J Algebr Comb DOI 10.1007/s10801-016-0681-y

Modular adjacency algebras, standard representations, and p-ranks of cyclotomic association schemes Akihide Hanaki1

Received: 12 May 2015 / Accepted: 15 March 2016 © Springer Science+Business Media New York 2016

Abstract In this paper, we consider cyclotomic association schemes S = Cyc( pa , d). We focus on the adjacency algebra of S over algebraically closed fields K of characteristic p. If p ≡ 1 (mod d), p ≡ −1 (mod d), or d ∈ {2, 3, 4, 5, 6}, we identify the adjacency algebra of S over K as a quotient of a polynomial ring over an admissible ideal. In several cases, we determine the indecomposable direct sum decomposition of the standard module of S. As a consequence, we are able to compute the p-rank of several specific elements of the adjacency algebra of S over K . Keywords

Association scheme · Cyclotomic scheme · Representation · p-Rank

1 Introduction Two association schemes are said to be algebraically isomorphic if the intersection numbers coincide. In this case, their algebraic properties are the same but combinatorial properties are not, in general. For example, distance-regular graphs with the same intersection array give algebraically isomorphic association schemes. In general, it is not so easy to distinguish them. In some papers, for example [3,7,9,10], it was shown that p-ranks, ranks of matrices over a field of characteristic p, of adjacency matrices can distinguish algebraically isomorphic association schemes for some examples. In [7], Yoshikawa and the author considered the structure of adjacency algebras and standard modules (representations) over a field of characteristic p and relation with the p-ranks. Structures of modular adjacency algebras were studied in [11–13].

This work was supported by JSPS KAKENHI Grant Number 25400011.

B 1

Akihide Hanaki [email protected] Faculty of Science, Shinshu University, Matsumoto 390-8621, Japan

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In this paper, we consider the adjacency algebra and representations of the cyclotomic (association) scheme Cyc( pa , d) over a field of characteristic p. We will give an effective method to determine the structure of the adjacency algebra (Theorem 3.4). In general, the structure seems to be complicated. So we will give concrete structures for some special cases : p ≡ 1 (mod d) (Theorem 3.7), p ≡ −1 (mod d) (Theorem 3.8), and d = 2, 3, 4, 5, 6 (theorems in Sect. 3.4). We will describe the algebra as a quotient of a polynomial ring by an admissible ideal. Also we determine the structures of standard modules for the case p ≡ 1 (mod d) (Theorem 4.2) and d = 3 (Theorem 4.6). We will determine the indecomposable direct sum decomposition of the module. For these cases, we also determine the p-ranks of some matrices (Corollaries 4.4 and 4.7) and compare them with some algebraically isomorphic association schemes (Examples 4.5, 4.8, and 4.9). A cyclotomic scheme is defined by the action of a subgroup G of the affine group AGL(1, pa ). The standard module is a G-module, and its G-submodules are linear codes invariant under G. Linear codes invariant under AGL(1, pa ) were classified in [8].

2 Preliminaries 2.1 Association schemes and adjacency algebras Following the book [1] or [14], we will define association schemes and adjacency algebras. Let X be a finite set. For s ⊂ X × X , we define a matrix σs whose rows and columns are indexed by the set X and the (x, y)-entry of σs is 1 if (x, y) ∈ s and 0 otherwise. We call σs the adjacency matrix of s. Let X × X = s∈S s be a partition. We call the pair (X, S) an association scheme if the following three conditions hold : (1) (2) (3)

The diagonal relation {(x, x) | x ∈ X } is in S, we denote it by 1. For every s ∈ S, the transposition {(y, x) | (x, y) ∈ s} is in S, we denote it by s ∗ . u For  all us, t, u ∈ S, there are non-negative integers pst such that σs σt = p σ , where the product is the usual matrix product. u∈S st u

u in the condition (3) is called an intersection number. For s ∈ S, we The number pst 1 call n s = pss ∗ the valency of s. Two association schemes (X, S) and (X  , S  ) are isomorphic if there are bijections ϕ : X → X  and ψ : S → S  such that (x, y) ∈ s if and only if (ϕ(x), ϕ(y)) ∈ ψ(s). They are algebraically isomorphic if there is a u = p ψ(u) bijection ψ : S → S  such that pst ψ(s)ψ(t) for all s, t, u ∈ S. Let K be a field. We regard σs (s ∈S) as matrices over K . Then, by the condition (3), we can define a K -algebra KS = s∈S K σs of dimension |S|. We call the algebra KS the adjacency algebra of (X, S) over K . We regard KS as a subalgebra of the full matrix algebra Mat X (K ). Let K X be the K -vector space with basis X . Since we regard KS as a subalgebra of Mat X (K ), we can see that K X is a right KS-module. We call this module the standard KS-module. The corresponding representation is KS → Mat X (K ) (σs → σs ), and we call this the standard representation of (X, S) over K .

