p -Ranks of quasi-symmetric designs and standard modules of coherent - PowerPoint PPT Presentation

p -Ranks of quasi-symmetric designs and standard modules of coherent configurations Akihide Hanaki Shinshu University June 2, 2014, Villanova University. A. Hanaki (Shinshu Univ.) p -Ranks of quasi-symmetric designs June 2, 2014 1 / 22

Motivation and definition 1 Adjacency algebras 2 Standard modules 3 Characteristic 3 , 5 , and 7 for 2 - (15 , 3 , 1) -designs 4 A. Hanaki (Shinshu Univ.) p -Ranks of quasi-symmetric designs June 2, 2014 2 / 22

Motivation and definition Let C be an incidence matrix of a combinatorial design. The p -ranks, the ranks of matrices in characteristic p > 0 , of designs with same parameters are not constant, in general. We want to know what is p -ranks from a view point of representation theory. For 80 nonisomorphic 2 - (15 , 3 , 1) -designs, the 2 -ranks of incidence matrices are 11 , 12 , 13 , 14 , and 15 . We will focus on the 2 - (15 , 3 , 1) -designs and p = 2 . A. Hanaki (Shinshu Univ.) p -Ranks of quasi-symmetric designs June 2, 2014 3 / 22

Motivation and definition A combinatorial design is said to be quasi-symmetric if there are integers a and b ( a > b ) such that two blocks are incident with either a or b points. For example, 2 - ( v, ℓ, 1) -designs are quasi-symmetric for a = 1 and b = 0 . By a quasi-symmetric design, we can construct a coherent configuration of type (2 , 2; 3) . Let ( P, B ) be a quasi-symmetric design, where P is the set of points and B is the set of blocks. For b, b ′ ∈ B , b � = b ′ , we can see that | b ∩ b ′ | = a or b . We can define a graph with point set B and b is adjacent to b ′ iff | b ∩ b ′ | = a . Then the graph is strongly regular. A. Hanaki (Shinshu Univ.) p -Ranks of quasi-symmetric designs June 2, 2014 4 / 22

Motivation and definition Now we can define a coherent configuration ( X, S ) of type (2 , 2; 3) . complete graph quasi-symmetric design relations : s 1 , s 3 relations : s 6 , s 7 t quasi-symmetric design strongly regular graph relations : s 8 , s 9 relations : s 2 , s 4 , s 5 Put X = X 1 ∪ X 2 , X 1 = P , and X 2 = B . The configuration has two fibers X 1 and X 2 . Put S 11 = { s 1 , s 3 } , S 12 = { s 6 , s 7 } , S 21 = { s 8 , s 9 } , and S 22 = { s 2 , s 4 , s 5 } . We denote by σ i the adjacency matrix of s i . Then FS = � 9 i =1 Fσ i ⊂ Mat X ( F ) is the adjacency algebra of ( X, S ) over a field F . We will consider representations of FS . A. Hanaki (Shinshu Univ.) p -Ranks of quasi-symmetric designs June 2, 2014 5 / 22

Motivation and definition The parameters of a 2 - ( v, ℓ, 1) -design and strongly regular graph defined by the design are : v − 1 r = ℓ − 1 , v ( v − 1) b = ℓ ( ℓ − 1) , n = b, � v − 1 � k = ℓ ℓ − 1 − 1 , v − 1 ℓ − 1 − 2 + ( ℓ − 1) 2 , a = ℓ 2 . c = A. Hanaki (Shinshu Univ.) p -Ranks of quasi-symmetric designs June 2, 2014 6 / 22

Motivation and definition We can compute the table of multiplications : σ 1 σ 3 σ 6 σ 7 σ 1 σ 1 σ 3 σ 6 σ 7 σ 3 σ 3 ( v − 1) σ 1 ( ℓ − 1) σ 6 ( v − ℓ ) σ 6 +( v − 2) σ 3 + ℓσ 7 +( v − ℓ − 1) σ 7 σ 8 σ 8 ( ℓ − 1) σ 8 ℓσ 2 + ℓσ 9 + σ 4 ( ℓ − 1) σ 4 + ℓσ 5 σ 9 ( v − ℓ ) σ 8 ( v − ℓ ) σ 2 σ 9 +( v − ℓ − 1) σ 9 ( ℓ − 1) σ 4 +( v − 2 ℓ + 1) σ 4 + ℓσ 5 +( v − 2 ℓ ) σ 5 A. Hanaki (Shinshu Univ.) p -Ranks of quasi-symmetric designs June 2, 2014 7 / 22

