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Representations of association schemesand coherent configurations
Akihide Hanaki
Shinshu University
June, 2014, Summer School
A. Hanaki (Shinshu Univ.) Representations June, 2014 1 / 52
1 Introduction
2 Coherent configurations and association schemes
3 Ordinary representations
4 Modular representationsp-Modular systemsDecomposition matrices and Cartan matricesModular irreducible representations of commutative schemesIrreducible representations of coherent configurations and their fibers
5 Examples – Coherent configurations defined by quasi-symmetric designsCharacteristic 3, 5, and 7 for 2-(15, 3, 1)-designs
A. Hanaki (Shinshu Univ.) Representations June, 2014 2 / 52
Introduction
We will consider representations of association schemes and coherentconfigurations.
Usually, “representations” mean representations of adjacencyalgebras. But it is not enough. For example, it is known that thereare many strongly regular graphs with same parameters. They haveisomorphic adjacency algebras.
The standard module can contain more combinatorial informations.But standard modules over the complex number field are alwaysisomorphic.
We will consider modular representations, representations over apositive characteristic field.
As an application, we will consider p-ranks of designs.
A. Hanaki (Shinshu Univ.) Representations June, 2014 3 / 52
Coherent configurations and association schemes
Let X be a finite set.
Let S be a partition of X ×X. Namely, X ×X =⋃s∈S s is disjoint
and s 6= ∅ for s ∈ S.
For s ⊂ X ×X, put s∗ = {(y, x) | (x, y) ∈ s}.A pair (X,S) is called a coherent configuration if
there is a subset {11, · · · , 1r} of S such that⋃ri=1 1i = {(x, x) | x ∈ X},
if s ∈ S, then s∗ ∈ S, andthere are nonnegative integers pust (s, t, u ∈ S) such that
pust = ]{z ∈ X | (x, z) ∈ s, (z, y) ∈ t}
when (x, y) ∈ u.
Put Xi = {x ∈ X | (x, x) ∈ 1i} (i = 1, · · · , r) and call Xi a fiber.
(X,S) is said to be homogeneous if r = 1. A homogeneous coherentconfiguration is also called an association scheme (noncommutaive).
A. Hanaki (Shinshu Univ.) Representations June, 2014 4 / 52
Let (X,S) be a coherent configuration with fibers X1, · · ·Xr.
For s ∈ S, we denote by σs the adjacency matrix of s.
By definition, ZS =⊕
s∈S Zσs is a subring of MatX(Z).
σsσt =∑u∈S
pustσu
(pust : intersection number, structure constant)
For a commutative ring R with unity, define RS = R⊗Z ZS and callthis R-algebra the adjacency algebra of (X,S) over R. We often usethe notation σs for the corresponding element in RS.
A. Hanaki (Shinshu Univ.) Representations June, 2014 5 / 52
For s ∈ S, there is a unique pair (i, j) such that σ1iσsσ1j = σs.
PutSij = {s ∈ S | σ1iσsσ1j = σs}.
Then S =⋃i
⋃j S
ij is a partition of S.
(Xi, Sii) is a homogeneous coherent configuration and
RSii =⊕s∈Sii
Rσs
is a subalgebra of RS (with non-common identity).
A. Hanaki (Shinshu Univ.) Representations June, 2014 6 / 52
Example
Let G be a permutation group on a finite set X.
Let S be the set of G-orbits on X ×X by diagonal action of G.
Then (X,S) is a coherent configuration and the G-orbits of X arefibers.
Put G = 〈(12), (34)〉 and H = 〈(12)(34)〉.Then the coherent configurations are
0 1 4 41 0 4 45 5 2 35 5 3 2
0 1 4 51 0 5 46 7 2 37 6 3 2
A. Hanaki (Shinshu Univ.) Representations June, 2014 7 / 52
Ordinary representations
Let (X,S) be a coherent configuration.
We will consider ordinary representations, representations over thecomplex number field C, equivalently representations of CS.
Lemma 3.1
Let A be a subalgebra of Matn(C) closed under transposed conjugate.Namely x ∈ A implies tx ∈ A. Then A is semisimple.
Proposition 3.2
The adjacency algebra CS of a coherent configuration (X,S) issemisimple.
