2007/06/30 by Masato Okado, Anne Schilling · 2 citations
Mathematics · Physics and Astronomy · #Advanced Topics in Algebra #Algebraic structures and combinatorial models #Nonlinear Waves and Solitons #math.QA #math.RT #msc:17B37 #msc:81R10 #msc:81R50
paper · pdf · doi:10.1090/s1088-4165-08-00329-4
published as Represent.Theory12:186-207,2008 · 23 pages; version that appeared in Representation Theory and erratum added
openalex publication_date 2008/04/14 · arxiv created 2008/11/11 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Using the methods of Kang et al. and recent results on the characters of Kirillov–Reshetikhin modules by Nakajima and Hernandez, the existence of Kirillov–Reshetikhin crystals <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper B Superscript r comma s"> <mml:semantics> <mml:msup> <mml:mi>B</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>r</mml:mi> <mml:mo>,</mml:mo> <mml:mi>s</mml:mi> </mml:mrow> </mml:msup> <mml:annotation encoding="application/x-tex">Br,s</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is established for all nonexceptional affine types. We also prove that the crystals <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper B Superscript r comma s"> <mml:semantics> <mml:msup> <mml:mi>B</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>r</mml:mi> <mml:mo>,</mml:mo> <mml:mi>s</mml:mi> </mml:mrow> </mml:msup> <mml:annotation encoding="application/x-tex">Br,s</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of type <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper B Subscript n Superscript left-parenthesis 1 right-parenthesis"> <mml:semantics> <mml:msubsup> <mml:mi>B</mml:mi> <mml:mi>n</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">(</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:msubsup> <mml:annotation encoding="application/x-tex">Bn(1)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper D Subscript n Superscript left-parenthesis 1 right-parenthesis"> <mml:semantics> <mml:msubsup> <mml:mi>D</mml:mi> <mml:mi>n</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">(</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:msubsup> <mml:annotation encoding="application/x-tex">Dn(1)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A Subscript 2 n minus 1 Superscript left-parenthesis 2 right-parenthesis"> <mml:semantics> <mml:msubsup> <mml:mi>A</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>2</mml:mn> <mml:mi>n</mml:mi> <mml:mo> − </mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">(</mml:mo> <mml:mn>2</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:msubsup> <mml:annotation encoding="application/x-tex">A2n-1(2)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are isomorphic to recently constructed combinatorial crystals for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="r"> <mml:semantics> <mml:mi>r</mml:mi> <mml:annotation encoding="application/x-tex">r</mml:annotation> </mml:semantics> </mml:math> </inline-formula> not a spin node.