Differential operators and crystals of extremal weight modules
type: Item
instance of: scholarly article
title: Differential operators and crystals of extremal weight modules
zbMATH Open document ID: 1189.17015
author: Jae-Hoon Kwon
published in: Advances in Mathematics
publication date: 2009-10-14T00:00:00Z
review text: 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.
reviewed by: Stefano Capparelli
Mathematics Subject Classification ID: 17B37
Mathematics Subject Classification ID: 05E05
zbMATH DE Number: 5614875
zbMATH Keywords: quantized enveloping algebras
zbMATH Keywords: extremal weight crystal
zbMATH Keywords: symmetric function
MaRDI profile type: Publication
full work available at URL: j.aim.2009.06.012
OpenAlex ID: W2032505040
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DOI: 10.1016/J.AIM.2009.06.012
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description: scientific article; zbMATH DE number 5614875