Diamond representations of 𝔰𝔩 ( n )

Didier Arnal[1]; Nadia Bel Baraka[1]; Norman J. Wildberger[2]

  • [1] Institut de Mathématiques de Bourgogne UMR CNRS 5584 Université de Bourgogne U.F.R. Sciences et Techniques B.P. 47870 F-21078 Dijon Cedex France
  • [2] School of Mathematics University of New South Wales Sydney 2052 Australia

Annales mathématiques Blaise Pascal (2006)

  • Volume: 13, Issue: 2, page 381-429
  • ISSN: 1259-1734

Abstract

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In [6], there is a graphic description of any irreducible, finite dimensional 𝔰𝔩 ( 3 ) module. This construction, called diamond representation is very simple and can be easily extended to the space of irreducible finite dimensional 𝒰 q ( 𝔰𝔩 ( 3 ) ) -modules.In the present work, we generalize this construction to 𝔰𝔩 ( n ) . We show it is in fact a description of the reduced shape algebra, a quotient of the shape algebra of 𝔰𝔩 ( n ) . The basis used in [6] is thus naturally parametrized with the so called quasi standard Young tableaux. To compute the matrix coefficients of the representation in this basis, it is possible to use Groebner basis for the ideal of reduced Plücker relations defining the reduced shape algebra.

How to cite

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Arnal, Didier, Bel Baraka, Nadia, and Wildberger, Norman J.. "Diamond representations of $\mathfrak{sl}(n)$." Annales mathématiques Blaise Pascal 13.2 (2006): 381-429. <http://eudml.org/doc/10535>.

@article{Arnal2006,
abstract = {In [6], there is a graphic description of any irreducible, finite dimensional $\mathfrak\{sl\}(3)$ module. This construction, called diamond representation is very simple and can be easily extended to the space of irreducible finite dimensional $\{\mathcal\{U\}\}_q(\mathfrak\{sl\}(3))$-modules.In the present work, we generalize this construction to $\mathfrak\{sl\}(n)$. We show it is in fact a description of the reduced shape algebra, a quotient of the shape algebra of $\mathfrak\{sl\}(n)$. The basis used in [6] is thus naturally parametrized with the so called quasi standard Young tableaux. To compute the matrix coefficients of the representation in this basis, it is possible to use Groebner basis for the ideal of reduced Plücker relations defining the reduced shape algebra.},
affiliation = {Institut de Mathématiques de Bourgogne UMR CNRS 5584 Université de Bourgogne U.F.R. Sciences et Techniques B.P. 47870 F-21078 Dijon Cedex France; Institut de Mathématiques de Bourgogne UMR CNRS 5584 Université de Bourgogne U.F.R. Sciences et Techniques B.P. 47870 F-21078 Dijon Cedex France; School of Mathematics University of New South Wales Sydney 2052 Australia},
author = {Arnal, Didier, Bel Baraka, Nadia, Wildberger, Norman J.},
journal = {Annales mathématiques Blaise Pascal},
keywords = {diamonds; representations; shape algebra},
language = {eng},
month = {7},
number = {2},
pages = {381-429},
publisher = {Annales mathématiques Blaise Pascal},
title = {Diamond representations of $\mathfrak\{sl\}(n)$},
url = {http://eudml.org/doc/10535},
volume = {13},
year = {2006},
}

TY - JOUR
AU - Arnal, Didier
AU - Bel Baraka, Nadia
AU - Wildberger, Norman J.
TI - Diamond representations of $\mathfrak{sl}(n)$
JO - Annales mathématiques Blaise Pascal
DA - 2006/7//
PB - Annales mathématiques Blaise Pascal
VL - 13
IS - 2
SP - 381
EP - 429
AB - In [6], there is a graphic description of any irreducible, finite dimensional $\mathfrak{sl}(3)$ module. This construction, called diamond representation is very simple and can be easily extended to the space of irreducible finite dimensional ${\mathcal{U}}_q(\mathfrak{sl}(3))$-modules.In the present work, we generalize this construction to $\mathfrak{sl}(n)$. We show it is in fact a description of the reduced shape algebra, a quotient of the shape algebra of $\mathfrak{sl}(n)$. The basis used in [6] is thus naturally parametrized with the so called quasi standard Young tableaux. To compute the matrix coefficients of the representation in this basis, it is possible to use Groebner basis for the ideal of reduced Plücker relations defining the reduced shape algebra.
LA - eng
KW - diamonds; representations; shape algebra
UR - http://eudml.org/doc/10535
ER -

References

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  1. D. Cox, J. Little, D. O’shea, Ideals, varieties, and algorithms, (1996), Springer-Verlag, New York Zbl0861.13012
  2. W. Fulton, J. Harris, Representation theory, (1991), Springer-Verlag, New York Zbl0744.22001MR1153249
  3. M. Kashiwara, Bases cristallines des groupes quantiques, (2002), Soc. Math. France, Paris Zbl1066.17007MR1997677
  4. G. Lancaster, J. Towber, Representation-functors and flag-algebras for the classical groups, J. Algebra 59 (1979) Zbl0441.14013MR541667
  5. V.S. Varadarajan, Lie groups, Lie algebras, and their representations, (1984), Springer-Verlag, New York, Berlin Zbl0955.22500MR746308
  6. N. Wildberger, Quarks, diamonds and representation of 𝔰𝔩 ( 3 ) , (2005) 

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