Separation of Variables for Symplectic Characters

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1、LettMathPhysDOI10.1007/s11005-011-0467-zSeparationofVariablesforSymplecticCharactersJANDEGIERandANITAPONSAINGDepartmentofMathematicsandStatistics,TheUniversityofMelbourne,Melbourne,VIC3010,Australia.e-mail:jdgier@unimelb.edu.au;a.ponsaing@ms.unimelb.edu.auReceived:17Septe

2、mber2010/Revised:14January2011/Accepted:26January2011©Springer2011Abstract.WeperformseparationofvariablesforthesymplecticWeylcharacterusingSklyaninsscheme.Viewingthecharactersaseigenfunctionsofaquantumintegrablesys-tem,weexplicitlyconstructtheseparatingoperatorusingtheQ-o

3、peratormethod.Wealsoconstructtheinverseoftheseparatingoperator,aswellasthefactorisedHamiltonian.MathematicsSubjectClassification(2010)33,32W99,82B23.Keywords.integrablesystems,integraloperators.1.IntroductionSymplecticcharacters,orSchurpolynomialsfortherootsystemoftypeC,pl

4、ayanimportantroleintherepresentationtheoryoftheclassicalgroups[9].Theyarealsousedasgeneratingfunctionsforcountingproblemswithboundariesinenumerativecombinatorics,seee.g.[3,15].Inrecentyears,symplecticcharactershaveappearedinsquarelatticecriticalbondpercolationonlatticestr

5、ipswithboundaries[2,5,7,21],withapplicationstothespinquantumHalleffect[4].Inthesecases,theyarisefrompolynomialsolutionsoftheq-deformedKnizhnik-Zamolodchikovequation[8,10,11,19],whereqisthethirdrootofunity,andappearintheexactfinitesizeexpressionsfornormalisationsandcorrelat

6、ionfunctions.Fortheseapplications,onenaturallywouldliketounderstandtheasymptoticbehaviourofthesymplecticcharacterasthenumberofvariablestendstoinfin-ity,andinparticularinthelimitwhenallbutafinitenumberofvariablesaresetto1.OkounkovandOlshanskistudiedthislimitforbothtypeAJackp

7、olynomi-alsandtypeBCorthogonalpolynomials[16,17].Forbothoftheseproblems,theauthorsconsideredpolynomialswithadegreegrowinglinearlywiththenumberofvariables.Incontrast,thesymplecticcharacterswhichariseintheaforementionedbondpercolationmodelshaveadegreethatisquadraticinthenum

8、berofvariablesand,asfarasweareaware,theirasymptoticbehaviourisanopenproblem.Inthispaper,westudya