Enumerating coherent configurations of small order Matan Ziv-Av - PowerPoint PPT Presentation

Enumerating coherent configurations of small order Matan Ziv-Av Ben Gurion University of the Negev MTAGT, Villanova, June 2-5 2014

Coherent configurations A partition of Ω 2 , M = R 1 , . . . , R r is a coherent configuration if CC1 ∀ i ∈ [1 , r ] ∃ i ′ ∈ [1 , r ] R − 1 = R i ′ i CC2 ∃ I ′ ⊆ [1 , r ] � R i = ∆ i ∈ I ′ CC3 ∀ i , j , k ∈ [1 , r ] ∀ ( x , y ) ∈ R k |{ z ∈ X | ( x , z ) ∈ R i ∧ ( z , y ) ∈ R j }| = p k ij If R i = ∆ then M is called an association scheme. The partition of ∆ defines a partition of Ω.

Structure of coherent configurations    AS 1 CB 1 CB 2 CB 3  � AS 1 � CB AS 1 CB ∗ AS 2 CB 4 CB 5         ∗ AS 2 ∗ ∗ AS 3 CB 6     ∗ AS 2 ∗ ∗ ∗ AS 4

� � � � � � � � � � �� � � � � � � � � � � � � � � � � � � � � � � � � Structure of coherent configurations    AS 1 CB 1 CB 2 CB 3  � AS 1 � CB AS 1 CB ∗ AS 2 CB 4 CB 5         ∗ AS 2 ∗ ∗ AS 3 CB 6     ∗ AS 2 ∗ ∗ ∗ AS 4 By CB I mean a color partition of the complete (directed) bipartite graph into biregular subgraphs: • • • • • • • • • • • • • • • • • • • • •

Similar enumerations Hanaki and Miyamoto enumerated all association schemes of order up to 30, as well as those of orders 32, 33, 34 and 38. http://math.shinshu-u.ac.jp/~hanaki/as/ All SRGs (rank 3 symmetric association schemes) on up t0 48 vertices are enumerated http: //www.win.tue.nl/~aeb/graphs/srg/srgtab1-50.html All S-rings (association scheme with a regular subgroup of automorphisms) of order up to 63 were enumerated by Pech & Reichard, Z. http://my.svgalib.org/s-rings/wschur.tar.gz

Using enumeration of AS to enumerate CCs   AS 1 CB       ∗ AS 2

Previous attempts Shiratsuchi (1997) and Nagatomo & Junichi (2009) enumerated CCs of order up to 13. “Most of the classification was done by hand, and we also use ‘Mathematica’ partially.”

Previous attempts Shiratsuchi (1997) and Nagatomo & Junichi (2009) enumerated CCs of order up to 13. “Most of the classification was done by hand, and we also use ‘Mathematica’ partially.” Sven Riechard (2012) enumerated CCs of order up to 13.

Previous attempts Shiratsuchi (1997) and Nagatomo & Junichi (2009) enumerated CCs of order up to 13. “Most of the classification was done by hand, and we also use ‘Mathematica’ partially.” Sven Riechard (2012) enumerated CCs of order up to 13. The results don’t agree.

Results Size 1 2 3 4 5 6 7 8 CCs 1 2 4 10 15 38 57 143 Scuhurian 1 2 4 10 15 38 57 143 Size 9 10 11 12 13 CCs 228 492 769 1845 2806 Scuhurian 228 492 769 1845 2806

Non-schurian CCs The smallest non-Schurian strongly regular graph is Shrikhande graph of order 16.

Non-schurian CCs The smallest non-Schurian strongly regular graph is Shrikhande graph of order 16. The smallest non-Schurian association scheme is a tournament of order 15.

Non-schurian CCs The smallest non-Schurian strongly regular graph is Shrikhande graph of order 16. The smallest non-Schurian association scheme is a tournament of order 15. The smallest known non-Schurian coherent configuration is of order 15 (the same association scheme).

Non-schurian CCs The smallest non-Schurian strongly regular graph is Shrikhande graph of order 16. The smallest non-Schurian association scheme is a tournament of order 15. The smallest known non-Schurian coherent configuration is of order 15 (the same association scheme). Natural questions arise: Is there a non-Schurian coherent configuration of smaller order? Are there other non-Schurian coherent configurations of order 15?

Confidence in results Size 1 2 3 4 5 6 7 8 CCs 1 2 4 10 15 38 57 143 Scuhurian 1 2 4 10 15 38 57 143 Size 9 10 11 12 13 CCs 228 492 769 1845 2806 Scuhurian 228 492 769 1845 2806

Fully confident I Size 1 2 3 4 5 6 7 8 CCs 1 2 4 10 15 38 57 143 Scuhurian 1 2 4 10 15 38 57 143 Size 9 10 11 12 13 CCs 228 492 769 1845 2806 Scuhurian 228 492 769 1845 2806 These results can be achieved with a GAP “one-liner”: n:=8;;Sn:=SymmetricGroup(n);; Size(Filtered( List(ConjugacyClassesSubgroups(Sn),Representative), x->x=Intersection( List(Orbits(x,Tuples([1..n],2),OnTuples), y->Stabilizer(Sn,Set(y),OnSetsTuples))) ));

Fully confident II Size 1 2 3 4 5 6 7 8 CCs 1 2 4 10 15 38 57 143 Scuhurian 1 2 4 10 15 38 57 143 Size 9 10 11 12 13 CCs 228 492 769 1845 2806 Scuhurian 228 492 769 1845 2806 All three efforts agree on these results. Can be easily calculated by calculating all mergings of trivial (rank n 2 ) configuration.

A little bit confident Size 1 2 3 4 5 6 7 8 CCs 1 2 4 10 15 38 57 143 Scuhurian 1 2 4 10 15 38 57 143 Size 9 10 11 12 13 CCs 228 492 769 1845 2806 Scuhurian 228 492 769 1845 2806 Each effort got different results. Others provide only “intereseting” data, so it is hard to say if the disagreement is in intereseting or in trivial results.

A little bit confident Size 1 2 3 4 5 6 7 8 CCs 1 2 4 10 15 38 57 143 Scuhurian 1 2 4 10 15 38 57 143 Size 9 10 11 12 13 CCs 228 492 769 1845 2806 Scuhurian 228 492 769 1845 2806 Each effort got different results. Others provide only “intereseting” data, so it is hard to say if the disagreement is in intereseting or in trivial results. Third time lucky?

Thank you for your attention • • • • • • • • • •