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2.2 Schurian (association) schemes Let G be a finite transitive permutation group on a finite set X . Then G also acts on X × X diagonally. The set S of orbits of G on X × X define a partition of X × X . It is known that (X, S) becomes an association scheme [1, II, Example 2.1]. An association scheme obtained in this way is said to be schurian. For x ∈ X , let H = G x be the stabilizer of x in G. Then we can identify X with H \G. So we can say that a schurian scheme is obtained by a finite group and its subgroup. We denote it by X(G, H ). For a schurian scheme, the adjacency algebra is understood by the following way. Let T be the permutation representation of G over a field K . Then the adjacency algebra is just the centralizer algebra {A ∈ Mat X (K ) | AT (g) = T (g)A for any g ∈ G}. Let N be a finite group, and let H be a subgroup of the automorphism group of N . We can define the semidirect product N  H . We consider the schurian scheme (X, S) = X(N  H, H ). For this case, N is a regular normal subgroup of N  H and we can apply [1, II, Theorem 6.1]. Let K be a field. The group H also acts on the group algebra KN. By [1, II, Theorem 6.1], the adjacency algebra is just the ring of fixed points (KN) H = {α ∈ KN | α h = α for any h ∈ H }, where we regard KN as a subalgebra of Mat N (K ) by a regular permutation representation. Since (KN) H ⊂ KN, KN becomes a right (KN) H -module. This is just the standard module of the scheme X(N  H, H ). 2.3 Cyclotomic (association) schemes Let p be a prime number. We denote the finite field of pa elements by F pa . The a multiplicative group F× pa = F pa \{0} is a cyclic group of order p − 1. We fix a × primitive element ζ of F pa (a generator of F pa ). Let N be an elementary abelian group of order pa and fix a group isomorphism F pa → N (α → [α]). Let H = h 0

be a cyclic group of order pa − 1 and define the action of H on N by [α]h 0 = [ζ α]. Let d be a divisor of pa − 1. There is a unique subgroup Hd = h d0 of H of index d. Now we can define a schurian scheme X(N  Hd , Hd ). We call this the cyclotomic (association) scheme and denote it by Cyc( pa , d). For details, see [4]. Adjacency matrices of Cyc( pa , d) are obtained as follows. Put X 0 = {0} and X i = {ζ j | 0 ≤ j < pa , j ≡ i (mod d)} (i = 1, . . . , d). Then X 0 , X 1 , . . . , X d are Hd -orbits of F pa . Let T be a regular permutation representation of N . Then the adjacency matrices are σi =



T (α) (i = 0, 1, . . . , d).

α∈X i

We prove one easy lemma here. Lemma 2.1 Let s and t be non-trivial relations of the cyclotomic scheme Cyc( pa , d), and let f be a polynomial over an arbitrary field K . The f (σs ) and f (σt ) have the same rank over K .

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Proof Let P be the permutation matrix on N defined by the multiplication of ζ . Then P −1 σi P = σi+1 for i = 1, . . . , d − 1 and P −1 σd P = σ1 , where σi are defined as above. So the lemma holds.



3 Adjacency algebras of cyclotomic schemes Let K be an algebraically closed field of characteristic p. In this section, we will give an effective method to determine the structure of the adjacency algebra of a cyclotomic scheme Cyc( pa , d) over K . In general, the structure seems to be complicated. So we will give concrete structures for some spacial cases in Sects. 3.2, 3.3, and 3.4. We use notations in Sect. 2.3. First, we will consider the KH-module structure of KN. The following facts are well known.   (1) KN is a local algebra with the Jacobson radical { n∈N cn n | n∈N cn = 0}. (2) Let {n 1 , . . . , n a } be a set of generators of N , and put xi = n i − 1 for i = 1, . . . , a. p Then KN = K [x1 , . . . , xa ] with relations xi = 0 (i = 1, . . . , a). ∼ C pa −1 is a p  -group and so KH is semisimple. We remark that the group H = h 0 = Since J (KN) is fixed by automorphisms, we have KN ∼ =

a( p−1)

J i−1 (KN)/J i (KN)

i=1

as a KH-module. We consider J (KN)/J 2 (KN). It has basis {xi | i = 1, . . . , a}, where xi = xi + J 2 (KN). Put αi ∈ F pa with [αi ] = n i for i = 1, . . . , a. a a Lemma 3.1 We have [ i=1 ci αi ] − [0] ≡ i=1 ci xi (mod J 2 (KN)) for ci ∈ F p (i = 1, . . . , a). Proof Remark that [α][β] = [α + β]. So [cα] = [α]c for a non-negative integer c. We show that [cαi ] − [0] ≡ cxi (mod J 2 (KN)) by induction on 0 ≤ c < p. If c = 0, 1, then this is clear. Suppose 2 ≤ c < p. By inductive hypothesis, we have [cαi ] − [0] = [αi ]c − [0] = ([αi ]−[0])([αi ]c−1 +· · · + [αi ] + [0]) = xi (([αi ]c−1 − [0])+· · ·+([αi ] − [0]) + c[0]) ≡ cxi (mod J 2 (KN)). Now we have 

a 

 ci αi

=

i=1

≡

a

[ci αi ] ≡

i=1 a 

a

(ci xi + [0])

i=1

ci xi + [0] (mod J 2 (KN)).

i=1

The assertion holds.