We denote by Irr(S) the set of all irreducible characters of (X,S),where a character is the trace function of a representation.
A. Hanaki (Shinshu Univ.) Representations June, 2014 8 / 52
For s ∈ Sii, define the valency ns = p1iss∗ .
Define a map by σs 7→ ns if s ∈ Sii and σs 7→ 0 if s ∈ Sij (i 6= j).Then the map is the character of an irreducible representation. Wecall this the trivial representation. The trivial representation hasdegree r.
The map ΓS : CS → MatX(C), ΓS(σs) = σs is also a representation.We call this the standard representation. By γS we denote thecharacter of ΓS . Then γS(σs) = |Xi| if s = 1i, and 0 otherwise.
Consider the irreducible decomposition of γS :
γS =∑
χ∈Irr(S)
mχχ.
We call mχ the multiplicity of χ.
A. Hanaki (Shinshu Univ.) Representations June, 2014 9 / 52
Proposition 3.3
Let (X,S) be a homogeneous coherent configuration. For χ ∈ Irr(S), wedenote by eχ the primitive central idempotent corresponding to χ. Then
eχ =mχ
|X|∑s∈S
1
nsχ(σs∗)σs.
Proposition 3.4 (Orthogonality relation)
Let (X,S) be a homogeneous coherent configuration. For χ, ϕ ∈ Irr(S),
mχ
|X|χ(1)
∑s∈S
1
nsχ(σs∗)ϕ(σs) = δχϕ.
There are similar formulas also for non-homogeneous coherentconfigurations.Note that the multiplicity mχ is determined by character values. So,if two coherent configurations have the same intersection numbers,then the standard modules are isomorphic.
A. Hanaki (Shinshu Univ.) Representations June, 2014 10 / 52
Problem 3.5
Find a good algorithm to compute character tables of (homogeneous)coherent configurations.
For finite groups, Dixon–Schneider algorithm is good.
When (X,S) is commutative, the intersections of eigenspacesdetermine the table. But it is not so easy to compute.
Question 3.6 (Bannai-Ito)
Let (X,S) be a (commutative) coherent configuration. Is it true that alleigenvalues of σs (s ∈ S) are cyclotomic numbers ?
A. Hanaki (Shinshu Univ.) Representations June, 2014 11 / 52
Suppose that (X,S) be a homogeneous coherent configuration with|X| prime.
If an eigenvalue of σs (s ∈ S) is non-cyclotomic, then the fieldgenerated by the value have some strong properties.
Toru Komatsu (2006) constructed such field and possible intersectionnumbers (character table).
By the same method, Sho Teranishi (2012) constructed manyexamples.
A. Hanaki (Shinshu Univ.) Representations June, 2014 12 / 52
Example : p = 2875
f1(X) = X4 +X3 − 1071X2 − 7321X − 8850,
f2(X) = X4 +X3 − 1071X2 − 4464X + 102573,
f3(X) = X4 +X3 − 1071X2 + 1250X − 279,
f4(X) = X4 +X3 − 1071X2 + 1250X + 85431
B1 =
0 1 0 0 0
714 185 162 188 1780 162 186 183 1830 188 183 166 1770 178 183 177 176
, B2 =
0 0 1 0 00 162 186 183 183
714 186 182 180 1650 183 180 177 1740 183 165 174 192
,
B3 =
0 0 0 1 00 188 183 166 1770 183 180 177 174
714 166 177 176 1940 177 174 194 169
, B4 =
0 0 0 0 10 178 183 177 1760 183 165 174 1920 177 174 194 169
714 176 192 169 176
Question 3.7
Is there an association scheme with the above intersection numbers ?
A. Hanaki (Shinshu Univ.) Representations June, 2014 13 / 52
Modular representations
“Modular represntation” means representation of an adjacencyalgebra over a positive characteristic field F with non-semisimple FS.
Theorem 4.1
For a homogeneous coherent configuration (X,S) and a field ofcharacteristic p, FS is semisimple if and only if p does not divide F(S).Here F(S) is the Frame number :
F(S) = |X||S|∏s∈S ns∏
χ∈Irr(S)mχ(χ(1)2)
.
We also consider relations between representations over C and F .
A. Hanaki (Shinshu Univ.) Representations June, 2014 14 / 52
p-Modular systems
Let p be a prime number.