Motivation and definition σ 2 σ 4 σ 5 σ 8 σ 9 σ 2 σ 2 σ 4 σ 5 σ 8 σ 9 σ 4 kσ 2 ( r − 1) σ 8 ( k − r + 1) σ 8 σ 4 + aσ 4 ( k − a − 1) σ 4 + ℓσ 9 +( k − ℓ ) σ 9 + ℓ 2 σ 5 +( k − ℓ 2 ) σ 5 σ 5 ( b − k − 1) σ 2 ( b − k − 1) σ 8 ( k − a − 1) σ 4 +( b + a − 2 k ) σ 4 ( r − ℓ ) σ 9 +( b − r − k + ℓ − 1) σ 9 +( k − ℓ 2 ) σ 5 +( b − 2 k − 2 + ℓ 2 ) σ 5 σ 5 σ 6 σ 6 ( r − 1) σ 6 rσ 1 + ℓσ 7 ( r − ℓ ) σ 7 + σ 3 ( r − 1) σ 3 ( k − r + 1) σ 6 ( b − k − 1) σ 6 ( b − r ) σ 1 σ 7 +( k − ℓ ) σ 7 +( b − r − k + ℓ − 1) σ 7 ( r − 1) σ 3 +( b − 2 r + 1) σ 3 σ 7 We remark that the coefficients are polynomial of v , ℓ , k , a , r , and b . Lemma 1.1 If ℓ and r = ( v − 1) / ( ℓ − 1) are odd, then v , a , and b are odd and k is even. A. Hanaki (Shinshu Univ.) p -Ranks of quasi-symmetric designs June 2, 2014 8 / 22

Motivation and definition Theorem 1.2 Let F be a field of characteristic 2 . Let A be the adjacency algebra of a coherent configuration defined by a 2 - (15 , 3 , 1) -design over F . Suppose that ℓ and r = ( v − 1) / ( ℓ − 1) are odd. Then the adjacency algebra of a coherent configuration defined by a 2 - ( v, ℓ, 1) -design over F is isomorphic to A . Let FX be the standard module of ( X, S ) . Namely, FX is a right FS -module defined by a natural action of FS ⊂ Mat X ( F ) . We will determine the structure of adjacency algebras and standard modules of coherent configuration defined by 2 - ( v, ℓ, 1) -designs such that ℓ and r are odd. Also we will consider what is 2 -ranks of the designs. A. Hanaki (Shinshu Univ.) p -Ranks of quasi-symmetric designs June 2, 2014 9 / 22

Adjacency algebras Let F be a field of characteristic 2 . Let ( X, S ) be a coherent configuration defined by a 2 - ( v, ℓ, 1) -design. Suppose that ℓ and r = ( v − 1) / ( ℓ − 1) are odd. The modular character table of FS is : s 1 s 3 s 2 s 4 s 5 multiplicity A 1 0 1 0 0 1 B 1 1 0 0 0 v − 1 C 0 0 1 1 0 b − 1 We will see that A , B , and C are simple FS -modules. Remark that dim A = 2 and dim B = dim C = 1 . The multiplicity is the cardinality of the simple module in FX as simple components. Note that A is in a simple block B 1 ∼ = Mat 2 ( F ) and B and C are in the same block B 2 . A. Hanaki (Shinshu Univ.) p -Ranks of quasi-symmetric designs June 2, 2014 10 / 22

� Adjacency algebras Let Q be the following quiver α � • y x • β and consider the quiver algebra FQ . Theorem 2.1 Let F be a field of characteristic 2 . Let ( X, S ) be a coherent configuration defined by a 2 - ( v, ℓ, 1) -design. Suppose that ℓ and r = ( v − 1) / ( ℓ − 1) are odd. Then FS ∼ = Mat 2 ( F ) ⊕ FQ/ ( αβ ) . The projective covers of simple modules are : � B   C �  . P ( A ) = [ A ] , P ( B ) = , P ( C ) = B  C C A. Hanaki (Shinshu Univ.) p -Ranks of quasi-symmetric designs June 2, 2014 11 / 22