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Lemma3.2 Recall that ζ is a primitive element of F pa and h 0 is a generator of H . Put αi ζ = nj=1 ci j α j (ci j ∈ F p ). Namely ζ → (ci j ) is a regular representation of F pa  over F p . Then xih 0 ≡ nj=1 ci j x j (mod J 2 (KN)). This means that J (KN)/J 2 (KN) is a KH-module given by the regular representation of H ∼ = F× pa . Proof By Lemma 3.1, we have xih 0 = [αi ζ ] − [0] = a j=1 ci j x j .

 a

j=1 ci j α j



− [0] ≡



Lemma 3.3 There exist v1 , . . . , va ∈ J (KN) such that the following statements holds. a f f (1) The set {v1 1 · · · va a + J  (KN) | 0 ≤ f i < p (1 ≤ i ≤ a), i=1 f i = } is f f  +1 a basis of J (KN)/J (KN) for  = 0, 1, . . . , a( p − 1). Hence, {v1 1 · · · va a | 0 ≤ f i < p (1 ≤ i ≤ a)} is a basis of KN. f f (2) If 0 ≤ f i < p for all 1 ≤ i ≤ a, then K v1 1 · · · va a is a one-dimensional right a f1 fa h 0 f f 1 KH-module such that (v1 · · · va ) = ζ f v1 · · · va a , where f = i=1 f i pi−1 . Proof As in Lemma 3.2, J (KN)/J 2 (KN) is an KN-module given by the regular representation of F× pa over F p . Define C = (ci j ) as in the proof of Lemma 3.2. Since ζ is an 2

a−1

eigenvalue of C, their conjugates ζ p , ζ p , . . . , ζ p are also eigenvalues of C. Since KH is semisimple and commutative, we have KN ∼ = KN/J (KN) ⊕ J (KN)/J 2 (KN) ⊕ 2 2 J (KN) as a KH-module and J (KN)/J (KN) is a direct sum of one-dimensional submodules. There are v1 , . . . , va ∈ J (KN) such that {v1 , . . . , va } is a K -basis of i−1 J (KN)/J 2 (KN) and vih 0 = ζ p vi . This shows that (1) holds for  = 1. p p−1 = 0 Since vi = 0 (i = 1, . . . , a) and v1 , . . . , va generate KN, we can see that vi (i = 1, . . . , a). It is easy to see that the all assertions hold.

We use the elements v1 , . . . , va obtained in Lemma 3.3 throughout this paper. We fix one more notation. For 0 ≤ f ≤ pa − 1, we can write f =

a 

f i pi−1 , 0 ≤ f i ≤ p − 1 (i = 1, . . . , a)

i=1

uniquely. We put v ( f ) = v1 1 . . . va a . f

f

Then Lemma 3.3 says that {v ( f ) | 0 ≤ f ≤ pa − 1} is a basis of KN. Now we can show the most important theorem in this section. Theorem 3.4 We consider a cyclotomic scheme (X, S) = Cyc( pa , d) = X(N  Hd , Hd ), where d | pa − 1. Let K be an algebraically closed field of characteristic p. Put e = ( pa − 1)/d. Then {v (ie) | 0 ≤ i ≤ d} is a basis of the adjacency algebra KS. The product v (ie) v ( je) is v ((i+ j)e) or 0.

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Proof By definition, we know that dim K KS = |S| = d + 1. So it is enough to show that v (ie) is fixed by Hd = h d0 . By Lemma 3.3 (2), we have

h d a 0 v (ie) = ζ ied v (ie) = ζ i( p −1) v (ie) = v (ie) .



The last statement is clear.

a j−1 for i = 0, 1, . . . , d. For an We determine the expression ie = j=1 f j p integer m, we denote by ρ(m) the smallest non-negative integer such that m ≡ ρ(m) (mod d). Put ρ(m) p − ρ(mp) . d

αm =

Obviously, αm = αm  if m ≡ m  (mod d). Lemma 3.5 For every integer m, αm is a non-negative integer and 0 ≤ αm < p. We have ie =

a 

αi pa− j p j−1

j=1

for i = 1, . . . , d − 1, and for i = 0, d, 0e =

a 

0p

j−1

a  , de = ( p − 1) p j−1 .

j=1

j=1

Proof By the definition of ρ, αm is a non-negative integer and 0 ≤ αm < p. The equations for i = 0, d are trivial. Suppose that 0 < i < d. Remark that ρ(i) = i and pa ≡ 1 (mod d). We have a 

αi pa− j p

j=1

j−1

    a  ρ i pa− j p − ρ i pa− j+1 j−1 p = d j=1

 a−1  ρ(i) p a−1  ρ i pa− j p j−1 p p = + d d j=1   a  ρ i pa− j+1 j−1 ρ(i pa ) − − p d d j=2

= Now the lemma holds.

