Schur's theory for partial projective representations

Partial generalized crossed products and a seven-term exact sequence

“…The starting point of [14] is the replacement of global G-actions on abelian groups (G-modules) by unital partial actions of G on commutative semigroups (partial G-modules), in particular, on commutative rings. The partial group cohomology found applications to partial projective group representations [14], [18] and to the study of ideals of (global) reduced C * -crossed products [24]. It also motivated the treatment of partial cohomology from the point of view of Hopf algebras [4].…”

Recent developments around partial actions

“…In particular, computations of the partial Schur multiplier of concrete groups in [131], [226], [243], [258], [260] show that each component is, in fact, isomorphic to a direct power of κ * , suggesting that this should be true for all groups. This motivated the recent preprint [137], in which this conjecture was confirmed for all finite groups over an algebraically closed field. This surprisingly gives a better understanding of the structure of pM (G) than one has for that of the usual Schur Multiplier.…”

“…The key idea in [137] is to replace κ * by an arbitrary abelian group A and define pre-cocycles which are functions σ : G×G → A obeying condition (6). They form a group denoted by pZ 2 (G, A).…”

“…Then for any σ ∈ Z 2 (G, I; A) it is possible to construct a central extension A → E → M, a partial homomorphism ψ : G → M and its lifting (section) ϕ : G → E, such that the diagram ( 7) is commutative upon the replacement of κ * by A, and σ plays the role of the factor set related to ϕ as in (3), ( 4), (5). This leads to a bijection between the elements of H 2 (G, I; A) and the equivalence classes of certain appropriately defined central extensions [137,Theorem 3.3]. This way the partial Schur multiplier pM (G) over an arbitrary field κ coincides with the second partial cohomology semilattice of groups with values in A = κ * , the components of which being second relative partial cohomology groups.…”

“…Theorem 4.3 em [137] gives a precise structure of the pre-cohomology group for a finite group G of order n an any abelian group A :…”