Idempotent Orthogonal Class System Of A Group And Its Applications

Group symmetry, one of the soul of contemporary mathematics, is becoming the most efficient tool for many symmetry problems in recent sciences. To study symmetry of a group, a basic implement is group representation, the central category of which, irreducible character of a group, naturally become one of the central problems in the group representation theory. In cite [1], a usable partition of a group and idempotent orthogonal class system are defined, followed by its characterization and relative numerical method, dexterously applying the simultaneous diagonalization of matrices. In fact, a usable partition of a group and idempotent orthogonal class system are respectively generalizations of conjugate partition and irreducible character of a group.Following [1], this thesis investigate the usable partition of a group and idempotent orthogonal class system by the numbers, present many new definitions and methods, find that this idempotent orthogonal class system can give voice to some symmetry properties as well as the irreducible character of a group.Recapitulations as follows:(1) Get a character theorem for usable partition of a group.(2) Introduce some concepts, such as matrix representation for a finite group G corresponding to some matrix, bi-class function, inverse closed partition, Her-mite faction, uniform intersection partition, uniform commutative partition, symmetry usable partition and so on, and discuss their relations; present six equivalent character for symmetry usable partition; find out the relation between symmetry idempotent orthogonal class system and its matrix representation; and solve the supposition brought forward by [1] partly.(3) Give an exact formula for usable matrix of usable partition as well as for symmetry usable partition. Open out the relation between irreducible character of a group and idempotent orthogonal class system, and give several numerical methods for irreducible character of a group. (4) Prove the orthogonal theorem for symmetry idempotent orthogonal class system. Give an important theorem of irreducible character about a finite group.(5) Study the structure of a symmetry usable partition which can be written into the product of two Abel subgroups, and in addition give many symmetry usable partition of this kind of groups. Define Kronecker product and generalized direct sum in idempotent orthogonal class system, and applying these two definitions, present two methods for constructing such system.(6) Find out that the idempotent orthogonal class system can disclose the symmetry of multi-function, the idempotent property of matrix and the symmetry of a multilinear map. Give the generalized symmetric tensor map decomposition of tensor map and the generalized symmetric class decomposition of tensor space.