A crystal to rigged configuration bijection and the filling map for type

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• 作者：Travis Scrimshaw¹ ; ^{tscrimsh@umn.edu" class="auth_mail" title="E-mail the corresponding author}
• 关键词：05E10 ; 17B37
• 刊名：Journal of Algebra
• 出版年：2016
• 出版时间：15 February 2016
• 年：2016
• 卷：448
• 期：Complete
• 页码：294-349
• 全文大小：832 K

文摘

We give a bijection Φ from rigged configurations to a tensor product of Kirillov&ndash;Reshetikhin crystals of the form an id="mmlsi2" class="mathmlsrc">an class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869315005189&_mathId=si2.gif&_user=111111111&_pii=S0021869315005189&_rdoc=1&_issn=00218693&md5=9de0373fcaaee140772f169e58c3022e" title="Click to view the MathML source">B^r,1an>an class="mathContainer hidden">an class="mathCode">ath altimg="si2.gif" overflow="scroll">Br,1ath>an>an>an> and an id="mmlsi3" class="mathmlsrc">an class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869315005189&_mathId=si3.gif&_user=111111111&_pii=S0021869315005189&_rdoc=1&_issn=00218693&md5=6aed4f805465e651a2018370be96f0b4" title="Click to view the MathML source">B^1,san>an class="mathContainer hidden">an class="mathCode">ath altimg="si3.gif" overflow="scroll">B1,sath>an>an>an> in type an id="mmlsi1" class="mathmlsrc"><a title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869315005189&_mathId=si1.gif&_user=111111111&_pii=S0021869315005189&_rdoc=1&_issn=00218693&md5=e072649b727c5ea023cecbfc097e1135">ass="imgLazyJSB inlineImage" height="21" width="31" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0021869315005189-si1.gif">a>an class="mathContainer hidden">an class="mathCode">ath altimg="si1.gif" overflow="scroll">D4alse">(3alse">)ath>an>an>an>. We show that the cocharge statistic is sent to the energy statistic for tensor products an id="mmlsi4" class="mathmlsrc"><a title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869315005189&_mathId=si4.gif&_user=111111111&_pii=S0021869315005189&_rdoc=1&_issn=00218693&md5=e99b16fd6a23dfed94f3164bfc88d1b3">ass="imgLazyJSB inlineImage" height="21" width="78" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0021869315005189-si4.gif">a>an class="mathContainer hidden">an class="mathCode">ath altimg="si4.gif" overflow="scroll">⨂i=1NBri,1ath>an>an>an> and an id="mmlsi5" class="mathmlsrc"><a title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869315005189&_mathId=si5.gif&_user=111111111&_pii=S0021869315005189&_rdoc=1&_issn=00218693&md5=9df5719c6a0f6cb233cd619416256bd5">ass="imgLazyJSB inlineImage" height="21" width="78" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0021869315005189-si5.gif">a>an class="mathContainer hidden">an class="mathCode">ath altimg="si5.gif" overflow="scroll">⨂i=1NB1,siath>an>an>an>. We extend this bijection to a single an id="mmlsi117" class="mathmlsrc">an class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869315005189&_mathId=si117.gif&_user=111111111&_pii=S0021869315005189&_rdoc=1&_issn=00218693&md5=a5d5e53d5b206035781dca56d681b0af" title="Click to view the MathML source">B^r,san>an class="mathContainer hidden">an class="mathCode">ath altimg="si117.gif" overflow="scroll">Br,sath>an>an>an>, show that it preserves statistics, and obtain the so-called Kirillov&ndash;Reshetikhin tableaux model for an id="mmlsi117" class="mathmlsrc">an class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869315005189&_mathId=si117.gif&_user=111111111&_pii=S0021869315005189&_rdoc=1&_issn=00218693&md5=a5d5e53d5b206035781dca56d681b0af" title="Click to view the MathML source">B^r,san>an class="mathContainer hidden">an class="mathCode">ath altimg="si117.gif" overflow="scroll">Br,sath>an>an>an>. Additionally, we show that Φ commutes with the virtualization map and that an id="mmlsi3" class="mathmlsrc">an class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869315005189&_mathId=si3.gif&_user=111111111&_pii=S0021869315005189&_rdoc=1&_issn=00218693&md5=6aed4f805465e651a2018370be96f0b4" title="Click to view the MathML source">B^1,san>an class="mathContainer hidden">an class="mathCode">ath altimg="si3.gif" overflow="scroll">B1,sath>an>an>an> is naturally a virtual crystal in type an id="mmlsi281" class="mathmlsrc"><a title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869315005189&_mathId=si281.gif&_user=111111111&_pii=S0021869315005189&_rdoc=1&_issn=00218693&md5=6a063e631a6ce52b4a3ef0762e1dcc57">ass="imgLazyJSB inlineImage" height="21" width="31" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0021869315005189-si281.gif">a>an class="mathContainer hidden">an class="mathCode">ath altimg="si281.gif" overflow="scroll">D4alse">(1alse">)ath>an>an>an>, thus defining an affine crystal structure on rigged configurations corresponding to an id="mmlsi3" class="mathmlsrc">an class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869315005189&_mathId=si3.gif&_user=111111111&_pii=S0021869315005189&_rdoc=1&_issn=00218693&md5=6aed4f805465e651a2018370be96f0b4" title="Click to view the MathML source">B^1,san>an class="mathContainer hidden">an class="mathCode">ath altimg="si3.gif" overflow="scroll">B1,sath>an>an>an>.