Let R be a complete discrete valuation ring with valuation ideal πR.
Let K be the quotient field of R, and let F be the residue field R/πR.
Suppose that K is of characteristic 0 and F is of characteristic p.
Then we say that (K,R, F ) is a p-modular system.
For a coherent configuration (X,S), we say that (K,R, F ) is asplitting p-modular system of (X,S) if all adjacency algebras ofsubconfigurations and quotient configurations over K and F aresplitting algebras. (In general, a finite dimensional k-algebra A iscalled a splitting algebra if A/J(A) is isomorphic to a direct sum offull matrix k-algebras.)
It is enough to consider the case that K and F are big enough.
(If you do not understand, consider (Q,Z, GF (p)), though this is nota p-modular system.)
A. Hanaki (Shinshu Univ.) Representations June, 2014 15 / 52
Let (X,S) be a coherent configuration, and let (K,R, F ) is asplitting p-modular system of (X,S).
For a KS-module M , there is an RS-lattice M̃ such thatK ⊗R M̃ ∼= M . We call M̃ an R-form of M . (An R-form is notuniue.)
This means that, for every representation Φ of KS, there exists asimilar representation Φ′ such that Φ′(σs) ∈ Matd(R) (s ∈ S).
Through an R-form, we can get an FS-module M∗ = M̃/πM̃ .
M∗ is not uniquely determined, but its composition factors areunique.
So the modular character, the trace of the representation, isdetermined.
For FS-modules V and W , we write V ↔W if they have the samecomposition factors.
A. Hanaki (Shinshu Univ.) Representations June, 2014 16 / 52
Question 4.2
Suppose that we know the (ordinary) character table of (X,S). Can wedetermine modular irreducible characters (modules) ?
The answer is NO, in general.
It is difficult even for finite groups. (Problem to determinedecomposition numbers.)
If a group G is solvable, then every simple FG-module has a formM∗ for some simple KG-module (Fong-Swan’s Theorem).
Problem 4.3
Consider a good definition of “solveble coherent configurations”.(Definition by French-Zieschang ?)
A. Hanaki (Shinshu Univ.) Representations June, 2014 17 / 52
Decomposition matrices and Cartan matrices
Let M be a simple KS-module.
Determine dM,V by M∗ ↔⊕
V ∈IRR(FS) dM,V V .
We call dM,V the decomposition number.
Put D = (dM,V )IRR(KS)×IRR(FS) and call this matrix thedecomposition matrix.
Let V be a simple FS-module.
There is a primitive idempotent eV such that eV FS/eV J(FS) ∼= V .
In this case, eV FS is the projective cover P (V ) of V .
For V , W ∈ IRR(FS), definecV,W = dimF HomFS(eV FS, eWFS) = dimF eWFSeV .
We call cV,W the Cartan invariant.
Remark that cV,W is the number of V in P (W ) as simpleconstituents.
Put C = (cV,W )IRR(FS)×IRR(FS) and call this matrix the Cartanmatrix.
A. Hanaki (Shinshu Univ.) Representations June, 2014 18 / 52
It is known thattDD = C.
Especially, the Cartan matrix C is symmetic.
Problem 4.4
Consider properties of decomposition matrices and Cartan matrices.
For group representations,
C is non-singular, andelementary divisors of C are p-power.
But they are not true for (homogeneous) coherent configurations.
A. Hanaki (Shinshu Univ.) Representations June, 2014 19 / 52
Modular irreducible representations of commutativeschemes
Let (X,S) be a commutative association scheme (homogeneouscoherent configuration).
Every irreducible representation over K has degree 1 and has valuesin R.
Taking values modulo πR, we can get modular irreducible characters.
Every modular irreducible character is obtained in this way.
A. Hanaki (Shinshu Univ.) Representations June, 2014 20 / 52
Example
Let (X,S) be an association scheme defined by a Fano plane.
