation” ◮ E.g., binary codes in ◮ E.g, t -( v , k , λ ) designs Hamming scheme H ( n , q ) William J. Martin The Ideal of E 1

Prelude Definitions from a Simple Example The 6-cycle Some Theory The Ideal Six Vectors in R 2 We will start by looking at a very simple example. William J. Martin The Ideal of E 1

Prelude Definitions from a Simple Example The 6-cycle Some Theory The Ideal Spherical Code A spherical code is simply a finite non-empty subset of the unit sphere. X ⊂ S m − 1 (We’ll set v = | X | and assume v > m .) Example: m = 2, v = 6 √ √ � � � � � 1 3 − 1 3 X = (1 , 0) , 2 , , 2 , , ( − 1 , 0) , 2 2 √ √ � � � �� − 1 3 1 3 2 , − , 2 , − 2 2 William J. Martin The Ideal of E 1

Prelude Definitions from a Simple Example The 6-cycle Some Theory The Ideal Gram Matrix   1 0 � 1 1 − 1 − 1 1 − 1 � √ 2 2 2 2   1 3 √ √ √ √   2 2 3 3 3 3   0 0 − − √   2 2 2 2 − 1 3   2 2      2 1 − 1 − 2 − 1 1  − 1 0     1 2 1 − 1 − 2 − 1 √     − 1 3  −    = 1 − 1 1 2 1 − 1 − 2  2 2    =: G √     − 2 − 1 1 2 1 − 1 2 1 3 −   2 2   − 1 − 2 − 1 1 2 1   1 − 1 − 2 − 1 1 2 William J. Martin The Ideal of E 1

Prelude Definitions from a Simple Example The 6-cycle Some Theory The Ideal Schur (Hadamard) Multiplication G ◦ G =  2 1 − 1 − 2 − 1 1   2 1 − 1 − 2 − 1 1  1 2 1 − 1 − 2 − 1 1 2 1 − 1 − 2 − 1         1 − 1 1 2 1 − 1 − 2 − 1 1 2 1 − 1 − 2     ◦     − 2 − 1 1 2 1 − 1 − 2 − 1 1 2 1 − 1 4         − 1 − 2 − 1 1 2 1 − 1 − 2 − 1 1 2 1     1 − 1 − 2 − 1 1 2 1 − 1 − 2 − 1 1 2  1 1  1 1 1 1 4 4 4 4 1 1 1 1 1 1   4 4 4 4   1 1 1 1 1 1 G ◦ 2 =   4 4 4 4   1 1 1 1 1 1   4 4 4 4   1 1 1 1 1 1   4 4 4 4 1 1 1 1 1 1 4 4 4 4 William J. Martin The Ideal of E 1

Prelude Definitions from a Simple Example The 6-cycle Some Theory The Ideal Schur Multiplication Again G ◦ G ◦ 2 =  2 1 − 1 − 2 − 1 1   1 1 1 1 1 1  4 4 4 4 1 2 1 − 1 − 2 − 1 1 1 1 1 1 1     4 4 4 4     1 − 1 1 2 1 − 1 − 2 1 1 1 1 1 1     ◦ 4 4 4 4     − 2 − 1 1 2 1 − 1 1 1 2 1 1 1 1     4 4 4 4     − 1 − 2 − 1 1 2 1 1 1 1 1 1 1     4 4 4 4 1 − 1 − 2 − 1 1 2 1 1 1 1 1 1 4 4 4 4  1 − 1 − 1 − 1  1 1 8 8 8 8 1 − 1 − 1 − 1 1 1   8 8 8 8   − 1 1 1 1 − 1 − 1 G ◦ 3 =   8 8 8 8   − 1 − 1 1 − 1 1 1   8 8 8 8   − 1 − 1 − 1 1 1 1   8 8 8 8 − 1 − 1 − 1 1 1 1 8 8 8 8 William J. Martin The Ideal of E 1

Prelude Definitions from a Simple Example The 6-cycle Some Theory The Ideal Entrywise Powers of G Span a Vector Space Consider the vector space A spanned by J , G , G ◦ 2 , G ◦ 3 , G ◦ 4 , . . . � � where the all-ones matrix J is G ◦ 0 and G = G ◦ 1 . Clearly, in our case, this space has dimension four and admits a basis of 01-matrices. William J. Martin The Ideal of E 1

Prelude Definitions from a Simple Example The 6-cycle Some Theory The Ideal Symmetric Association Scheme Let us say that the set X determines an association scheme if this vector space A is closed under matrix multiplication. Observe: ◮ A is closed under Schur multiplication; ◮ A contains the identity, J , for Schur multiplication; ◮ A is closed under ordinary multiplication; ◮ Since the points in X are distinct, A contains the identity, I , for ordinary multiplication; ◮ Since the matrices in A are all symmetric, they commute. William J. Martin The Ideal of E 1

