Normal cyclotomic schemes over a Galois ring

Transcription

1 St.Petersburg Department of Steklov Mathematical Institute Russia (mostly joint work with Ilia Ponomarenko) The 11th International Conference on Finite Fields Friday, July 26, Magdeburg

2 Outline Cyclotomic schemes over a field 1 Cyclotomic schemes over a field Definition Automorphism group 2 Finite commutative rings Normality 3 Main Theorem Sketch of proof

3 Definition (Delsarte, 1973) Definition Automorphism group Let F = F q be a finite field of order q > 1 and K a subgroup of index m in its multiplicative group F. For each a F set E a = {(x, y) F F : x y ak }. Under a cyclotomic scheme C = Cyc(K, F) over F we mean the pair (F, {E a : a F}). Any such C is an association scheme, rk(c) = 1 + m. Moreover, C is a Cayley scheme over the additive group of F. The intersection numbers of C are cyclotomic numbers. The explicit evaluation of them is a hard number-theoretic problem.

4 McConnel Theorem (1963) Definition Automorphism group For a cyclotomic scheme C = Cyc(K, F) set Aut(C) = {f Sym(F) : (E a ) f = E a, a F}. Denote by AΓL 1 (F) the group consisting of all 1-dimensional semi-affine transformations of F: AΓL 1 (F) = {x ax σ + b : a F, b F, σ Aut(F)}. Theorem If rk(c) > 2, then Aut(C) AΓL 1 (F). Note. If rk(c) = 2, then Aut(C) = Sym(F).

5 Finite commutative rings Normality Decompositions in a finite commutative ring Let R be a finite commutative ring with identity. Then R = i R i where R i is a local ring for all i. If R is a local ring, then R/ rad(r) =: F is a field. Moreover, R = T U where T is the Teichmüller group of R, and U is the group of its principal units. The group T is isomorphic to F whereas U = 1 + rad(r) is a p-group where p = char(f).

6 Galois rings Cyclotomic schemes over a field Finite commutative rings Normality Definition A finite local ring R is Galois if rad(r) = pr. All Galois rings with char(r) = p n and F = p d are isomorphic. Any of them is denoted by GR(p n, d). R + is a homocyclic p-group of exponent p n and rank d, I(R) = {p i R : 0 i n}, the quotient homomorphism Aut(R) Aut(F) is a bijection, R is additively generated by T, n 1 x = x i p i, x i T {0}. i=0 Examples: GR(p n, 1) = Z p n, GR(p, d) = F p d.

7 Finite commutative rings Normality Multiplicative group of a Galois ring Let C m denote a cyclic group of order m. Theorem If R = GR(p n, d), then R = T U where T = C p d 1 If p is odd, then U = (C p n 1) d, If p = 2, then { U 1, if n = 1 = C 2 C 2 n 2 (C 2 n 1) d 1, if n > 1. Example: If R = GR(2 n, 1) = Z 2 n, then R = 1 5.

8 Finite commutative rings Normality Cyclotomic schemes over a ring (Goldbach, Claasen) Let R be a finite commutative ring with identity and let K be a subgroup of index m in its multiplicative group R. For each a R set E a = {(x, y) R R : x y ak }. Under a cyclotomic scheme C = Cyc(K, R) over R we mean the pair (R, {E a : a R}). Any such C is an association scheme, rk(c) = 1 + m. Moreover, C is a Cayley scheme over the additive group of R.

9 Normal schemes Cyclotomic schemes over a field Finite commutative rings Normality Let C = Cyc(K, R) with K R be a cyclotomic scheme over a finite commutative ring R. Definition The scheme C is normal if Aut(C) AΓL 1 (R). McConnel s Theorem implies the following statement. Theorem Let R = F q be a field. Then the scheme C is normal if and only if rk(c) > 2 or q = 2, 3, 4. Proof. If rk(c) = 2, then Aut(C) = Sym(F q ), and consequently Aut(C) AΓL 1 (F q ) q = 2, 3, 4.

