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We study the modular representation theory of the symmetric and alternatinggroups. One of the most natural ways to label the irreducible representationsof a given group or algebra in the modular case is to show the unitriangularityof the decomposition matrices, that is, the existence of a unitriangular basicset. We study several ways to obtain such sets in the general case of asymmetric algebra. We apply our results to the symmetric groups and to theirHecke algebras and thus obtain new ways to label the simple modules for theseobjects. Finally, we show that these sets do not always exist in the case ofthe alternating groups by studying two explicit cases in characteristic 3.

On unitriangular basic sets for symmetric and alternating groups