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ρ(i pa ) i( pa − 1) ρ(i) pa − = = ie. d d d



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We remark that v (ie) v ( je) = v ((i+ j)e) = 0 if and only if αi pa− + α j pa− < p for all  = 1, . . . , a by Lemma 3.5. Easily we can see that v (0) v (ie) = v (ie) = 0, v (ie) v ((d−i)e) = v (de) = 0, and v (ie) v ( je) = 0 if i + j > d. By Theorem 3.4, it is easy to determine the structure of the adjacency algebra for a given parameters pa and d. But it seems to be complicated to state the structure in general. So we determine the structure for some special cases. We denote by K Cyc( pa , d) the adjacency algebra of Cyc( pa , d) over a field K . We remark that K Cyc( pa , d) is a commutative local symmetric algebra if the characteristic of K is p by [5]. 3.1 K Cyc( p a , d) and K Cyc( p b , d) are isomorphic For coprime numbers d and p, we denote by ordd ( p) the smallest positive integer such that p ordd ( p) ≡ 1 (mod d). Then pa ≡ 1 (mod d) if and only if ordd ( p) | a. We will show the next theorem. Theorem 3.6 Let K be an algebraically closed field of characteristic p. Suppose that d | pa − 1 and d | p b − 1. Then K Cyc( pa , d) ∼ = K Cyc( p b , d). Proof We may suppose that a = ordd ( p) and a | b. Put e = ( pa − 1)/d and e = ( p b − 1)/d. For j = 0, 1, . . . , d, write je =

a−1 

ei, j pi ,

je =

i=0

b−1 

ei, j pi ,

i=0

where 0 ≤ ei, j , ei, j ≤ p − 1. Then b/a−1 a−1   =0 i=0

ei, j p a+i =

b/a−1  =0

p a

a−1  i=0

ei, j pi =

pb − 1 · je pa − 1

pb − 1 pa − 1 · = je . = j· a p −1 d So  en+i, j = ei, j

for 0 ≤ i ≤ a − 1 and 0 ≤  ≤ b/a − 1 by uniqueness of the expression. Now it   is easy to see that v ( je) → v ( je ) is an isomorphism, where {v ( je) } and {v ( je ) } are

bases of K Cyc( pa , d) and K Cyc( p b , d), respectively. By Theorem 3.6, it is enough to consider K Cyc( p ordd ( p) , d) to determine the structure of the adjacency algebra K Cyc( pa , d).

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Remark In [12], Yoshikawa proved that adjacency algebras of Hamming schemes are isomorphic for some parameters. For this case, intersection numbers (structure u are congruent modulo p. constants) pst For Theorem 3.6, intersection numbers are not necessarily congruent modulo p. For example, consider intersection numbers of Cyc(7, 3) and Cyc(72 , 3) modulo 7. 3.2 K Cyc( p a , d) with p ≡ 1 (mod d) Theorem 3.7 Let K be an algebraically closed field of characteristic p. Suppose that p ≡ 1 (mod d). Then K Cyc( pa , d) ∼ = K [x]/(x d+1 ). Proof We may assume that a = ordd ( p) = 1. Put e = ( p − 1)/d. Then v (ie) = v1ie for i = 1, 2, . . . , d, and we have the result.

3.3 K Cyc( p a , d) with p ≡ −1 (mod d) Theorem 3.8 Let K be an algebraically closed field of characteristic p. Suppose that d = 2 and p ≡ −1 (mod d). Then K Cyc( pa , d) ∼ = K [x1 , . . . , xd−1 ]/I , where I is the ideal generated by xi x j (i + j = d), xi x j − xk x (i + j = k +  = d). Proof We may assume that a = ordd ( p) = 2. In Lemma 3.5, put αi =

i p − (d − i) d

for i = 1, 2, . . . , d −1. Then αi is an integer and 1 ≤ αi ≤ p−1 by p ≡ −1 (mod d). α Easily we have αi + αd−i p = ie, where e = ( p 2 − 1)/d. So v (ie) = v1αi v2 d−i . Since (ie) ( je) (ie) ((d−i)e) = 0 if i + j = d and v v = v (de) (= 0). αi + αd−i = p − 1, v v Comparing the dimensions, we have the result.

Remark The cases p ≡ ±1 (mod d) give two extreme structures of K Cyc( pa , d). By Theorem 3.4, K Cyc( pa , d) is determined by v (ie) v ( je) (0 ≤ i, j ≤ d). Always v (0) v ( je) = v ( je) = 0, v (ie) v ((d−i)e) = v (de) = 0, and v (ie) v ( je) = 0 if i + j > d. We proved (1) v (ie) v ( je) = 0 for all i + j < d, if p ≡ 1 (mod d), and (2) v (ie) v ( je) = 0 for all i ≥ 1, j ≥ 1, i + j < d, if p ≡ −1 (mod d). Easily we can see that (1) occurs only if p ≡ 1 (mod d). But (2) can occur for the other case (e.g. see Theorem 3.12). 3.4 K Cyc( p a , d) for d = 2, 3, 4, 5, 6 In this subsection, we will determine the structure of Cyc( pa , d) for d = 2, 3, 4, 5, 6. They are not so difficult by Theorems 3.4, 3.6, 3.7, and 3.8.

