Group rings of finite strongly monomial groups: central units and primitive idempotents

Transcription

1 Group rings of finite strongly monomial groups: central units and primitive idempotents joint work with E. Jespers, G. Olteanu and Á. del Río Inneke Van Gelder Vrije Universiteit Brussel Recend Trends in Rings and Algebras, Murcia, June 3-7, 2013

2 Outline Inneke Van Gelder (Brussels) Central units and primitive idempotents Murcia, June 3-7, / 26

3 Outline Background information on units Inneke Van Gelder (Brussels) Central units and primitive idempotents Murcia, June 3-7, / 26

4 Outline Background information on units Virtual basis of Z(U(ZG)) Inneke Van Gelder (Brussels) Central units and primitive idempotents Murcia, June 3-7, / 26

5 Outline Background information on units Virtual basis of Z(U(ZG)) Matrix units of each simple component in the rational group algebra QG Inneke Van Gelder (Brussels) Central units and primitive idempotents Murcia, June 3-7, / 26

6 Outline Background information on units Virtual basis of Z(U(ZG)) Matrix units of each simple component in the rational group algebra QG Description of subgroup of finite index in U(ZG) for a class of metacyclic groups Inneke Van Gelder (Brussels) Central units and primitive idempotents Murcia, June 3-7, / 26

7 Group rings Background Let G be a group and R a ring. Inneke Van Gelder (Brussels) Central units and primitive idempotents Murcia, June 3-7, / 26

8 Group rings Background Let G be a group and R a ring. RG is the set of all linear combinations r g g, g G with r g R and r g 0 for only finitely many coefficients. Inneke Van Gelder (Brussels) Central units and primitive idempotents Murcia, June 3-7, / 26

9 Group rings Background Let G be a group and R a ring. RG is the set of all linear combinations r g g, g G with r g R and r g 0 for only finitely many coefficients. Operations: g G r g g + s g g = g + s g )g g G g G(r r g g s g g = r g s h gh g G g G g,h G Inneke Van Gelder (Brussels) Central units and primitive idempotents Murcia, June 3-7, / 26

10 Units Background Let R be a ring with unity 1. Inneke Van Gelder (Brussels) Central units and primitive idempotents Murcia, June 3-7, / 26

11 Units Background Let R be a ring with unity 1. U(R) is the set of all units of R: U(R) = {r R s R : rs = 1 = sr}. Inneke Van Gelder (Brussels) Central units and primitive idempotents Murcia, June 3-7, / 26

12 Background Units Let R be a ring with unity 1. U(R) is the set of all units of R: U(R) = {r R s R : rs = 1 = sr}. Example Let G be a finite group. U(ZG) =? Inneke Van Gelder (Brussels) Central units and primitive idempotents Murcia, June 3-7, / 26

13 Example: Bass units Background Let G be a group, g an element of G of order n and k and m positive integers such that k m 1 mod n. The Bass (cyclic) unit with parameters g, k, m is the element in ZG u k,m (g) = (1 + g + + g k 1 ) m + (1 km ) (1 + g + g g n 1 ) n with inverse in ZG (1 + g k + + g k(i 1) ) m + (1 i m ) (1 + g + g g n 1 ) n where i is any integer such that ki 1 mod n. Inneke Van Gelder (Brussels) Central units and primitive idempotents Murcia, June 3-7, / 26

14 Background Example: Bass units Let G be a group, g an element of G of order n and k and m positive integers such that k m 1 mod n. The Bass (cyclic) unit with parameters g, k, m is the element in ZG u k,m (g) = (1 + g + + g k 1 ) m + (1 km ) (1 + g + g g n 1 ) n with inverse in ZG (1 + g k + + g k(i 1) ) m + (1 i m ) (1 + g + g g n 1 ) n where i is any integer such that ki 1 mod n. Theorem [Bass-Milnor] If G is a finite abelian group, then the Bass units generate a subgroup of finite index in U(ZG). Inneke Van Gelder (Brussels) Central units and primitive idempotents Murcia, June 3-7, / 26

