Abstract
We give a combinatorial realization of extremal weight crystals over the quantum group of type A + ∞ and their Littlewood–Richardson rule. Based on this description, we show that the Grothendieck ring generated by the isomorphism classes of extremal weight A + ∞ -crystals is isomorphic to the Weyl algebra of infinite rank, and hence each isomorphism class is realized as a differential operator or non-commutative Schur function acting on the algebra of symmetric functions. We also find a duality between extremal weight A + ∞ -crystals and generalized Verma A ∞ -crystals appearing in the crystal of the Fock space with infinite level, which recovers the generalized Cauchy identity for Schur operators in a bijective and crystal theoretic way.
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