0 1 1 1 1 1 1 2 2 2 3 3 3 31 0 1 1 1 1 1 3 3 2 2 2 3 31 1 0 1 1 1 1 2 3 3 3 2 2 31 1 1 0 1 1 1 2 3 3 2 3 3 21 1 1 1 0 1 1 3 2 3 3 2 3 21 1 1 1 1 0 1 3 3 2 3 3 2 21 1 1 1 1 1 0 3 2 3 2 3 2 32 3 2 2 3 3 3 0 1 1 1 1 1 12 3 3 3 2 3 2 1 0 1 1 1 1 12 2 3 3 3 2 3 1 1 0 1 1 1 13 2 3 2 3 3 2 1 1 1 0 1 1 13 2 2 3 2 3 3 1 1 1 1 0 1 13 3 2 3 3 2 2 1 1 1 1 1 0 13 3 3 2 2 2 3 1 1 1 1 1 1 0
A. Hanaki (Shinshu Univ.) Representations June, 2014 21 / 52
The character table is
g0 g1 g2 g3 mi
χ1 1 6 3 4 1χ2 1 6 −3 −4 1
χ3 1 −1√
2 −√
2 6
χ4 1 −1 −√
2√
2 6
Let p = 7.
Then
χ∗1 = χ∗
4 and χ∗2 = χ∗
3 if we choose a prime ideal containing 3 +√
2,andχ∗1 = χ∗
3 and χ∗2 = χ∗
4 if we choose a prime ideal containing 3−√
2.
It depends on the choice of the prime ideal.
A. Hanaki (Shinshu Univ.) Representations June, 2014 22 / 52
Irreducible representations of coherent configurations andtheir fibers
Irreducible representations of (non-homogeneous) a coherentconfiguration can be determined by them of its fibers.
We denote by IRR(A) the set of representatives of isomorphismclasses of simple right-A modules.
Theorem 4.5
Let K be an algebraically closed field. Let (X,S) be a coherentconfiguration with fibers Xi (i = 1, 2, · · · , r). Put A =
⊕ri=1KS
ii. Thenthere are injections Φi : IRR(KSii)→ IRR(S) (i = 1, 2, · · · , r) satisfyingthe following properties.
1 Define Φ :⋃ri=1 IRR(KSii)→ IRR(KS) by Φ(W ) = Φi(W ) if
W ∈ IRR(KSii). Then Φ is surjective.
2 For V ∈ IRR(KS), V ↓A∼=⊕
W∈Φ−1(V )W .
3 The map Φ preserves the multiplicities.
A. Hanaki (Shinshu Univ.) Representations June, 2014 23 / 52
What is “multiplicity”.
It is the number of the simple module in the standard module KX assimple constituents.
Namely,
mV = dimK HomKS(P (V ),KX) = dimK KSe = rank(e),
where P (V ) is the projective cover of V and e is a primitiveidempotent corresponding to V .
If K = C, then the multiplicity is just the usual multiplicity of acharacter.
A. Hanaki (Shinshu Univ.) Representations June, 2014 24 / 52
How can we get a partition of⋃ri=1 IRR(KSii) ?
Proposition 4.6
For V , W ∈ IRR(A), let e, f be primitive idempotents corresponding toV , W , respectively. Namely eA/eJ(A) ∼= V and fA/fJ(A) ∼= V . ThenΦ(V ) = Φ(W ) if and only if eKSf 6⊂ J(KS).
A. Hanaki (Shinshu Univ.) Representations June, 2014 25 / 52
Examples – Coherent configurations defined byquasi-symmetric designs
Higman defined the type of coherent configurations.
Let (X,S) be a coherent configuration with fibers Xi (i = 1, · · · , r).
The type of (X,S) is an r × r matrix T = (tij) with tij = |Sij |.It is clear that T is symmetric. We omit entries below the maindiagonal.
For example, a symmetric design defines a coherent configuration of
type
(2 2
2
).
We also write the type as (2, 2; 2).
A. Hanaki (Shinshu Univ.) Representations June, 2014 26 / 52
A design is said to be quasi-symmetric if any two blocks intersect ineither x of y points (x 6= y).
For example, a 2-(v, k, 1)-design is quasi-symmetric with x = 0 andy = 1.
The block graph of a quasi-symmetric design is defined as follows :
Points of the graph are blocks of the design.Two blocks are adjacent in the graph if they intersect in y points.
The block graph of a quasi-symmetric design is strongly-regular.
The incidence matrix of a quasi-symmetric design defines a coherentconfiguration of type (2, 2, ; 3).
