Crystal bases and combinatorics of infinite rank quantum groups

Lecouvey, Cédric

doi:10.1090/S0002-9947-08-04480-2

Crystal bases and combinatorics of infinite rank quantum groups
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Trans. Amer. Math. Soc. 361 (2009), 297-329 Request permission

Abstract:

The tensor powers of the vector representation associated to an infinite rank quantum group decompose into irreducible components with multiplicities independent of the infinite root system considered. Although the irreducible modules obtained in this way are not of highest weight, they admit a crystal basis and a canonical basis. This permits us in particular to obtain for each family of classical Lie algebras a Robinson-Schensted correspondence on biwords defined on infinite alphabets. We then depict a structure of bicrystal on these biwords. This RSK-correspondence yields also a plactic algebra and plactic Schur functions distinct for each infinite root system. Surprisingly, the algebras spanned by these plactic Schur functions are all isomorphic to the algebra of symmetric functions.

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