p-Ranks of quasi-symmetric designs and standard modules of ... · A combinatorial design is said to be quasi-symmetric if there are integers aand b(a>b) such that two blocks are incident

p-Ranks of quasi-symmetric designs and standardmodules 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

1 Motivation and definition

2 Adjacency algebras

3 Standard modules

4 Characteristic 3, 5, and 7 for 2-(15, 3, 1)-designs


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 designswith same parameters are not constant, in general.

We want to know what is p-ranks from a view point of representationtheory.

For 80 nonisomorphic 2-(15, 3, 1)-designs, the 2-ranks of incidencematrices are 11, 12, 13, 14, and 15.

We will focus on the 2-(15, 3, 1)-designs and p = 2.



A combinatorial design is said to be quasi-symmetric if there areintegers a and b (a > b) such that two blocks are incident with eithera or b points.

For example, 2-(v, `, 1)-designs are quasi-symmetric for a = 1 andb = 0.

By a quasi-symmetric design, we can construct a coherentconfiguration of type (2, 2; 3).

Let (P,B) be a quasi-symmetric design, where P is the set of pointsand B is the set of blocks.

For b, b′ ∈ B, b 6= 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.



Now we can define a coherent configuration (X,S) of type (2, 2; 3).

complete graph quasi-symmetric designrelations : s1, s3 relations : s6, s7

tquasi-symmetric design strongly regular graphrelations : s8, s9 relations : s2, s4, s5

Put X = X1 ∪X2, X1 = P , and X2 = B.

The configuration has two fibers X1 and X2.

Put S11 = {s1, s3}, S12 = {s6, s7}, S21 = {s8, s9}, andS22 = {s2, s4, s5}.We denote by σi the adjacency matrix of si.

Then FS =⊕9

i=1 Fσi ⊂ MatX(F ) is the adjacency algebra of(X,S) over a field F .

We will consider representations of FS.



The parameters of a 2-(v, `, 1)-design and strongly regular graphdefined by the design are :

r =v − 1

`− 1,

b =v(v − 1)

`(`− 1),

n = b,

k = `

(v − 1

`− 1− 1

),

a =v − 1

`− 1− 2 + (`− 1)2,

c = `2.



We can compute the table of multiplications :

σ1 σ3 σ6 σ7

σ1 σ1 σ3 σ6 σ7σ3 σ3 (v − 1)σ1 (`− 1)σ6 (v − `)σ6

+(v − 2)σ3 +`σ7 +(v − `− 1)σ7

σ8 σ8 (`− 1)σ8 `σ2+`σ9 +σ4 (`− 1)σ4

+`σ5σ9 (v − `)σ8 (v − `)σ2

σ9 +(v − `− 1)σ9 (`− 1)σ4 +(v − 2`+ 1)σ4+`σ5 +(v − 2`)σ5



σ2 σ4 σ5 σ8 σ9

σ2 σ2 σ4 σ5 σ8 σ9σ4 kσ2 (r − 1)σ8 (k − r + 1)σ8

σ4 +aσ4 (k − a− 1)σ4 +`σ9 +(k − `)σ9+`2σ5 +(k − `2)σ5

σ5 (b− k − 1)σ2 (b− k − 1)σ8(k − a− 1)σ4 +(b + a− 2k)σ4 (r − `)σ9 +(b− r − k + `− 1)σ9

σ5 +(k − `2)σ5 +(b− 2k − 2 + `2)σ5

σ6 σ6 (r − 1)σ6 rσ1+`σ7 (r − `)σ7 +σ3 (r − 1)σ3

σ7 (k − r + 1)σ6 (b− k − 1)σ6 (b− r)σ1σ7 +(k − `)σ7 +(b− r − k + `− 1)σ7 (r − 1)σ3 +(b− 2r + 1)σ3

We remark that the coefficients are polynomial of v, `, k, a, r, and b.

Lemma 1.1

If ` and r = (v − 1)/(`− 1) are odd, then v, a, and b are odd and k iseven.



Theorem 1.2

Let F be a field of characteristic 2. Let A be the adjacency algebra of acoherent configuration defined by a 2-(15, 3, 1)-design over F . Supposethat ` and r = (v − 1)/(`− 1) are odd. Then the adjacency algebra of acoherent configuration defined by a 2-(v, `, 1)-design over F is isomorphicto A.

Let FX be the standard module of (X,S). Namely, FX is a rightFS-module defined by a natural action of FS ⊂ MatX(F ).

We will determine the structure of adjacency algebras and standardmodules of coherent configuration defined by 2-(v, `, 1)-designs suchthat ` and r are odd.

Also we will consider what is 2-ranks of the designs.


Adjacency algebras

Let F be a field of characteristic 2.

Let (X,S) be a coherent configuration defined by a 2-(v, `, 1)-design.

Suppose that ` and r = (v − 1)/(`− 1) are odd.

The modular character table of FS is :

s1 s3 s2 s4 s5 multiplicity

A 1 0 1 0 0 1B 1 1 0 0 0 v − 1C 0 0 1 1 0 b− 1

We will see that A, B, and C are simple FS-modules. Remark thatdimA = 2 and dimB = dimC = 1.

The multiplicity is the cardinality of the simple module in FX assimple components.

Note that A is in a simple block B1 ∼= Mat2(F ) and B and C are inthe same block B2.


Adjacency algebras

Let Q be the following quiver

x•α //

•yβ

oo

and consider the quiver algebra FQ.

