Diamond Representations for Rank Two Semisimple Lie Algebras

The present work is a part of a larger program to construct explicit combinatorial models for the (indecomposable) regular representation of the nilpotent factor N in the Iwasawa decomposition of a semisimple Lie algebra g, using the restrictions to N of the simple finite dimensional modules of g. Such a description is given in Arnal, D., N. Bel Baraka, and N.-J. Wildberger, Diamond representations of sl(n), Annales Mathematiques Blaise Pascal 13 (2006), 381429 for the case g = sl(n). Here, we perform the same construction for the rank 2 semisimple Lie algebras (of type A(1) x A(1), A(2), C(2) and G(2)). The algebra C[N] of polynomial functions on N is a quotient, called the reduced shape algebra, of the shape algebra for g. Bases for the shape algebra are known, for instance the so-called semistandard Young tableaux give an explicit basis (see Alverson, L.-W., R.-G. Donnelly, S.-J. Lewis, M. McClard, R. Pervine, R.-A. Proctor, and N.-J. Wildberger, Distributive lattice defined for representations of rank two semisimple Lie algebras, SIAM J. Discrete Math. 23 (2008/09), no. 1, 527-559). We select among the semistandard tableaux, the so-called quasistandard ones which define a kind basis for the reduced shape algebra.