15 Background Wedderburn decomposition The Wedderburn decomposition of QG is the decomposition into simple algebras A 1 A k. Inneke Van Gelder (Brussels) Central units and primitive idempotents Murcia, June 3-7, / 26

16 Background Wedderburn decomposition The Wedderburn decomposition of QG is the decomposition into simple algebras A 1 A k. Each simple component is determined by a primitive central idempotent e i, i.e. A i = QGe i. Inneke Van Gelder (Brussels) Central units and primitive idempotents Murcia, June 3-7, / 26

17 Strong Shoda pairs Background If K H G and K H then ε(h, K) = ( K M) = K (1 M), where M runs through the set of all minimal normal subgroups of H containing K properly. Inneke Van Gelder (Brussels) Central units and primitive idempotents Murcia, June 3-7, / 26

18 Strong Shoda pairs Background If K H G and K H then ε(h, K) = ( K M) = K (1 M), where M runs through the set of all minimal normal subgroups of H containing K properly. We set ε(h, H) = Ĥ. Inneke Van Gelder (Brussels) Central units and primitive idempotents Murcia, June 3-7, / 26

19 Background Strong Shoda pairs If K H G and K H then ε(h, K) = ( K M) = K (1 M), where M runs through the set of all minimal normal subgroups of H containing K properly. We set ε(h, H) = Ĥ. A strong Shoda pair of G is a pair (H, K) of subgroups of G such that K H N G (K) H/K is cyclic and a maximal abelian subgroup of N G (K)/K the different G-conjugates of ε(h, K) are orthogonal. Inneke Van Gelder (Brussels) Central units and primitive idempotents Murcia, June 3-7, / 26

20 Background Strongly monomial groups Let χ be an irreducible (complex) character of G. Inneke Van Gelder (Brussels) Central units and primitive idempotents Murcia, June 3-7, / 26

21 Background Strongly monomial groups Let χ be an irreducible (complex) character of G. χ is strongly monomial if there is a strong Shoda pair (H, K) of G and a linear character θ of H with kernel K such that χ = θ G. Inneke Van Gelder (Brussels) Central units and primitive idempotents Murcia, June 3-7, / 26

22 Background Strongly monomial groups Let χ be an irreducible (complex) character of G. χ is strongly monomial if there is a strong Shoda pair (H, K) of G and a linear character θ of H with kernel K such that χ = θ G. The group G is strongly monomial if every irreducible character of G is strongly monomial. Inneke Van Gelder (Brussels) Central units and primitive idempotents Murcia, June 3-7, / 26

23 Background Strongly monomial groups Let χ be an irreducible (complex) character of G. χ is strongly monomial if there is a strong Shoda pair (H, K) of G and a linear character θ of H with kernel K such that χ = θ G. The group G is strongly monomial if every irreducible character of G is strongly monomial. Example Abelian-by-supersolvable groups are strongly monomial. Inneke Van Gelder (Brussels) Central units and primitive idempotents Murcia, June 3-7, / 26

24 Background The Wedderburn decomposition for strongly monomial groups Theorem [Olivieri-del Río-Simón] Let G be a finite strongly monomial group, then the Wedderburn decomposition is as follows: QG = QGe(G, H, K) = M n (Q(ξ k ) N G (K)/H) (H,K) (H,K) with (H, K) running on strong Shoda pairs of G. Inneke Van Gelder (Brussels) Central units and primitive idempotents Murcia, June 3-7, / 26

25 Virtual basis Virtual basis of the center It is well known that Z(U(ZG)) = ±Z(G) T, where T is a finitely generated free abelian group. Inneke Van Gelder (Brussels) Central units and primitive idempotents Murcia, June 3-7, / 26

26 Virtual basis Virtual basis of the center It is well known that Z(U(ZG)) = ±Z(G) T, where T is a finitely generated free abelian group. A virtual basis of Z(U(ZG)) is a set of multiplicatively independent elements of Z(U(ZG)) which generate a subgroup of finite index in Z(U(ZG)). Inneke Van Gelder (Brussels) Central units and primitive idempotents Murcia, June 3-7, / 26

