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
Akihide Hanaki
Shinshu University
June 2, 2014, Villanova University.
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
p-Ranks of quasi-symmetric designs June 2, 2014 2 / 22
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.
p-Ranks of quasi-symmetric designs June 2, 2014 3 / 22
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.
p-Ranks of quasi-symmetric designs June 2, 2014 4 / 22
Now we can define a coherent configuration (X, S) of type (2, 2; 3). complete graph quasi-symmetric design relations : s1, s3 relations : s6, s7
tquasi-symmetric design
strongly regular graph relations : 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}, and S22 = {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.
p-Ranks of quasi-symmetric designs June 2, 2014 5 / 22
The parameters of a 2-(v, ℓ, 1)-design and strongly regular graph defined by the design are : r = v − 1 ℓ − 1 , b = v(v − 1) ℓ(ℓ − 1) , n = b, k = ℓ v − 1 ℓ − 1 − 1
a = v − 1 ℓ − 1 − 2 + (ℓ − 1)2, c = ℓ2.
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We can compute the table of multiplications : σ1 σ3 σ6 σ7 σ1 σ1 σ3 σ6 σ7 σ3 σ3 (v − 1)σ1 (ℓ − 1)σ6 (v − ℓ)σ6 +(v − 2)σ3 +ℓσ7 +(v − ℓ − 1)σ7 σ8 σ8 (ℓ − 1)σ8 ℓσ2 +ℓσ9 +σ4 (ℓ − 1)σ4 +ℓσ5 σ9 (v − ℓ)σ8 (v − ℓ)σ2 σ9 +(v − ℓ − 1)σ9 (ℓ − 1)σ4 +(v − 2ℓ + 1)σ4 +ℓσ5 +(v − 2ℓ)σ5
p-Ranks of quasi-symmetric designs June 2, 2014 7 / 22
σ2 σ4 σ5 σ8 σ9 σ2 σ2 σ4 σ5 σ8 σ9 σ4 kσ2 (r − 1)σ8 (k − r + 1)σ8 σ4 +aσ4 (k − a − 1)σ4 +ℓσ9 +(k − ℓ)σ9 +ℓ2σ5 +(k − ℓ2)σ5 σ5 (b − k − 1)σ2 (b − k − 1)σ8 (k − a − 1)σ4 +(b + a − 2k)σ4 (r − ℓ)σ9 +(b − r − k + ℓ − 1)σ9 σ5 +(k − ℓ2)σ5 +(b − 2k − 2 + ℓ2)σ5 σ6 σ6 (r − 1)σ6 rσ1 +ℓσ7 (r − ℓ)σ7 +σ3 (r − 1)σ3 σ7 (k − r + 1)σ6 (b − k − 1)σ6 (b − r)σ1 σ7 +(k − ℓ)σ7 +(b − r − k + ℓ − 1)σ7 (r − 1)σ3 +(b − 2r + 1)σ3
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.
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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 ⊂ MatX(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.
p-Ranks of quasi-symmetric designs June 2, 2014 9 / 22
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 : s1 s3 s2 s4 s5 multiplicity A 1 1 1 B 1 1 v − 1 C 1 1 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 B1 ∼ = Mat2(F) and B and C are in the same block B2.
p-Ranks of quasi-symmetric designs June 2, 2014 10 / 22
Let Q be the following quiver
x• α
β
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
FS ∼ = Mat2(F) ⊕ FQ/(αβ). The projective covers of simple modules are : P(A) = [A], P(B) = B C
P(C) = C B C .
p-Ranks of quasi-symmetric designs June 2, 2014 11 / 22
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 ∼ = Mat2(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. [A] ,
C B
C B C ,
B C
p-Ranks of quasi-symmetric designs June 2, 2014 12 / 22
Since A has multiplicity one, we can write FX ∼ = [A] ⊕ g1 [C] ⊕ g2 C B
C B C ⊕ h1
B C
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.
p-Ranks of quasi-symmetric designs June 2, 2014 13 / 22
Since β in the quiver goes to σ7 and βα goes to σ5, we put s = 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 standard module FX. Remark that the usual 2-rank of the design is rank(σ6) and rank(σ6) = 1 + rank(σ7) = 1 + s.
p-Ranks of quasi-symmetric designs June 2, 2014 14 / 22
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
FX ∼ = [A] ⊕ g1 [C] ⊕ g2 C B
C B C ⊕ h1
B C
where (g1, g2, g3, h1, h2) = (b − 2s − 1, s − t, t, v − 2s + t − 1, s − t).
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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 4 11 1 11 8 12 3 8 3 12 5 12 10 10 2 10 2 13 15 13 12 8 1 12 1 14 58 14 14 6 14 15 I do not know why h1 = 0. If this is true in general, then the structure is determined only by the 2-rank of the design.
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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 (FX2)FS22 ∼ = [A] ⊕ (g1 + g2 + h2) [C] ⊕ g3 C C
This is just obtained by FX ∼ = [A] ⊕ g1 [C] ⊕ g2 C B
C B C ⊕ h1
B C
p-Ranks of quasi-symmetric designs June 2, 2014 17 / 22
Example 3.4
In [The CRC Handbook of Combinatorial Designs], we can find a list of designs with odd ℓ and r : No. v b r ℓ ♯ 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.
p-Ranks of quasi-symmetric designs June 2, 2014 18 / 22
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 of the projective covers of simple FS-modules are P(A) = A B C A , P(B) = B A
P(C) = C A C and the structures of standard FS-modules are FX ∼ = A B C A ⊕ 13 C A C ⊕ 7 [C] for all 80 designs. (By computation)
p-Ranks of quasi-symmetric designs June 2, 2014 19 / 22
Theorem 4.2
Let F be a field of characteristic 5. There are two simple FS-modules A and B with dimF A = 2 and dimF B = 1. The Loewy series of the projective covers of simple FS-modules are P(A) = A A
P(B) = [B] and the structure of the standard FS-module is FX ∼ = A A
p-Ranks of quasi-symmetric designs June 2, 2014 20 / 22
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 series
P(A) = A B
P(B) = B A B , P(C) = [C] and the structure of the standard FS-module is FX ∼ = B A B ⊕ 19 [B] ⊕ 14 [C] .
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