Differential operators and crystals of extremal weight modules (Q734906)

The author considers extremal weight crystals over \(U_q(\mathfrak{gl}_{>0})\) and its crystal base \(\mathcal B(\lambda)\) and gives a combinatorial realization of it via a set of bitableaux \(\mathcal B_{\mu,\nu}\). He then gives a decomposition of \(\mathcal B_{\mu,\nu}\otimes \mathcal B_{\sigma,\tau}\) into components each of which is isomorphic to a \(\mathcal B_{\zeta,\eta}\). Next, he considers a category \(\mathcal C\) of \(\mathfrak{gl}_{>0}\)-crystals where each object is isomorphic to a finite disjoint union of extremal weight crystals and defines the associated Grothendieck ring \(\mathcal K\) showing that it is a free \(\mathbb Z\)-module, giving a basis for it as well as an explicit isomorphism to an algebra of differential operators. Finally, using the affine algebra \(\mathfrak{gl}_{\infty}\) of type \(A_{\infty}\) and the crystal base of the level 1 fermionic Fock space, he obtains a crystal duality that allows him to recover the generalized Cauchy identity for Schur operators in a bijective and crystal theoretic way.