A Hodge algebra (or «Algebra with Straightening Law») is an commutative algebra having a special basis which allows one to determine many features of its structure by relatively simple combinatorial study of its generators and relations. This foundational text describes the basic theory and surveys the examples (e.g., coordinate rings of determinantal loci, Schubert cycles, varieties of complexes, etc...). Recent applications, mentioned in the text, have been to the rationality and rigidity of certain singularities, Brill-Noether theory for Riemann surfaces, and various combinatorial questions.
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