Theorem 2.1

Let F be a field of characteristic 2. Let (X,S) be a coherent configurationdefined by a 2-(v, `, 1)-design. Suppose that ` and r = (v − 1)/(`− 1) areodd. Then

FS ∼= Mat2(F )⊕ FQ/(αβ).

The projective covers of simple modules are :

P (A) = [A], P (B) =

[BC

], P (C) =

CBC

.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 coherentconfiguration defined by a 2-(v, `, 1)-design with odd ` and r.

SoFS ∼= Mat2(F )⊕ FQ/(αβ).

Easily, we can see that the algebra has finite representation type.Namely there are finitely many isomorphism classes ofindecomposable FS-modules.

[A] ,[C],

[CB

],

CBC

, [B],

[BC

].


Standard modules

Since A has multiplicity one, we can write

FX ∼= [A]⊕ g1 [C]⊕ g2[CB

]⊕ g3

CBC

⊕ h1 [ B ]⊕ h2

[BC

].

By the multiplicities, we have

g1 + g2 + 2g3 + h2 = b− 1,

g2 + g3 + h1 + h2 = v − 1.

Since the standard module is self-contragredient, we have

g2 = h2.


Standard modules

Since β in the quiver goes to σ7 and βα goes to σ5, we puts = rank(σ7) and t = rank(σ5).

We can see that rank(σ7) = g2 + g3 and rank(σ5) = g3.

We have

(g1, g2, g3, h1, h2) = (b− 2s− 1, s− t, t, v − 2s+ t− 1, s− t).

So the parameters s and t determine the structure of standardmodule FX.

Remark that the usual 2-rank of the design is rank(σ6) and

rank(σ6) = 1 + rank(σ7) = 1 + s.


Standard modules

Theorem 3.1

Let F be a field of characteristic 2. Let (X,S) be a coherent configurationdefined by a 2-(v, `, 1)-design. Suppose that ` and r = (v − 1)/(`− 1) areodd. Put s = rank(σ7) and t = rank(σ5). Then

FX ∼= [A]⊕ g1 [C]⊕ g2[CB

]⊕ g3

CBC

⊕ h1 [ B ]⊕ h2

[BC

],

where (g1, g2, g3, h1, h2) = (b− 2s− 1, s− t, t, v − 2s+ t− 1, s− t).


Standard modules

Example 3.2

For 80 nonisomorphic 2-(15, 3, 1)-designs, we have following parameters(by computation) :

] rank(σ7) rank(σ5) g1 g2 g3 h1 h2 rank(σ6)

1 10 6 14 4 6 0 4 111 11 8 12 3 8 0 3 125 12 10 10 2 10 0 2 1315 13 12 8 1 12 0 1 1458 14 14 6 0 14 0 0 15

I do not know why h1 = 0. If this is true in general, then thestructure is determined only by the 2-rank of the design.


Standard modules

Remark 3.3

All 80 strongly-regular graphs obtained by 2-(15, 3, 1)-designs arenonisomorphic to each other. The structures of standard modules of thegraphs are

(FX2)FS22∼= [A]⊕ (g1 + g2 + h2) [C]⊕ g3

[CC

].

This is just obtained by

FX ∼= [A]⊕ g1 [C]⊕ g2[CB

]⊕ g3

CBC

⊕ h1 [ B ]⊕ h2

[BC

].


Standard modules

Example 3.4

In [The CRC Handbook of Combinatorial Designs], we can find a list ofdesigns with odd ` and r :

No. v b r ` ]

14 15 35 7 3 80

29 19 57 9 3 ≥ 1.1× 109

57 45 99 11 5 ≥ 16

86 27 117 13 3 ≥ 1011

114 31 155 15 3 ≥ 6× 1016

120 61 183 15 5 ≥ 10

129 91 195 15 7 ≥ 2

I want to compute p-ranks of them. But I do not have data.


Characteristic 3, 5, and 7 for 2-(15, 3, 1)-designs

Theorem 4.1

Let F be a field of characteristic 3. There are three simple FS-modules A,B, and C with dimF A = dimF B = dimF C = 1. The Loewy series ofthe projective covers of simple FS-modules are

P (A) =

AB CA

, P (B) =

[BA

], P (C) =

CAC

and the structures of standard FS-modules are

FX ∼=

AB CA

⊕ 13

CAC

⊕ 7 [C]

for all 80 designs. (By computation)



Theorem 4.2

Let F be a field of characteristic 5. There are two simple FS-modules Aand B with dimF A = 2 and dimF B = 1. The Loewy series of theprojective covers of simple FS-modules are

P (A) =

[AA

], P (B) = [B]

and the structure of the standard FS-module is

FX ∼=[AA

]⊕ 13 [A]⊕ 20 [B] .



Theorem 4.3

Let F be a field of characteristic 7. There are three simple FS-modules A,B, and C with dimF A = dimF B = 1 and dimF C = 2. The Loewy seriesof the projective covers of simple FS-modules are

P (A) =

[AB

], P (B) =

BAB

, P (C) = [C]

and the structure of the standard FS-module is

FX ∼=

BAB

⊕ 19 [B]⊕ 14 [C] .


Thank you very much !


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p-Ranks of quasi-symmetric designs and standard modules of ... · A combinatorial design is said to be quasi-symmetric if there are integers aand b(a>b) such that two blocks are incident