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International Mathematics Research Notices Advance Access originally published online on February 27, 2009
International Mathematics Research Notices (2009) 2009:2200-2247, doi:10.1093/imrn/rnp014 published on June 16, 2009
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© The Author 2009. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: journals.permissions@oxfordjournals.org.

A Quiver Construction of Symmetric Crystals

Naoya Enomoto

Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606-8502, Japan

Correspondence: Correspondence to be sent to: henon{at}kurims.kyoto-u.ac.jp

In the papers [4–6] with Masaki Kashiwara, the author introduced the notion of symmetric crystals and presented the Lascoux–Leclerc–Thibon–Ariki-type conjectures for the affine Hecke algebras of type B. Namely, we conjectured that certain composition multiplicities and branching rules for the affine Hecke algebras of type B are described by using the lower global basis of symmetric crystals of Formula . In the present paper, we prove the existence of crystal bases and global bases of Formula for any symmetric quantized Kac–Moody algebra by using a geometry of quivers (with a Dynkin diagram involution). This is analogous to George Lusztig's geometric construction of U–v and its lower global basis.

Communicated by Prof. Toshiyuki Kobayashi


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