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Theorem 3.9 (Case d = 2) Let K be an algebraically closed field of characteristic p. For an odd prime p, we have K Cyc( pa , 2) ∼ = K [x]/(x 3 ). Proof For this case, 2 | p − 1. So we can apply Theorem 3.7.



Theorem 3.10 (Case d = 3) Let K be an algebraically closed field of characteristic p. (1) If p ≡ 1 (mod 3), then K Cyc( pa , 3) ∼ = K [x]/(x 4 ). (2) If p ≡ 2 (mod 3), then

  K Cyc pa , 3 ∼ = K [x, y]/ x 2 , y 2 .



Proof (1) holds by Theorem 3.7. (2) holds by Theorem 3.8.

Theorem 3.11 (Case d = 4) Let K be an algebraically closed field of characteristic p. (1) If p ≡ 1 (mod 4), then K Cyc( pa , 4) ∼ = K [x]/(x 5 ). (2) If p ≡ 3 (mod 4), then

  K Cyc pa , 4 ∼ = K [x, y, z]/ x 2 , y 3 , z 2 , x z, yz .



Proof (1) holds by Theorem 3.7. (2) holds by Theorem 3.8.

Theorem 3.12 (Case d = 5) Let K be an algebraically closed field of characteristic p. (1) If p ≡ 1 (mod 5), then K Cyc( pa , 5) ∼ = K [x]/(x 6 ). (2) If p ≡ 2, 3, 4 (mod 5), then

  K Cyc pa , 5 ∼ = K [x, y, z, u]/ x 2 , y 2 , z 2 , u 2 , x y, x z, yu, zu, xu − yz . Proof (1) holds by Theorem 3.7. For the case p ≡ 4 (mod 5), we can apply Theorem 3.8. Suppose p ≡ 2 (mod 5). We may assume that a = ord5 ( p) = 4. Put α1 =

p−2 2p − 4 3p − 1 4p − 3 , α2 = , α3 = , α4 = . 5 5 5 5

Then e = α3 + α4 p + α2 p 2 + α1 p 3 , 2e = α1 + α3 p + α4 p 2 + α2 p 3 , 3e = α4 + α2 p + α1 p 2 + α3 p 3 , 4e = α2 + α1 p + α3 p 2 + α4 p 3 . We can check the relations.

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Suppose p ≡ 3 (mod 5). We may assume that a = ord5 ( p) = 4. Put α1 =

p−3 2p − 1 3p − 4 4p − 2 , α2 = , α3 = , α4 = . 5 5 5 5

Then e = α2 + α4 p + α3 p 2 + α1 p 3 , 2e = α4 + α3 p + α1 p 2 + α2 p 3 , 3e = α1 + α2 p + α4 p 2 + α3 p 3 , 4e = α3 + α1 p + α2 p 2 + α4 p 3 . We can check the relations.



Theorem 3.13 (Case d = 6) Let K be an algebraically closed field of characteristic p. (1) If p ≡ 1 (mod 6), then K Cyc( pa , 6) ∼ = K [x]/(x 7 ). (2) If p ≡ 5 (mod 6), then K Cyc( pa , 6) ∼ = K [x, y, z, u, v]/I, where I is an ideal generated by x 2 , y 2 , z 3 , u 2 , v 2 , x y, x z, xu, yz, yv, zu, zv, xv − z 2 , yu − z 2 . Proof (1) holds by Theorem 3.7. (2) holds by Theorem 3.8.



4 Standard modules of cyclotomic schemes and p-ranks In this section, we will consider structures of standard modules and p-ranks of elements of adjacency algebras. Let (X, S) = Cyc( pa , d) = X(N  Hd , Hd ) be a cyclotomic scheme, and let K be an algebraically closed field of characteristic p. We recall that the adjacency algebra is KS = (KN) Hd and the standard module is K X = KN. So we want to know (KN) Hd -module structure of KN. Also we remark that the adjacency algebra KS is local. So, if an element of KS is not in the Jacobson radical, then it is invertible and has full rank. We are interested in elements in the Jacobson radical. The ranks are closely related to the standard module. Theorem 3.4 gives good information to consider the structure of standard modules. But it is not so easy to determine the structure of the standard module, in general. We restrict our attention only to the cases (1) p ≡ 1 (mod d) and (2) d = 3. 4.1 Case p ≡ 1 (mod d) Let K be an algebraically closed field of characteristic p. Suppose that p ≡ 1 (mod d). By Theorem 3.7, the adjacency algebra K Cyc( pa , d) has basis {v (ie) | i = 0, 1, . . . , d}, where e = ( pa − 1)/d, and the multiplication is v (ie) v ( je) = v ((i+ j)e) if i + j ≤ d and 0 otherwise. Also

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i v (e) = v (ie) = (v1 v2 . . . va )i( p−1)/d for i = 0, 1, . . . , d and (v (e) )d+1 = 0. Proposition 4.1 We have rank K v (ie) = ( p − i( p − 1)/d)a for i = 0, 1, . . . , d. Proof We can see that rank K v (ie) = dim K (KN)v (ie) and (KN)v (ie) has a basis 

 v11 · · · vaa | i( p − 1)/d ≤  j ≤ p − 1 ( j = 1, . . . , a) .