27 Rank Virtual basis of the center Strategy: We know how many elements a virtual basis of Z(U(ZG)) should have. Inneke Van Gelder (Brussels) Central units and primitive idempotents Murcia, June 3-7, / 26

28 Rank Virtual basis of the center Strategy: We know how many elements a virtual basis of Z(U(ZG)) should have. Theorem [Jespers-del Río-Olteanu-VG] Let G be a finite strongly monomial group. Then the rank of Z(U(ZG)) equals ( ) ϕ([h : K]) k (H,K) [N : H] 1, (H,K) where (H, K) runs through a complete and non-redundant set of strong Shoda pairs of G, h is such that H = h, K and { 1, if hh k (H,K) = n K for some n N G (K); 2, otherwise. Inneke Van Gelder (Brussels) Central units and primitive idempotents Murcia, June 3-7, / 26

29 Main strategy Virtual basis of the center Take an arbitrary central unit u in Z(U(ZG)) QG. Inneke Van Gelder (Brussels) Central units and primitive idempotents Murcia, June 3-7, / 26

30 Main strategy Virtual basis of the center Take an arbitrary central unit u in Z(U(ZG)) QG. We can write u as follows u = ue(g, H, K) = (1 e(g, H, K) + ue(g, H, K)). (H,K) (H,K) Inneke Van Gelder (Brussels) Central units and primitive idempotents Murcia, June 3-7, / 26

31 Main strategy Virtual basis of the center Take an arbitrary central unit u in Z(U(ZG)) QG. We can write u as follows u = ue(g, H, K) = (1 e(g, H, K) + ue(g, H, K)). (H,K) (H,K) Hence it is necessary and sufficient to construct a virtual basis in each Z(ZGe(G, H, K)+Z(1 e(g, H, K))) Z[ζ [H:K] ] N G (K)/H +Z(1 e(g, H, K)). Inneke Van Gelder (Brussels) Central units and primitive idempotents Murcia, June 3-7, / 26

32 Main strategy Virtual basis of the center Take an arbitrary central unit u in Z(U(ZG)) QG. We can write u as follows u = ue(g, H, K) = (1 e(g, H, K) + ue(g, H, K)). (H,K) (H,K) Hence it is necessary and sufficient to construct a virtual basis in each Z(ZGe(G, H, K)+Z(1 e(g, H, K))) Z[ζ [H:K] ] N G (K)/H +Z(1 e(g, H, K)). Strategy 1 Compute a basis of U(Z[ζ [H:K] ] N G (K)/H ) 2 Compute units in ZG projecting to the basis in U(Z[ζ [H:K] ] N G (K)/H ) and trivially in the other components. Inneke Van Gelder (Brussels) Central units and primitive idempotents Murcia, June 3-7, / 26

33 Virtual basis of the center Step 1: Cyclotomic units If n > 1 and k is an integer coprime with n then η k (ζ n ) = 1 ζk n = 1 + ζ n + ζn ζn k 1 1 ζ n is a unit of Z[ζ n ]. These units are called the cyclotomic units of Q(ζ n ). Inneke Van Gelder (Brussels) Central units and primitive idempotents Murcia, June 3-7, / 26

34 Virtual basis of the center Step 1: Cyclotomic units If n > 1 and k is an integer coprime with n then η k (ζ n ) = 1 ζk n = 1 + ζ n + ζn ζn k 1 1 ζ n is a unit of Z[ζ n ]. These units are called the cyclotomic units of Q(ζ n ). Theorem [Washington] {η k (ζ p n) 1 < k < pn 2, p k} generates a free abelian subgroup of finite index in U(Z[ζ p n]) when p is prime. Inneke Van Gelder (Brussels) Central units and primitive idempotents Murcia, June 3-7, / 26

35 Virtual basis of the center Step 1: Cyclotomic units For a subgroup A of Gal(Q(ζ p n)/q) and u Q(ζ p n), we define π A (u) to be σ A σ(u). Inneke Van Gelder (Brussels) Central units and primitive idempotents Murcia, June 3-7, / 26

