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Cometric Association Schemes

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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Cometric Association Schemes

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  1. Bill Martin Worcester Polytechnic Institute USA Cometric Association Schemes Geometric and Algebraic Combinatorics 4, Oisterwijk, Thursday 21 August 2008

  2. 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.

  3. 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

  4. My Real Goals • To make the next 45 minutes as pleasant as possible

  5. My Real Goals • To make the next 45 minutes as pleasant as possible (for both you and me)

  6. My Real Goals • To make the next 45 minutes as pleasant as possible (for both you and me) • To not look too dumb

  7. 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

  8. 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

  9. 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

  10. First, an Example E8 Root Lattice

  11. A Spherical 7-Design in R8

  12. A Curious Property

  13. Krein Parameters in Disguise

  14. The Polytope Definition

  15. The Polytope Definition

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

  17. Another Example

  18. The Usual Approach

  19. The Bose-Mesner Algebra

  20. Orthogonality Relations

  21. A Taste of Duality

  22. Polynomial Schemes Delsarte (1973):

  23. Some Natural Questions Concerning cometric association schemes . . .

  24. What do they look like? • I don’t know • The model I just showed you is my favorite definition so far

  25. Balanced Set Condition

  26. Balanced Set Condition Terwilliger (1987):

  27. 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

  28. How Many?

  29. Are They Mostly P-Polynomial?

  30. Cometric Schemes with Large d

  31. Cometric Schemes with Large d

  32. What do the Imprimitive Cometric Schemes Look Like?

  33. Duality and Imprimitivity w=3 fibres of size r=2 w=2 fibres of size r=3 A familiar dual pair of association schemes

  34. Duality and Imprimitivity Another dual pair of complete multipartite schemes

  35. Suzuki’s Theorem H. Suzuki (1998):

  36. Q-Bipartite Structure

  37. Diam. 3 Distance-Regular Graphs

  38. 3-Class Cometric Schemes

  39. 3-Class Cometric Schemes Edwin van Dam (1995)

  40. Diam. 4 Distance-Regular Graphs

  41. 4-Class Cometric Schemes

  42. Don Higman’sTriality Scheme

  43. Donald Higman (1928-2006)

  44. More 4-Class Q-Antipodal Schemes

  45. Hyperovals in PG(2,4) This is a 4-class Q-antipodal association scheme

  46. Schemes from Unbiased Bases

  47. Real MUBs

  48. Real MUBs

  49. A Surprising Result

  50. Four-Class Schemes from MOLS A Construction of Wocjan and Beth (2005)

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