Conversely, a coherent configuration of type (2, 2; 3) gives aquasi-symmetric design.
We will consider some 2-(v, k, 1)-designs, especially 80 isomorphismclasses of Steiner triple systems 2-(15, 3, 1).
A. Hanaki (Shinshu Univ.) Representations June, 2014 27 / 52
type
(2 2
2
)←→ symmetric design
type
(2 2
3
)←→ quasi-symmetric design
type
(3 2
3
)←→ strongly regular design
strongly regular design : Higman, Klin–Reichard
A. Hanaki (Shinshu Univ.) Representations June, 2014 28 / 52
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.
For 80 nonisomorphic 2-(15, 3, 1)-designs, the 2-ranks of incidencematrices are 11, 12, 13, 14, and 15.
The 3-ranks are 14, and p-ranks are 15 for p 6= 2, 3.
We will focus on the 2-(15, 3, 1)-designs and p = 2.
A. Hanaki (Shinshu Univ.) Representations June, 2014 29 / 52
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.
A. Hanaki (Shinshu Univ.) Representations June, 2014 30 / 52
Let (X1, X2, F ) be a 2-(v, `, 1)-design, where X1 is the set of points,X2 is the set of blocks, and F is the set of flags.
Put X = X1 ∪X2.
Define binary relations si (i = 1, · · · , 9) on X by
s1 = {(x, x) | x ∈ X1}, s2 = {(x, x) | x ∈ X2},s3 = X2
1 − s1,
s4 = {(x, y) ∈ X22 | ](x ∩ y) = 1},
s5 = {(x, y) ∈ X22 | ](x ∩ y) = 0},
s6 = F, s7 = X1 ×X2 − F,s8 = ts6 = {(y, x) | x, y ∈ s6},s9 = ts7 = {(y, x) | x, y ∈ s7}.
Put S = {s1, · · · , s9}.Then (X,S) is a coherent configuration.
Note that the subconfiguration (X2, {s2, s4, s5}) defines a stronglyregular graph.
A. Hanaki (Shinshu Univ.) Representations June, 2014 31 / 52
We can compute the table of multiplications :
σ1 σ3 σ6 σ7
σ1 σ1 σ3 σ6 σ7
σ3 σ3 (v − 1)σ1 (`− 1)σ6 (v − `)σ6
+(v − 2)σ3 +`σ7 +(v − `− 1)σ7
σ8 σ8 (`− 1)σ8 `σ2
+`σ9 +σ4 (`− 1)σ4
+`σ5
σ9 (v − `)σ8 (v − `)σ2
σ9 +(v − `− 1)σ9 (`− 1)σ4 +(v − 2`+ 1)σ4
+`σ5 +(v − 2`)σ5
A. Hanaki (Shinshu Univ.) Representations June, 2014 32 / 52
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 5.1
If ` and r = (v − 1)/(`− 1) are odd, then v, a, and b are odd and k iseven.
A. Hanaki (Shinshu Univ.) Representations June, 2014 33 / 52
Theorem 5.2
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.
We determine the structure of the algebra A.
What should we do ?
Give informations enough to apply representation theory of finitedimensional algebras.
A. Hanaki (Shinshu Univ.) Representations June, 2014 34 / 52
Morita equivalence – changing the dimensions of simple modules suchthat the module categories are equivalent.
Example : K and Matn(K) are Morita equivalent.
A finite dimensional algebra is said to be basic if every simplemodules are one-dimensional.
Every finite dimensional algebra is Morita equivalent to a basic algebra :
Put IRR(A) = {Vi | i = 1, · · · , `}. Let ei be a primitiveidempotent corresponding to Vi. Put e =
∑ri=1 ei. Then
eAe is basic and Morita equivelent to A.
For Matn(K), e11Matn(K)e11∼= K, where e11 is the
matrix unit.
A. Hanaki (Shinshu Univ.) Representations June, 2014 35 / 52
Let Q = (V, P, s, t) be a quiver where V is a set of vertices, P is a setof arrows, s : P → V , and t : P → V .
A quiver is a directed graph with loops and multiple edges.
The quiver algebra KQ is a K-algebra whose basis is the set of allpaths and multiplication is composition of paths.