36 Virtual basis of the center Step 1: Cyclotomic units For a subgroup A of Gal(Q(ζ p n)/q) and u Q(ζ p n), we define π A (u) to be σ A σ(u). Lemma [Jespers-del Río-Olteanu-VG] Let A be a subgroup of Gal(Q(ζ p n)/q). Let I be a set of coset representatives of U(Z/p n Z) modulo A, φ 1 containing 1. Then the set is a virtual basis of U ( Z[ζ p n] A). {π A (η k (ζ p n)) k I \ {1}} Inneke Van Gelder (Brussels) Central units and primitive idempotents Murcia, June 3-7, / 26

37 Virtual basis of the center Step 2: Generalized Bass units If G is a finite group, M a normal subgroup of G, g G and k and m positive integers such that gcd(k, g ) = 1 and k m 1 mod g. Inneke Van Gelder (Brussels) Central units and primitive idempotents Murcia, June 3-7, / 26

38 Virtual basis of the center Step 2: Generalized Bass units If G is a finite group, M a normal subgroup of G, g G and k and m positive integers such that gcd(k, g ) = 1 and k m 1 mod g. Then we have u k,m (1 M + g M) = 1 M + u k,m (g) M. Observe that any element b = u k,m (1 M + g M) is an invertible element of ZG(1 M) + ZG M. As this is an order in QG, there is a positive integer n such that b n U(ZG). Inneke Van Gelder (Brussels) Central units and primitive idempotents Murcia, June 3-7, / 26

39 Virtual basis of the center Step 2: Generalized Bass units If G is a finite group, M a normal subgroup of G, g G and k and m positive integers such that gcd(k, g ) = 1 and k m 1 mod g. Then we have u k,m (1 M + g M) = 1 M + u k,m (g) M. Observe that any element b = u k,m (1 M + g M) is an invertible element of ZG(1 M) + ZG M. As this is an order in QG, there is a positive integer n such that b n U(ZG). Let n G,M denote the minimal positive integer satisfying this condition for all g G. Then we call the element u k,m (1 M + g M) n G,M = u k,mng,m (1 M + g M) a generalized Bass unit based on g and M with parameters k and m. Inneke Van Gelder (Brussels) Central units and primitive idempotents Murcia, June 3-7, / 26

40 Virtual basis of the center Step 2: Generalized Bass units Let k be a positive integer coprime with p and let r be an arbitrary integer. Inneke Van Gelder (Brussels) Central units and primitive idempotents Murcia, June 3-7, / 26

41 Virtual basis of the center Step 2: Generalized Bass units Let k be a positive integer coprime with p and let r be an arbitrary integer. For every 0 j s n we construct recursively the following products of generalized Bass units of ZH: Inneke Van Gelder (Brussels) Central units and primitive idempotents Murcia, June 3-7, / 26

42 Virtual basis of the center Step 2: Generalized Bass units Let k be a positive integer coprime with p and let r be an arbitrary integer. For every 0 j s n we construct recursively the following products of generalized Bass units of ZH: c s s (H, K, k, r) = 1, Inneke Van Gelder (Brussels) Central units and primitive idempotents Murcia, June 3-7, / 26

43 Virtual basis of the center Step 2: Generalized Bass units Let k be a positive integer coprime with p and let r be an arbitrary integer. For every 0 j s n we construct recursively the following products of generalized Bass units of ZH: and, for 0 j s 1, c s j (H, K, k, r) = c s s (H, K, k, r) = 1, p s j 1 u k,op n (k)n H,K (g rpn s h K + 1 K) h H j ( s 1 j 1 ) cl s (H, K, k, r) 1 c s+l j l (H, K, k, r) 1. l=j+1 l=0 Inneke Van Gelder (Brussels) Central units and primitive idempotents Murcia, June 3-7, / 26

