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Bill Martin Worcester Polytechnic Institute USA. Cometric Association Schemes. Geometric and Algebraic Combinatorics 4, Oisterwijk, Thursday 21 August 2008. Several Collaborators. Jason Williford Misha Muzychuk Edwin van Dam Nick LeCompte (WPI student) Will Owens (WPI student)

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Bill martin worcester polytechnic institute usa

Bill Martin

Worcester Polytechnic Institute

USA

Cometric Association Schemes

Geometric and Algebraic Combinatorics 4, Oisterwijk, Thursday 21 August 2008


Several collaborators
Several Collaborators

  • Jason Williford

  • Misha Muzychuk

  • Edwin van Dam

  • Nick LeCompte (WPI student)

  • Will Owens (WPI student)

  • . . . and I’ve received valuable suggestions from many others.


Today s goals
Today’s Goals

  • Survey the known examples

  • Summarize the main results to date

  • Explore the structure of imprimitive

    Q-polynomial schemes, especially with

    3 or 4 classes

  • List some open problems, big and small


My real goals
My Real Goals

  • To make the next 45 minutes as pleasant as possible


My real goals1
My Real Goals

  • To make the next 45 minutes as pleasant as possible (for both you and me)


My real goals2
My Real Goals

  • To make the next 45 minutes as pleasant as possible (for both you and me)

  • To not look too dumb


My real goals3
My Real Goals

  • To make the next 45 minutes as pleasant as possible (for both you and me)

  • To not look too dumb

  • To get some smart people to work on these interesting problems


My real goals4
My Real Goals

  • To make the next 45 minutes as pleasant as possible (for both you and me)

  • To not look too dumb

  • To get some smart people to work on these interesting problems

  • To tell you as much as I reasonably can about the subject


My real goals5
My Real Goals

  • To make the next 45 minutes as pleasant as possible (for both you and me)

  • To not look too dumb

  • To get some smart people to work on these interesting problems

  • To tell you as much as I reasonably can about the subject

  • To avoid typesetting math in PowerPoint


First an example
First, an Example

E8 Root Lattice




Krein parameters in disguise
Krein Parameters in Disguise


The polytope definition
The Polytope Definition


The polytope definition1
The Polytope Definition


The polytope definition2
The Polytope Definition

Inner product of two zonal polynomials only depends on distance between the two base points and the single-variable polynomials.




The bose mesner algebra
The Bose-Mesner Algebra


Orthogonality relations
Orthogonality Relations



Polynomial schemes
Polynomial Schemes

Delsarte (1973):


Some natural questions
Some Natural Questions

Concerning cometric association schemes . . .


What do they look like
What do they look like?

  • I don’t know

  • The model I just showed you is my favorite definition so far



Balanced set condition1
Balanced Set Condition

Terwilliger (1987):


Sources of examples
Sources of Examples

  • Q-polynomial distance-regular graphs (e.g., all those with classical parameters)

  • Spherical designs / lattices

  • Extremal codes and block designs

  • Real mutually unbiased bases

  • Sporadic groups (e.g., triality)

  • linked systems of designs and geometries






What do the imprimitive cometric schemes look like
What do the Imprimitive Cometric Schemes Look Like?


Duality and imprimitivity
Duality and Imprimitivity

w=3 fibres of size r=2

w=2 fibres of size r=3

A familiar dual pair of association schemes


Duality and imprimitivity1
Duality and Imprimitivity

Another dual pair of complete multipartite schemes


Suzuki s theorem
Suzuki’s Theorem

H. Suzuki (1998):





3 class cometric schemes1
3-Class Cometric Schemes

Edwin van Dam (1995)




Don higman s triality scheme
Don Higman’sTriality Scheme


Donald higman 1928 2006
Donald Higman (1928-2006)



Hyperovals in pg 2 4
Hyperovals in PG(2,4)

This is a 4-class Q-antipodal association scheme



Real mubs
Real MUBs


Real mubs1
Real MUBs



Four class schemes from mols
Four-Class Schemes from MOLS

A Construction of Wocjan and Beth (2005)


Four class schemes from mols1
Four-Class Schemes from MOLS

A Construction of Wocjan and Beth (2005)





Linked system of symmetric designs
Linked System of Symmetric Designs

  • 48 vertices, split into three classes of size 16

  • Graph G1represents “incidence”, yielding a

    square (16,6,2)-design between any two

    Q-antipodal classes

  • “linked”: the number of common neighbors in the third class of a point chosen from Class One and a point chosen from Class Two depends on only whether or not these are incident (1 and 3, resp.)


Mubs from cameron seidel scheme
MUBs from Cameron-Seidel Scheme

  • Muzychuk, Williford, WJM introduced the extended Q-bipartite double

  • Applied to the subschemes of the Cameron-Seidel scheme, these are 4-class Q-bipartite, Q-antipodal schemes

  • So we have the same schemes that Bannai and Bannai found from mutually unbiased bases


More material in beamer format
More Material in Beamer Format

Check time available



Shortest vectors in leech lattice
Shortest Vectors in Leech Lattice

  • 196560 vectors in R24, all of squared length 8

  • only 7 possible inner products: ±8, ±4, ±2, 0

  • construct one graph for each inner product

  • we obtain a 7-class cometric scheme which is Q-bipartite

  • Krein array:

  • {24, 23, 288/13, 150/7, 104/5, 81/4;  

  • 1, 24/13, 18/7, 16/5, 15/4, 24 }


One more photo
One More Photo

Heather Lewis and …?