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Journal of Algebraic Combinatorics

, Volume 34, Issue 3, pp 427–449 | Cite as

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

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Abstract

We give a new bijective interpretation of the Cauchy identity for Schur operators which is a commutation relation between two formal power series with operator coefficients. We introduce a plactic algebra associated with the Kashiwara’s extremal weight crystals over the Kac–Moody algebra of type A +∞, and construct a Knuth type correspondence preserving the plactic relations. This bijection yields the Cauchy identity for Schur operators as a homomorphic image of its associated identity for plactic characters of extremal weight crystals, and also recovers Sagan and Stanley’s correspondence for skew tableaux as its restriction.

Keywords

Plactic algebra Crystal Schur operator 
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© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of SeoulSeoulKorea

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