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Why should the Littlewood-Richardson Rule be true?


Authors: Roger Howe and Soo Teck Lee
Journal: Bull. Amer. Math. Soc. 49 (2012), 187-236
MSC (2000): Primary 20G05; Secondary 05E15
DOI: https://doi.org/10.1090/S0273-0979-2011-01358-1
Published electronically: October 20, 2011
MathSciNet review: 2888167
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Abstract: We give a proof of the Littlewood-Richardson Rule for describing tensor products of irreducible finite-dimensional representations of $ \textrm {GL}_n$. The core of the argument uses classical invariant theory, especially $ (\textrm {GL}_n, \textrm {GL}_m)$-duality. Both of the main conditions (semistandard condition, lattice permutation/Yamanouchi word condition) placed on the tableaux used to define Littlewood-Richardson coefficients have natural interpretations in the argument.


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Additional Information

Roger Howe
Affiliation: Department of Mathematics, Yale University, New Haven, Connecticut 06520-8283
Email: howe@math.yale.edu

Soo Teck Lee
Affiliation: Department of Mathematics, National University of Singapore, Block S17, 10 Lower Kent Ridge Road, Singapore 119076, Singapore
Email: matleest@nus.edu.sg

DOI: https://doi.org/10.1090/S0273-0979-2011-01358-1
Keywords: Littlewood-Richardson Rule, Pieri Rule, $GL_{n}$ tensor product algebra, $(GL_{n}, GL_{m})$-duality.
Received by editor(s): March 30, 2009
Received by editor(s) in revised form: February 14, 2011
Published electronically: October 20, 2011
Additional Notes: The second named author is partially supported by NUS grant R-146-000-110-112.
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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