44 Virtual basis of the center Step 2: Generalized Bass units Proposition [Jespers-del Río-Olteanu-VG] Let H be a finite group and K a subgroup of H such that H/K = gk is cyclic of order p n. Let H = {L H K L} = {H j = g pn j, K 0 j n}. Let k be a positive integer coprime with p and let r be an arbitrary integer. Then { ρ Hj1 (cj s η (H, K, k, r)) = k (ζ r ) O p s j p n (k)ps 1 n H,K, if j = j 1 ; (1) 1, if j j 1. for every 0 j, j 1 s n. Inneke Van Gelder (Brussels) Central units and primitive idempotents Murcia, June 3-7, / 26

45 Main Theorem Virtual basis of the center Theorem [Jespers-del Río-Olteanu-VG] Let G be a strongly monomial group such that there is a complete and non-redundant set S of strong Shoda pairs (H, K) of G with the property that each [H : K] is a prime power. For every (H, K) S, let T K be a right transversal of N G (K) in G, let I (H,K) be a set of representatives of U(Z/[H : K]Z) modulo N G (K)/H, 1 containing 1 and let [H : K] = p n (H,K) (H,K), with p (H,K) prime. Then c n (H,K) 0 (H, K, k, x) t : (H, K) S, k I (H,K) \ {1} t T K x N G (K)/H is a virtual basis of Z(U(ZG)). Inneke Van Gelder (Brussels) Central units and primitive idempotents Murcia, June 3-7, / 26

46 Virtual basis of the center Examples of groups satisfying the conditions C q m C p n with C p n acting faithfully on C q m Inneke Van Gelder (Brussels) Central units and primitive idempotents Murcia, June 3-7, / 26

47 Virtual basis of the center Examples of groups satisfying the conditions C q m C p n with C p n acting faithfully on C q m A 4 Inneke Van Gelder (Brussels) Central units and primitive idempotents Murcia, June 3-7, / 26

48 Virtual basis of the center Examples of groups satisfying the conditions C q m C p n with C p n acting faithfully on C q m A 4 D 2n = a, b a n = b 2 = 1, a b = a 1 n is a power of a prime Inneke Van Gelder (Brussels) Central units and primitive idempotents Murcia, June 3-7, / 26

49 Virtual basis of the center Examples of groups satisfying the conditions C q m C p n with C p n acting faithfully on C q m A 4 D 2n = a, b a n = b 2 = 1, a b = a 1 n is a power of a prime Q 4n = x, y x 2n = y 4 = 1, x n = y 2, x y = x 1 n is a power of 2... Inneke Van Gelder (Brussels) Central units and primitive idempotents Murcia, June 3-7, / 26

50 Idea Matrix units Each simple component of QG is isomorphic to a matrix algebra. Inneke Van Gelder (Brussels) Central units and primitive idempotents Murcia, June 3-7, / 26

51 Idea Matrix units Each simple component of QG is isomorphic to a matrix algebra. We want to know which group ring elements represent the matrix units E ij = Inneke Van Gelder (Brussels) Central units and primitive idempotents Murcia, June 3-7, / 26

52 Idea Matrix units For a strongly monomial group G, each simple component of QG is of the form QGe(G, H, K) = M n (QHε(H, K) N G (K)/H) for a strong Shoda pair (H, K). Inneke Van Gelder (Brussels) Central units and primitive idempotents Murcia, June 3-7, / 26

53 Idea Matrix units For a strongly monomial group G, each simple component of QG is of the form QGe(G, H, K) = M n (QHε(H, K) N G (K)/H) for a strong Shoda pair (H, K). When the twisting of QN G (K)ε(H, K) N G (K)/H is trivial, Reiner provides an explicit isomorphism ψ : QHε(H, K) N G (K)/H M [NG (K):H](F ). Inneke Van Gelder (Brussels) Central units and primitive idempotents Murcia, June 3-7, / 26

54 Matrix units Theorem [Jespers-del Río-Olteanu-VG] Let (H, K) be a strong Shoda pair of a finite group G such that τ(nh, n H) = 1 for all n, n N G (K). Let P, A M n (F ) be the matrices P = Then and A =. {E x x := x T1 ε(h, K)x 1 x, x T 2 x e } is a complete set of matrix units of QGe(G, H, K) where x e = ψ 1 (PAP 1 ), T 1 is a transversal of H in N G (K) and T 2 is a right transversal of N G (K) in G.. Inneke Van Gelder (Brussels) Central units and primitive idempotents Murcia, June 3-7, / 26