The quiver algebra is not necessary finite dimensional.
Let J be the ideal of KQ generated by all arrows.
An ideal I of KQ is said to be admissible if J2 ⊃ I ⊃ Jn for some n.
A. Hanaki (Shinshu Univ.) Representations June, 2014 36 / 52
Let A be a basic algebra.
For V ∈ IRR(A), we denote by eV a primitive idempotentcorresponding to V .
We define the Gabriel quiver Q(A) of A.
The point set is IRR(A).
The number of arrows from V to W is dimK eV (J(A)/J2(A))eW .
Theorem 5.3 (Gabriel)
Let A be a basic algebra. Then there is an admissible ideal I of KQ(A)such that A ∼= KQ(A)/I.
We will compute the Gabriel quiver and the admissible ideal foradjacency algebra of a coherent configuration defined by a2-(15, 3, 1)-design.
A. Hanaki (Shinshu Univ.) Representations June, 2014 37 / 52
Let (X,S) be a coherent configuration defined by a2-(15, 3, 1)-design.
The character table of fibers (Xi, Sii) (i = 1, 2) are
s1 s3 multiplicity
ϕ1 1 14 1ϕ2 1 −1 14
s2 s4 s5 multiplicity
ψ1 1 18 16 1ψ2 1 3 −4 14ψ3 1 −3 2 20
The possibilities of irreducible characters of (X,S) are{ϕ1 + ψ1, ϕ2, ψ2, ψ3} and {ϕ1 + ψ1, ϕ2 + ψ2, ψ3}.Degrees are {2, 1, 1, 1} and {2, 2, 1}.Since |S| = 9 is the sum of squares of degrees, we have the charactertable.
s1 s3 s2 s4 s5 multiplicity
χ1 1 14 1 18 16 1χ2 1 −1 1 3 −4 14χ3 0 0 1 −3 2 20
A. Hanaki (Shinshu Univ.) Representations June, 2014 38 / 52
We consider the modular character table for p = 2.
The character table of fibers are
s1 s3 multiplicity
ϕ1 1 14 1ϕ2 1 −1 14
s2 s4 s5 multiplicity
ψ1 1 18 16 1ψ2 1 3 −4 14ψ3 1 −3 2 20
Consider them in characteristic 2. We have
s1 s3 multiplicity
1 0 11 1 14
s2 s4 s5 multiplicity
1 0 0 11 1 0 34
We can see that the modular character table of (X,S) is
s1 s3 s2 s4 s5 multiplicity
ξ1 1 0 1 0 0 1ξ2 1 1 0 0 0 14ξ3 0 0 1 1 0 34
A. Hanaki (Shinshu Univ.) Representations June, 2014 39 / 52
Bys1 s3 s2 s4 s5 multiplicity
χ1 1 14 1 18 16 1χ2 1 −1 1 3 −4 14χ3 0 0 1 −3 2 20
s1 s3 s2 s4 s5 multiplicity
ξ1 1 0 1 0 0 1ξ2 1 1 0 0 0 14ξ3 0 0 1 1 0 34
we can determine the decomposition matrix and the Cartan matrix.
D =
1 0 0
0 1 10 0 1
C = tDD =
1 0 0
0 1 10 1 2
A. Hanaki (Shinshu Univ.) Representations June, 2014 40 / 52
We write A, B, C for the simple modules corresponding to ξ1, ξ2,and ξ3.
By the Cartan matrix
1 0 0
0 1 10 1 2
, we have the Loewy structures of
projective covers P (A) and P (B).
P (A) = A, P (B) =
[BC
]
The projective cover P (C) is either
CBC
or
[C
B C
].
Note that the algebra has two blocks B1 and B2. The block B1
(containing A) is isomorphic to Mat2(F ). So we want to know B2.
Remark that the algebra is not basic but B2 is basic.
A. Hanaki (Shinshu Univ.) Representations June, 2014 41 / 52
Now we consider the Gabriel quiver.
We can choose primitive idempotent eB = σ3 and eC = σ4 + σ5.
Since eBFSeC = Fσ7 and eCFSeB = Fσ9, we put α = σ7 andβ = σ9.
We have
eB = σ3, eBα = σ7, (e2α)β = 0
eC = σ4 + σ5, eCβ = σ9, (eCβ)α = σ5.