10 Reduction to the local case Finite commutative rings Normality Let R = i R i be a finite commutative ring and C = Cyc(K, R) a cyclotomic scheme over R where K R. For each i set C i = Cyc(K i, R i ) where K i = (ϕ i ) 1 (K ) with ϕ i the natural embedding of R i into R : x (1,..., x,..., 1). Theorem The scheme C is normal if and only if the scheme C i is normal for all i.

11 Main Theorem Sketch of proof Pure and quasipure subgroups of R Let R be a commutative ring with identity. For a group K R denote by I K the largest of all ideals I of R such that K + I = K (equivalently, 1 + I K ). Definition The group K is said to be pure if I K = 0; it is said to be quasipure if I K I 0 where I 0 = ann(rad(r)). If R is a Galois ring of characteristic p n, then I 0 = p n 1 R.

12 Statements Cyclotomic schemes over a field Main Theorem Sketch of proof Theorem Let R = GR(p n, d) be a Galois ring other than a field. Then the scheme Cyc(K, R) is normal if and only if the group K is pure for q > 2 and quasipure for q = 2. Corollary If R is not isomorphic to Z 2 n, then Cyc(K, R) is normal if and only if K is pure.

13 Necessity Cyclotomic schemes over a field Main Theorem Sketch of proof Let R be a local ring, C = Cyc(K, R) and I 0 = ann(rad(r)). Theorem Suppose that C is a normal scheme and K = K + I for some ideal I of R. Then I = 0 whenever q > 2. Moreover, if q = 2, then I I 0. Proof. Assuming I 0, one can show that C is the generalized wreath product of smaller schemes and consequently the group Aut(C) contains a subgroup which is not inside AΓL 1 (R) unless q = 2 and I I 0. Cyclotomic S-ring over G = R + corresponding to C: A = span{σx; X Orb(K, G)} ZG.

14 Sufficiency: q = 2 Main Theorem Sketch of proof If q = 2, then R = Z 2 n and I 0 = 2 n 1 R. Theorem Let R = Z 2 n and I K I 0. Then K belongs to one of the following families of groups: {1}, n 1, {1, 1}, n 2, {1, 2 n 1 + 1}, n 3, {1, 2 n 1 1}, n 3, {1, 2 n 1 1, 2 n 1 + 1, 1}, n 4, {1, 2 n 2 1, 2 n 1 + 1, 3 2 n 2 1}, n 4. Moreover, the scheme Cyc(K, R) is normal.

15 Sufficiency: q > 2 Main Theorem Sketch of proof All we need to prove is the following statement: if I K = 0, then the scheme Cyc(K, R) is normal. Let π : R R/I 0 be the quotient epimorphism. Theorem Let K R be a pure group, C = Cyc(K, R), and C = Cyc(K, R ) where K = π(k ) and R = π(r). Then the scheme C is normal whenever so is the scheme C. Theorem If R = GR(p n, d), then the group K is pure whenever n > 2 for an odd p and n > 3 for p = 2. Thus the proof of sufficiency is reduced to the following cases: n = 2, p is arbitrary and n = 3, p = 2.

16 Sufficiency Cyclotomic schemes over a field Main Theorem Sketch of proof It suffices to prove that Aut(C) 0,1 Aut(R) where Aut(C) 0,1 = {γ Aut(C) : 0 γ = 0, 1 γ = 1}. Lemma Let R be a local commutative ring. Suppose that a permutation γ Sym(R) fixes 0 and 1 and normalizes the group AGL 1 (R). Then γ Aut(R). Multiplication S-ring of the scheme C: by definition it equals the S-ring over the group R that corresponds to the Cayley scheme ((C 0 ) R ) R. In general this S-ring is not cyclotomic.

17 Papers Papers I R. McConnel. Pseudo-ordered polynomials over a finite field. Acta Arith. 8: , B. R. McDonald. Finite rings with identity. Pure and Applied Mathematics, Vol. 28, Marcel Dekker Inc., New York, S. Evdokimov, I. Ponomarenko. Normal cyclotomic schemes over a finite commutative ring. Algebra & Analysis 19(6):58 84, 2007.