Prelude Definitions from a Simple Example The 6-cycle Some Theory The Ideal Bose-Mesner Algebra The vector space/ring/ring of matrices A is called the Bose-Mesner algebra . This is equivalent to a symmetric association scheme. We may always construct two canonical bases: { A 0 = I , A 1 , . . . , A d } (01-matrices which sum to J (pairwise disjoint support)); { E 0 = 1 v J , E 1 , . . . , E d } (pairwise orthogonal idempotents summing to I ). William J. Martin The Ideal of E 1

Prelude Definitions from a Simple Example The 6-cycle Some Theory The Ideal Cometric ( Q -polynomial) Association Scheme Let us say that the association scheme ( X , { A i } d i =0 ) is cometric with respect to X if ◮ for each k , the vector space � J , G , G ◦ 2 , . . . , G ◦ k � is closed under multiplication. Observe: Eigenvalues of G must be 0 and v / m , assuming X spans R m . Then we can take E 1 = m v G , E 2 = ω 2 ( G ◦ G ) + ω 1 G + ω 0 J and E j = q j ◦ ( E 1 ) where q j is a polynomial of degree exactly j (0 ≤ j ≤ d ) (Notation: f ◦ ( M ) is matrix obtained by applying f to each entry.) William J. Martin The Ideal of E 1

Prelude Definitions from a Simple Example The 6-cycle Some Theory The Ideal Back to the Example For the hexagon, we obtain     1 0 0 0 0 0 0 1 0 0 0 1 0 1 0 0 0 0 1 0 1 0 0 0         0 0 1 0 0 0 0 1 0 1 0 0     A 0 = , A 1 =     0 0 0 1 0 0 0 0 1 0 1 0         0 0 0 0 1 0 0 0 0 1 0 1     0 0 0 0 0 1 1 0 0 0 1 0     0 0 1 0 1 0 0 0 0 1 0 0 0 0 0 1 0 1 0 0 0 0 1 0         1 0 0 0 1 0 0 0 0 0 0 1     A 2 = , A 3 =     0 1 0 0 0 1 1 0 0 0 0 0         1 0 1 0 0 0 0 1 0 0 0 0     0 1 0 1 0 0 0 0 1 0 0 0 William J. Martin The Ideal of E 1

Prelude Definitions from a Simple Example The 6-cycle Some Theory The Ideal Symmetric 01-Matrices are Graphs William J. Martin The Ideal of E 1

Prelude Definitions from a Simple Example The 6-cycle Some Theory The Ideal Back to the Example For the hexagon, we obtain E 0 = 1 E 1 = 1 6 J , 3 G , E 2 = 1 E 3 = 1 6(3 A 0 + 3 A 3 − J ) , 6( A 0 − A 1 + A 2 − A 3 ) William J. Martin The Ideal of E 1

Prelude Definitions from a Simple Example The 6-cycle Some Theory The Ideal Another Example: E 8 Root Lattice ◮ even unimodular lattice in R 8 ◮ kissing number 240 (optimal) ◮ can be identified with the integral Cayley numbers We will focus on the spherical code consisting of the 240 (scaled) shortest vectors. William J. Martin The Ideal of E 1

Prelude Definitions from a Simple Example The 6-cycle Some Theory The Ideal Shortest vectors √ The 240 norm 8 vectors: ◮ (0 6 , ± 2) – any two positions, all possible signs (4 · 28 = 112 vectors) ◮ ( ± 1 , ± 1 , ± 1 , ± 1 , ± 1 , ± 1 , ± 1 , ± 1) – even number of minus signs (2 7 = 128 vectors) Scale these to unit vectors to get X ⊂ S 7 . Among these vectors, there are only 4 non-zero angles. This gives us a 4-class cometric association scheme. William J. Martin The Ideal of E 1

Prelude Main Parameters Definitions from a Simple Example Main Results and Conjectures Some Theory The known examples The Ideal Dismantlability Orthogonality relations d d E j = 1 � � A i = P ji E j Q ij A i v j =0 i =0 The change-of-basis matrices P and Q are called the “first and second eigenmatrices” of the scheme. A scaled version of P is called the “character table”: PQ = vI MP = Q ⊤ K where M is a diagonal matrix of multiplicities m j = rank E j and K is a diagonal matrix of valencies v i = rowsum A i . William J. Martin The Ideal of E 1