Standard modules Again, let F be a field of characteristic 2 and let ( X, S ) be a coherent configuration defined by a 2 - ( v, ℓ, 1) -design with odd ℓ and r . So FS ∼ = Mat 2 ( F ) ⊕ FQ/ ( αβ ) . Easily, we can see that the algebra has finite representation type. Namely there are finitely many isomorphism classes of indecomposable FS -modules. � C � B   C � �  , � � � � [ A ] , C , , B B , .  B C C A. Hanaki (Shinshu Univ.) p -Ranks of quasi-symmetric designs June 2, 2014 12 / 22

Standard modules Since A has multiplicity one, we can write � C � B   C � � FX ∼  ⊕ h 1 � � = [ A ] ⊕ g 1 [ C ] ⊕ g 2 ⊕ g 3 B B ⊕ h 2 .  B C C By the multiplicities, we have g 1 + g 2 + 2 g 3 + h 2 = b − 1 , g 2 + g 3 + h 1 + h 2 = v − 1 . Since the standard module is self-contragredient , we have g 2 = h 2 . A. Hanaki (Shinshu Univ.) p -Ranks of quasi-symmetric designs June 2, 2014 13 / 22

Standard modules Since β in the quiver goes to σ 7 and βα goes to σ 5 , we put s = rank( σ 7 ) and t = rank( σ 5 ) . We can see that rank( σ 7 ) = g 2 + g 3 and rank( σ 5 ) = g 3 . We have ( g 1 , g 2 , g 3 , h 1 , h 2 ) = ( b − 2 s − 1 , s − t, t, v − 2 s + t − 1 , s − t ) . So the parameters s and t determine the structure of standard module FX . Remark that the usual 2 -rank of the design is rank( σ 6 ) and rank( σ 6 ) = 1 + rank( σ 7 ) = 1 + s. A. Hanaki (Shinshu Univ.) p -Ranks of quasi-symmetric designs June 2, 2014 14 / 22

Standard modules Theorem 3.1 Let F be a field of characteristic 2 . Let ( X, S ) be a coherent configuration defined by a 2 - ( v, ℓ, 1) -design. Suppose that ℓ and r = ( v − 1) / ( ℓ − 1) are odd. Put s = rank( σ 7 ) and t = rank( σ 5 ) . Then � C � B   C � � FX ∼  ⊕ h 1 � � = [ A ] ⊕ g 1 [ C ] ⊕ g 2 ⊕ g 3 B B ⊕ h 2 ,  B C C where ( g 1 , g 2 , g 3 , h 1 , h 2 ) = ( b − 2 s − 1 , s − t, t, v − 2 s + t − 1 , s − t ) . A. Hanaki (Shinshu Univ.) p -Ranks of quasi-symmetric designs June 2, 2014 15 / 22

Standard modules Example 3.2 For 80 nonisomorphic 2 - (15 , 3 , 1) -designs, we have following parameters (by computation) : ♯ rank( σ 7 ) rank( σ 5 ) g 1 g 2 g 3 h 1 h 2 rank( σ 6 ) 1 10 6 14 4 6 0 4 11 1 11 8 12 3 8 0 3 12 5 12 10 10 2 10 0 2 13 15 13 12 8 1 12 0 1 14 58 14 14 6 0 14 0 0 15 I do not know why h 1 = 0 . If this is true in general, then the structure is determined only by the 2 -rank of the design. A. Hanaki (Shinshu Univ.) p -Ranks of quasi-symmetric designs June 2, 2014 16 / 22

Standard modules Remark 3.3 All 80 strongly-regular graphs obtained by 2 - (15 , 3 , 1) -designs are nonisomorphic to each other. The structures of standard modules of the graphs are � C � ( FX 2 ) FS 22 ∼ = [ A ] ⊕ ( g 1 + g 2 + h 2 ) [ C ] ⊕ g 3 . C This is just obtained by � C � B   C � � FX ∼  ⊕ h 1 � � = [ A ] ⊕ g 1 [ C ] ⊕ g 2 ⊕ g 3 B B ⊕ h 2 .  B C C A. Hanaki (Shinshu Univ.) p -Ranks of quasi-symmetric designs June 2, 2014 17 / 22