So rank K v (ie) = ( p − i( p − 1)/d)a .



We remark that rank K v (de) = 1. We determine the structure of the standard module KN. Since KS = (KN) Hd ∼ = K [x]/(x d+1 ), indecomposable KS-modules are uniserial of length 1 to d + 1 (corresponding to the Jordan normal form of v (e) ). We denote them by U j ( j = 1, . . . , d +1). Also we denote the multiplicity of U j in the standard module KN by m j . Namely KN ∼ =

d+1 

m jUj.

j=1

Since v (ie) has rank j − i on U j ( j > i), we have  p−

i( p − 1) d

a

= rank K v (ie) =

d+1 

( j − i)m j .

j=i+1

The following theorem holds. Theorem 4.2 Let K be an algebraically closed field of characteristic p. Suppose p ≡ 1 (mod d). Then K Cyc( pa , d) ∼ = K [x]/(x d+1 ). We denote by U j the uniserial d+1 a K Cyc( p , d)-module of length j for j = 1, 2, . . . , d + 1. Then K X ∼ = j=1 m j U j , where the multiplicities m j are determined by equations 

i( p − 1) p− d

a =

d+1 

( j − i)m j (i = 1, . . . , d).

j=i+1

Remark that always m d+1 = 1. If a = 1, then we can compute all m j . Corollary 4.3 Suppose a = 1 in Theorem 4.2. Then m 1 = · · · = m d−1 = 0, m d = ( p − 1)/d − 1, and m d+1 = 1. Proof We know m d+1 = 1. By p − (d − 1)( p − 1)/d = m d + 2m d+1 , we have m d = ( p − 1)/d − 1. Then dm d + (d + 1)m d+1 = p = dim K KN and so m 1 = · · · =

m d−1 = 0.

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We can determine ranks of adjacency matrices. The p-rank means the rank of a matrix over a field of characteristic p. Corollary 4.4 Let (X, S) = Cyc( pa , d) be a cyclotomic scheme with p ≡ 1 (mod d). For s ∈ S\{1}, the p-rank of (σs − n s σ1 )i is ( p − i( p − 1)/d)a for i = 0, 1, . . . , d. Proof We know that {σs − n s σ1 | s ∈ S\{1}} is a basis of the Jacobson radical of KS, where K is a field of characteristic p. So σs − n s σ1 ∈ J (KS)\J 2 (KS) for some s ∈ S\{1}. Then rank K (σs −n s σ1 )i = rank K (v (e) )i (i = 0, 1, . . . , d). By Lemma 2.1,

rank K (σs − n s σ1 )i = rank K (v (e) )i for every s ∈ S\{1}. If d = 2 in Corollary 4.4, the result is just [3, Proposition in 4.1]. If d = 2 and a = 1, then the ranks are the same for all symmetric association schemes algebraically isomorphic to Cyc( p, 2) [9, Corollary 3.1]. Example 4.5 (Association schemes algebraically isomorphic to Cyc(52 , 2)) There are 8 isomorphism classes of association schemes algebraically isomorphic to Cyc(52 , 2) (order 25, No. 4 to 11 in the list [6]). Let (X, S) be one of them. Set w = σs − 12σ1 for some s ∈ S\{1}. Then rank(w) = 12 for No. 4, 5, 6, 7, 8, rank(w) = 11 for No. 9, 10, and rank(w) = 9 for No. 11. By Corollary 4.4, the cyclotomic scheme Cyc(52 , 2) is No. 11. It is characterized by rank(w) and has the smallest value, in this case. 4.2 Case d = 3 We consider (X, S) = Cyc( pa , 3). If p ≡ 1 (mod 3), then we can apply results in Sect. 4.1. We assume that p ≡ 2 (mod 3). Remark that a is even. Then K Cyc( pa , 3) ∼ = K [x, y]/(x 2 , y 2 ) by Theorem 3.10. There are infinitely many indecomposable modules and they are classified (see [2, 4.3], for example). We put α=

p−2 2p − 1 , β= 3 3

and x = v (e) =

a/2

β α v2i−1 v2i ,

y = v (2e) =

i=1

a/2

β α v2i−1 v2i . i=1

To describe results, we use diagrams like the following. (1 , 2 , . . .) ll l l lll lll vlll (1 + α, 2 + β, . . .)

( (1 + β, 2 + α, . . .)

