Bill Martin Worcester Polytechnic Institute USA Geometric and
Algebraic Combinatorics 4, Oisterwijk, Thursday 21 August 2008
Slide 2 Jason Williford Misha Muzychuk Edwin van Dam Nick LeCompte
(WPI student) Will Owens (WPI student) ... and Ive received
valuable suggestions from many others. Slide 3 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 Slide 4 To make the next 45
minutes as pleasant as possible Slide 5 To make the next 45 minutes
as pleasant as possible (for both you and me) Slide 6 To not look
too dumb Slide 7 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 Slide 8 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 Slide 9 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 Slide 10 E 8 Root Lattice Slide 11 Slide 12
Slide 13 Slide 14 Slide 15 Slide 16 Inner product of two zonal
polynomials only depends on distance between the two base points
and the single-variable polynomials. Slide 17 Slide 18 Slide 19
Slide 20 Slide 21 Slide 22 Delsarte (1973): Slide 23 Concerning
cometric association schemes... Slide 24 I dont know The model I
just showed you is my favorite definition so far Slide 25 Slide 26
Terwilliger (1987): Slide 27 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 Slide 28 Slide 29 Slide 30 Slide 31 Slide 32 Slide
33 A familiar dual pair of association schemes w=3 fibres of size
r=2w=2 fibres of size r=3 Slide 34 Another dual pair of complete
multipartite schemes Slide 35 H. Suzuki (1998): Slide 36 Slide 37
Slide 38 Slide 39 Edwin van Dam (1995) Slide 40 Slide 41 Slide 42
Slide 43 Slide 44 Slide 45 This is a 4-class Q-antipodal
association scheme Slide 46 Slide 47 Slide 48 Slide 49 Slide 50 A
Construction of Wocjan and Beth (2005) Slide 51 Slide 52 Slide 53
Slide 54 Slide 55 vertices, split into three classes of size Graph
represents incidence, yielding a square -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 (
and resp.) Slide 56 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 Slide 57 Check time available Slide 58
Slide 59 vectors in R , all of squared length only possible inner
products: construct one graph for each inner product we obtain a
-class cometric scheme which is Q-bipartite Krein array: Slide 60
Heather Lewis and ?
