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We give a new bijective interpretation of the Cauchy identity for Schuroperators which is a commutation relation between two formal power series withoperator coefficients. We introduce a plactic algebra associated with theKashiwara's extremal weight crystals over the Kac-Moody algebra of type$A_{+\infty}$, and construct a Knuth type correspondence preserving the placticrelations. This bijection yields the Cauchy identity for Schur operators as ahomomorphic image of its associated identity for plactic characters of extremalweight crystals, and also recovers the Sagan and Stanley's correspondence forskew tableaux as its restriction.

A plactic algebra of extremal weight crystals and the Cauchy identity for Schur operators