In the diagram, (1 , 2 , . . .) means v11 v22 . . ., the arrow means multiplying x, and the dotted arrow means multiplying y. The diagram defines a representation of

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 +β

K Cyc( pa , d). Take a basis v11 v22 · · · , v11 +α v22 · · · , v11 v22 +α · · · . Then the representation is ⎛ ⎛ ⎞ ⎞ 010 001 x → ⎝0 0 0⎠ , y → ⎝0 0 0⎠ . 000 000 We define six modules (representations) : • ⎛ ⎞ ⎛ ~~ ~ 0100 001 ~ ~ ⎜0 0 0 0 ⎟ ⎜0 0 0  ~~ ⎟ ⎜ • x → ⎜ M1 : • ⎝0 0 0 1⎠ , y → ⎝0 0 0 ~ ~ ~ 0000 000 ~~  ~~ • ⎛ ⎛ ⎞ ⎞ • 010 001 ~ ~ ~ x → ⎝0 0 0⎠ , y → ⎝0 0 0⎠ M2 : ~~  ~~ 000 000 • • ⎛ ⎛ ⎞ ⎞ • • 0 0 1 0 0 0 ~ ~~ x  → ⎝0 0 0 ⎠ , y  → ⎝0 0 1 ⎠ M3 : ~~ ~  ~ 000 000 • •     ~ ~ 00 01 ~ M4 : , y → x → ~~ 00 0 0 ~ ~ • •     00 01 M5 : x → , y → . 0 0 00  •     • x → 0 , y → 0 M6 :

⎞ 0 1⎟ ⎟ 0⎠ 0

All of the modules (representations) above are indecomposable by [2, 4.3]. The vertices move over the range 0 ≤ i ≤ p − 1 (1 ≤ i ≤ a). If we take a connected diagram as large as possible, then the vector space spanned by the elements corresponding to the vertices is an KS-submodule of the standard module K X . So if we decompose all vertices into connected diagrams, we can get the indecomposable direct sum decomposition of K X . It is easy to see that only the six diagrams above are possible. For example, the diagram

•

~~ ~~ ~ ~ ~

• 

•

~~ ~~ ~ ~ ~

•

is impossible, because if we put the lower left vertex (1 , 2 , . . .), then the upper right vertex is (1 − p, 2 + 1, . . .), but 0 ≤ 1 , 1 − p ≤ p − 1 is impossible.

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Now we can determine the structure of the standard module. Theorem 4.6 Let K be an algebraically closed field of characteristic p. Suppose p ≡ 2 (mod 3). Then K Cyc( pa , d) ∼ = K [x, y]/(x 2 , y 2 ). The standard module is  6 i=1 m i Mi , where ( p + 1)a m 1 = 1, m 2 = m 3 = − 1, a  3 a   a/2  a/2+2 − 2 ( p + 1)a 2 − 2 ( p + 1) 2 a m4 = m5 = , m6 = p + 2 − . 3a 3a Proof Obviously, there is only one summand M1 with the top vertex (0, . . . , 0). The vertex (1 , . . . , a ) is a top vertex of M2 if and only if 0 ≤ i ≤ α for all i = 1, 2, . . . , a except the case M1 . So m 2 = (α +1)a −1 = ( p +1)a /3a −1. By symmetry, m 3 = m 2 . The vertex (1 , . . . , a ) is a top vertex of M4 if and only if 0 ≤ 2i−1 ≤ α, 0 ≤ 2i ≤ β for all i = 1, 2, . . . , a/2 except the cases M1 , M2 , M3 . So m 4 = (α + 1)

a/2

(β + 1)

a/2

   − 2 − 2 αa + 1 − 1 =



p+1 3

a

2a/2 − 2 .

By symmetry, we have m 5 = m 4 . Counting the number of all vertices, we have m 6 . We determine the p-rank of σs − n s σ1 for s ∈ S\{1}. Corollary 4.7 Let (X, S) be the cyclotomic scheme Cyc( pa , 3) with p ≡ 2 (mod 3). Then, for s ∈ S\{1}, the p-rank of σs − n s σ1 is 2a/2 ( p + 1)a /3a .  Proof Note that J 2 (KS) = K t∈S σt = K x y and σs − n s σ1 ∈ J (KS)\J 2 (KS). So we can write σs − n s σ1 = cx x + c y y + cx y x y with cx , c y , cx y ∈ K , (cx , c y ) = (0, 0). For the representation of M1 , we have ⎛ 0 ⎜0 σs − n s σ1 → ⎜ ⎝0 0

cx 0 0 0

cy 0 0 0

⎞ cx y cy ⎟ ⎟, cx ⎠ 0

and the rank is 2. Similarly, we have the rank 1 for M2 , M3 , and 0 for M6 . Since m 4 = m 5 , consider the sum M4 ⊕ M5 , ⎛ 0 ⎜0 σs − n s σ1 → ⎜ ⎝0 0

cx 0 0 0

0 0 0 0

⎞ 0 0⎟ ⎟. cy ⎠ 0

The rank for this representation is 1 if cx = 0 or c y = 0 and 2 otherwise. Since the ranks of σt − n t σ1 are constant for t ∈ S\{1} by Lemma 2.1 and |S| = 4, we can see

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that the rank is 2. By Theorem 4.6, the rank of σs −n s σ1 for the standard representation is  2+2



( p + 1)a 2a/2 ( p + 1)a ( p + 1)a a/2 − 1 + 2 2 − 2 = a a 3 3 3a



and we have the result.

