Representações parciais de grupos, seus domínios e o multiplicador de Schur parcial

Abstract

LIMA, H. G. G. de. Partial group representations, their domains and the partial Schur multiplier. 2014. 68 f. Tese (Doutorado) Instituto de Matematica e Estatistica, Universidade de Sao Paulo, Sao Paulo, 2014. The partial Schur multiplier of a group G is a commutative inverse semigroup pM(G) which, in the study of partial projective representations, plays a role analogous to the classical Schur multiplier M(G). There is a description of pM(G) as a union of abelian groups, in which each component pMD(G) is formed by the equivalence classes of certain partial functions (called partial factor sets), taking values in a field and having as its domain a subset D ⊆ G×G. The domains D form a lattice and were characterized as the T -invariant subsets of G × G, where T is a specific monoid acting on G×G. The total component pMG×G(G), which corresponds to the totally defined factor sets, is particularly interesting because it contains M(G) as one of its subgroups and, moreover, any other component is an epimorphic image of the total component. One of the objectives of this work is to determine the total component of the partial Schur multiplier for some important classes of groups, such as the dihedral groups, the dicyclic groups and the products of cyclic groups. Another topic which will be considered is the structure of the lattice of domains of partial factor sets, emphasizing properties of those domains that correspond to the so-called elementary partial representations, which play a relevant role in the theory. We shall prove that each domain can be represented in a unique way as a union of certain indecomposable domains, where the latter consists of the so-called blocks and minimal domains. The structure of the elementary domains also will be determined, and some numerical invariants of the partially ordered set of the elementary domains will be given. As a consequence of the obtained facts, the groups whose elementary domains are indecomposable will be characterized. We will also give an application of the theory of semigroup algebras to the partial group algebra, an algebra which is responsible for partial group representations.