Abstract
In \cite{W}, 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 this is in fact a description of the reduced shape algebra, a quotient of the shape algebra of $\mathfrak{sl}(n)$. The basis used in \cite{W} 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 Plucker relations defining the reduced shape algebra.
Highlights
In this paper, we consider the irreducible finite dimensional representations of the Lie algebra sl(n) = sl(n, C)
Sl(n) acts naturally on Cn, its fundamental representations are the natural actions on Cn, ∧2 Cn, . . . , ∧n−1 Cn, they have highest weights ω1, . . . , ωn−1
Wildberger gave a really different presentation of the simple sl(3)-modules. This description is based on the construction of the diamond cone for sl(3), it is an infinite dimensional indecomposable module for the Heisenberg Lie algebra with a very explicit basis
Summary
Wildberger gave a really different presentation of the simple sl(3)-modules This description is based on the construction of the diamond cone for sl(3), it is an infinite dimensional indecomposable module for the Heisenberg Lie algebra with a very explicit basis. The matrix coefficients are integral numbers and fixing the highest weight λ, it is easy to build the corresponding representation of sl(3), on the submodule generated by this vector in the diamond cone. The root space corresponding to −η is generated by lower triangular matrix: These matrices generate sl(n) as a Lie algebra. This matrix, acting by adjoint action generates the longest element of the Weyl group of SL(n). It corresponds to a change in the.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
