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The tensor powers of the vector representation associated to an infinite rankquantum group decompose into irreducible components with multiplicitiesindependant of the infinite root system considered. Although the irreduciblemodules obtained in this way are not of highest weight, they admit a crystalbasis and a canonical basis. This permits in particular to obtain for eachfamilly of classical Lie algebras a Robinson-Schensted correspondence onbiwords defined on infinite alphabets. We then depict a structure of bi-crystalon these bi-words. This RS-correspondence yields also a plactic algebra andplactic Schur functions distinct for each infinite root system. Surprisingly,the algebras spanned by these plactic Schur functions are all isomorphic to thealgebra of symmetric functions.

Crystal bases and combinatorics of infinite rank quantum groups