The Study Of Primitive Central Idempotents Of Schur Rings

In this thesis,we mainly study the primitive central idempotents of Schur rings.Through studying,when G is a finite group,we give the expression of primitive central idempotents of a Schur ring over Z[G]and Z[ω][G]by the expression of primitive central idempotents of C[G].When G is a finite cyclic group,we also describe the expression of primitive idempotents of Z[1/p][G]y the expression of primitive idempotents of Q[G]and then we describe the expression of primitive idempotents of a Schur ring over Z[1/p][G]by the expression of primitive idempotents of Z[1/p][G].The specific conclusions are as follows:Theorem 1.3.Let G be a finite group.Suppose Z is the ring of integers and S is a Schur ring over Z[G].Then 1 is the only primitive central idempotent of S.Theorem 1.4.Let G be a finite group and ω is a |G|-th primitive root of unity.Suppose Z[ω]is the subring of C generated by Z and ω and S is a Schur ring over Z[ω][G].Then 1 is the unique primitive central idempotent of S.Theorem 1.5.Let G be a finite cyclic group and pa ‖ |G| where p is a prime and a E N+.Suppose Z[1/p]is the subring of Q generated by Z and 1/p.Then the primitive idempotents of Z[1/p][G]are as follows:where Si={Hi≤G | pi ‖ |Hi|}(i∈ {0,1,…,a}).Theorem 1.6.The notation is as mentioned in the introduction.Let G be a finite cyclic group and pa ‖ |G| where p is a prime and a ∈N+.Suppose S is a Schur ring over Z[1/p][G].Then(?)j∈L,∑l∈N(L,j)εl is the primitive idempotent of S.In particular,{∑l∈N(L,j)εl| j∈L} is a complete set of primitive idempotents in S.Moreover,this thesis also studys the lattice Schur ring and gives a general method for finding the basis sets of the lattice Schur ring.And using lattice Schur ring,it states that there must be a non-traditional Schur ring over F[ZP×Zp]where p,more than 3,is a prime and F is a field with charF≠p.