# Mathematics Colloquia and Seminars

### Crystal graph of the modified quantum group of type $A_{+\infty}$ and RSK algorithm
We give a new combinatorial realization of the crystal base of the modified quantized enveloping algebras of affine type $A_{+\infty}$ (arXivmathRT/1002.1509). 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 algebra and hence its Peter-Weyl type decomposition generalizing the classical RSK correspondence.