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We classify globally irreducible representations of alternating groups anddouble covers of symmetric and alternating groups. In order to achieve thisclassification we also completely characterise irreducible representations ofsuch groups which reduce almost homogeneously in every characteristic. Thisalso allows us to classify irreducible representations that remain irreduciblein every characteristic. In particular we show that, apart from finitely manyexceptions, for any of these three questions such representations are either$1$-dimensional or basic spin representations.

Representations of symmetric and alternating groups and their double covers that remain irreducible modulo every prime