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Non-commutative rings and algebras, non-associative algebras, universal algebra and lattice theory, linear algebra, semigroups. For questions specific to commutative algebra (that is, rings that are assumed both associative and commutative), rather use the tag ac.commutative-algebra.
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Central idempotents from characters in Frobenius algebras (generalizing Lusztig arXiv:math/0...
I think you should check out Chapter 7 of Geck and Pfeiffer's book "Characters of Finite Coxeter Groups and Iwahori-Hecke Algebras". There, many properties of symmetric algebras are worked out in quit …