Insertion Scheme for the Classical Lie Algebras

Abstract

In this paper, we give an insertion scheme for the tableaux of Kashiwara– Nakashima realizing the crystal bases of the irreducible highest weight modules over the classical Lie algebras. It gives an explicit combinatorial description of the decomposition of the tensor product V(λ) ⊗ V(μ) into irreducible modules.

Key Words:

• Classical Lie algebra
• Crystal base
• Insertion scheme

Mathematics Subject Classification:

• Primary 05E15
• Secondary 81R50

Acknowledgments

This research was supported by KOSEF Grant # 98-0701-01-5-L and BK21 Mathematical Sciences Division, Seoul National University.

Part of this work was completed while the authors were visiting the Department of Mathematics of University of Wisconsin, Madison. We would like to express our sincere gratitude to the Department of Mathematics of University of Wisconsin, Madison for their hospitality and support during our visit. Special thanks should be given to Professor Georgia Benkart, Professor Chanyoung Lee Shader and Professor Itaru Terada for their interest in this work and many valuable discussions. We would also like to express our sincere gratitude to Professor S.-J. Kang for his guidance, encouragement, and many valuable suggestions.

Note. After we wrote this paper, we were informed that T. Baker and C. Lecouvey have obtained essentially the same insertion scheme independently by different approach (Baker, 2000a 2000b; Lecouvey, 2002).

Notes

^#Communicated by K. Misra.