This means that the Gabriel quiver is
Q : eBα //
eCβ
oo
and the relation is αβ = 0.
A. Hanaki (Shinshu Univ.) Representations June, 2014 42 / 52
Theorem 5.4
Let F be a field of characteristic 2. Under the above notations,
FS ∼= Mat2(F )⊕ FQ/(αβ)
where Q is the following quiver :
Q : •α //
•β
oo
There are three simple module A, B, C and the Loewy structures of theprojective covers are
P (A) = [A] , P (B) =
[BC
], P (C) =
CBC
.
A. Hanaki (Shinshu Univ.) Representations June, 2014 43 / 52
Now we apply representation theory of finite dimensional algebras andconsider the standard module FX.
There are three representation types (very rough definition!):
finite : there are finitely many indecomposable modules(infinite) tame : there are infinitely many indecomposable modules butthey can be classified(infinite) wild : there are infinitely many indecomposable modules butthey can not be classified
Problem 5.5
Consider representation types of adjacency algebras of coherentconfigurations (association schemes).
Our case is very easy and we can see that the representation type isfinite.
A. Hanaki (Shinshu Univ.) Representations June, 2014 44 / 52
All indecomposable B2-modules are
M1 =[C], M2 =
[CB
], M3 =
CBC
,N1 =
[B], N2 =
[BC
].
So we can write
FX ∼= [A]⊕ g1M1 ⊕ g2M2 ⊕ g3M3 ⊕ h1N1 ⊕ h2N2.
for some nonnegative integers g1, g2, g3, h1, h2.
A. Hanaki (Shinshu Univ.) Representations June, 2014 45 / 52
Now we consider a 2-(v, `, 1)-design with odd ` and r. As we saw, theadjacency algebras are isomorphic to the algebra defined by a2-(15, 3, 1)-design.
The multiplicities are mA = 1, mB = v − 1, mC = b− 1.
We have
g2 + g3 + h1 + h2 = b− 1, g1 + g2 + gs3 + h2 = v − 1.
Since σ9 is the transposed matrix of σ7, their ranks are equal. Thismeans g2 = h2.
Put s = rank(σ7) and t = rank(σ5). Then we have
(g1, g2, g3, h1, h2) = (b− 2s− 1, s− t, t, v − 2s+ t− 1, s− t).
Remark that the usual 2-rank of the design is rank(σ6) andrank(σ6) = 1 + rank(σ7) = 1 + s.
The structures of standard modules “can” contain more informationthan p-ranks.
A. Hanaki (Shinshu Univ.) Representations June, 2014 46 / 52
Example 5.6
For 80 nonisomorphic 2-(15, 3, 1)-designs, we have the followingparameters (by computation) :
] s = rank(σ7) t = 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
Question 5.7
Is one parameter enough ?
A. Hanaki (Shinshu Univ.) Representations June, 2014 47 / 52
Example 5.8
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 2-ranks of them. But I do not have data.
A. Hanaki (Shinshu Univ.) Representations June, 2014 48 / 52
Characteristic 3, 5, and 7 for 2-(15, 3, 1)-designers
Theorem 5.9
Let F be a field of characteristic 3. Under the above notations, the Loewyseries of the projective covers of simple FS-modules of coherentconfigurations obtained by 2-(15, 3, 1)-designs 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)
A. Hanaki (Shinshu Univ.) Representations June, 2014 49 / 52
Characteristic 3, 5, and 7 for 2-(15, 3, 1)-designers
Theorem 5.10
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 of coherent configurationsobtained by 2-(15, 3, 1)-designs are
P (A) =
[AA
], P (B) = [B]
and the structure of the standard FS-module is
FX ∼=[AA
]⊕ 13 [A]⊕ 20 [B] .
A. Hanaki (Shinshu Univ.) Representations June, 2014 50 / 52
Characteristic 3, 5, and 7 for 2-(15, 3, 1)-designers
Theorem 5.11
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 of coherent configurationsobtained by 2-(15, 3, 1)-designs 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] .
A. Hanaki (Shinshu Univ.) Representations June, 2014 51 / 52
Thank you very much !
A. Hanaki (Shinshu Univ.) Representations June, 2014 52 / 52