Crystal bases of modified quantized enveloping algebras and a double RSK correspondence

We give a new combinatorial realization of the crystal base of the modified quantized enveloping algebras of type A + ∞ or A ∞ . It is obtained by describing the decomposition of the tensor product of a highest weight crystal and a lowest weight crystal into extremal weight crystals, and taking its limit using a tableaux model of extremal weight crystals. This realization induces in a purely combinatorial way a bicrystal structure of the crystal base of the modified quantized enveloping algebras and hence its Peter–Weyl type decomposition generalizing the classical RSK correspondence.