55 Matrix units Examples of groups satisfying the conditions C q m C p n with C p n acting faithfully on C q m Inneke Van Gelder (Brussels) Central units and primitive idempotents Murcia, June 3-7, / 26

56 Matrix units Examples of groups satisfying the conditions C q m C p n with C p n acting faithfully on C q m S 4 Inneke Van Gelder (Brussels) Central units and primitive idempotents Murcia, June 3-7, / 26

57 Matrix units Examples of groups satisfying the conditions C q m C p n with C p n acting faithfully on C q m S 4 A 4 Inneke Van Gelder (Brussels) Central units and primitive idempotents Murcia, June 3-7, / 26

58 Matrix units Examples of groups satisfying the conditions C q m C p n with C p n acting faithfully on C q m S 4 A 4 D 2n = a, b a n = b 2 = 1, a b = a 1 Inneke Van Gelder (Brussels) Central units and primitive idempotents Murcia, June 3-7, / 26

59 Matrix units Examples of groups satisfying the conditions C q m C p n with C p n acting faithfully on C q m S 4 A 4 D 2n = a, b a n = b 2 = 1, a b = a 1... Inneke Van Gelder (Brussels) Central units and primitive idempotents Murcia, June 3-7, / 26

60 Subgroup of finite index Main ingredients Let O be an order in a division algebra D. For an ideal Q of O we denote by E(Q) the subgroup of SL n (O) generated by all Q-elementary matrices, that is E(Q) = I + qe ij q Q, 1 i, j n, i j, E ij a matrix unit. Theorem [Bass-Vaseršteĭn-Liehl-Venkataramana] If n 3 then [SL n (O) : E(Q)] <. If U(O) is infinite then [SL 2 (O) : E(Q)] <. Inneke Van Gelder (Brussels) Central units and primitive idempotents Murcia, June 3-7, / 26

61 Subgroup of finite index Main ingredients Let O be an order in a division algebra D. For an ideal Q of O we denote by E(Q) the subgroup of SL n (O) generated by all Q-elementary matrices, that is E(Q) = I + qe ij q Q, 1 i, j n, i j, E ij a matrix unit. Theorem [Bass-Vaseršteĭn-Liehl-Venkataramana] If n 3 then [SL n (O) : E(Q)] <. If U(O) is infinite then [SL 2 (O) : E(Q)] <. Since GL n (O) is generated by SL n (O) and its center, our construction of central units and matrix units is sufficient to describe a subgroup of finite index in U(ZG) for some groups G. Inneke Van Gelder (Brussels) Central units and primitive idempotents Murcia, June 3-7, / 26

62 Subgroup of finite index Theorem [Jespers-del Río-Olteanu-VG] Let G = C q m C p n be a finite metacyclic group with C p n = b acting faithfully on C q m = a and with p and q different primes. Assume that either q 3, or n 1 or p 2. Then the following two groups are finitely generated nilpotent subgroups of U(ZG): V + j = 1 + p n tj 2 yxj h k bx j y a b, h, k {1,..., p n }, h < k, V j = 1 + p n tj 2 yxj h k bx j y a b, h, k {1,..., p n }, h > k. Hence V + = m j=1 V + j and V = m j=1 V j are nilpotent subgroups of U(ZG). Furthermore, the group U, V +, V, with U a virtual basis of Z(U(ZG)), is of finite index in U(ZG). Inneke Van Gelder (Brussels) Central units and primitive idempotents Murcia, June 3-7, / 26

63 Reference Subgroup of finite index E. Jespers, G. Olteanu, Á. del Río, I. Van Gelder, Group rings of finite strongly monomial groups: Central units and primitive idempotents, Journal of Algebra, Volume 387, 1 August 2013, Pages Inneke Van Gelder (Brussels) Central units and primitive idempotents Murcia, June 3-7, / 26