Example 4.8 (Association schemes algebraically isomorphic to Cyc(24 , 3)) In the list in [6], Order 16 No. 20 is the cyclotomic scheme Cyc(24 , 3) and No. 21 is a nonschurian scheme algebraically isomorphic to Cyc(24 , 3). For both cases, rank K (σs − n s σs ) = dim K K X (σs − n s σs ) = 6 for any s ∈ S\{1}. But for s, t ∈ S\{1}, s = t, dim K (K X )J (KS) = dim K (K X (σs − n s σs ) + K X (σt − n s σt )) = 7 for No. 20 and 8 for No. 21. The difference comes from the structure of the standard modules. For the cyclotomic scheme No. 20, we can compute the dimension of K X (σs − n s σs ) + K X (σt −n s σt ) to be 7 by Theorem 4.6. The standard module is M1 ⊕2M4 ⊕2M5 ⊕4M6 , and dim K (M1 (σs − n s σs ) + M1 (σt − n s σt )) = 3, dim K (M4 (σs − n s σs ) + M4 (σt − n s σt )) = 1, dim K (M5 (σs − n s σs ) + M5 (σt − n s σt )) = 1, and dim K (M6 (σs − n s σs ) + M6 (σt − n s σt )) = 0. So K X (σs − n s σs ) + K X (σt − n s σt ) = 7. Example 4.9 (Association schemes algebraically isomorphic to Cyc(52 , 3)) In the list in [6], Order 25 No. 18 is the cyclotomic scheme Cyc(52 , 3) and No. 17 is a cases, rank K (σs − schurian scheme algebraically isomorphic to Cyc(52 , 3). For both n s σs ) = dim K K X (σs − n s σs ) = 8 for any s ∈ S\{1}. But dim K ( s∈S\{1} K X (σs − n s σ1 )) = 3 for No. 17 and 4 for No. 18. The difference comes from the structure of the standard modules. For the cyclotomic scheme No. 18, we can compute the dimension to be 4 by Theorem 4.6. The standard module  is M1 ⊕ 3M2 ⊕ 3M3 ⊕ dim ( M (σ − n σ )) = 1, dim ( 3M6 , and K 1 s s 1 K s∈S\{1} s∈S\{1} M2 (σs − n s σ1 )) = 0,   dim K (s∈S\{1} M3 (σs − n s σ1 )) = 1, and dim K ( s∈S\{1} M6 (σs − n s σ1 )) = 0. So dim K ( s∈S\{1} K X (σs − n s σ1 )) = 4. Acknowledgments

The author would like to thank the referee and the editor for their comments.

References 1. Bannai, E., Ito, T.: Algebraic Combinatorics. I. The Benjamin/Cummings Publishing Co. Inc., Menlo Park, CA (1984) 2. Benson, D.J.: Representations and Cohomology. I, Cambridge Studies in Advanced Mathematics, Vol. 30. Cambridge University Press, Cambridge (1991) 3. Brouwer, A.E., van Eijl, C.A.: On the p-rank of the adjacency matrices of strongly regular graphs. J. Algebr. Combin. 1(4), 329–346 (1992) 4. Goldbach, R.W., Claasen, H.L.: Cyclotomic schemes over finite rings. Indag. Math. (N.S.) 3(3), 301– 312 (1992) 5. Hanaki, A.: Locality of a modular adjacency algebra of an association scheme of prime power order. Arch. Math. (Basel) 79(3), 167–170 (2002) 6. Hanaki, A., Miyamoto, I.: Classification of association schemes with small vertices. http://math. shinshu-u.ac.jp/~hanaki/as/ 7. Hanaki, A., Yoshikawa, M.: On modular standard modules of association schemes. J. Algebr. Combin. 21(3), 269–279 (2005)

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J Algebr Comb 8. Kasami, T., Lin, S., Peterson, W.W.: Some results on cyclic codes which are invariant under the affine group and their applications. Inf. Control 11, 475–496 (1967) 9. Peeters, R.: Uniqueness of strongly regular graphs having minimal p-rank. Linear Algebra Appl. 226/228, 9–31 (1995). MR 1344551 10. Peeters, R.: On the p-ranks of the adjacency matrices of distance-regular graphs. J. Algebr. Combin. 15(2), 127–149 (2002) 11. Shimabukuro, O., Yoshikawa, M.: Modular adjacency algebras of Grassmann graphs. Linear Algebra Appl. 466, 208–217 (2015) 12. Yoshikawa, M.: Modular adjacency algebras of Hamming schemes. J. Algebr. Combin. 20(3), 331–340 (2004) 13. Yoshikawa, M.: The modular adjacency algebras of non-symmetric imprimitive association schemes of class 3. J. Fac. Sci. Shinshu Univ. 40(2005), 27–36 (2006) 14. Zieschang, P.-H.: An algebraic approach to association schemes. Lecture Notes in Mathematics, Vol. 1628. Springer, Berlin (1996)

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