\magnification=1200
\hsize=4in
\overfullrule=0pt
\input amssym
%\def\frac#1 #2 {{#1\over #2}}
\def\emph#1{{\it #1}}
\def\em{\it}
\nopagenumbers
\noindent
%
%
{\bf L. Wyatt Alverson II, Robert G. Donnelly, Scott J. Lewis and Robert Pervine}
%
%
\medskip
\noindent
%
%
{\bf Constructions of Representations of Rank Two Semisimple Lie Algebras with Distributive Lattices}
%
%
\vskip 5mm
\noindent
%
%
%
%
We associate one or two posets (which we call ``semistandard po\-sets'')
to any given irreducible representation of a rank two semi\-simple Lie
algebra over ${\Bbb C}$. Elsewhere we have shown how the
distributive lattices of order ideals taken from semistandard posets
(we call these ``semistandard lattices'') can be used to obtain
certain information about these irreducible representations. Here we
show that some of these semistandard lattices can be used to present
explicit actions of Lie algebra generators on weight bases (Theorem
5.1), which implies these particular semistandard lattices are
supporting graphs. Our descriptions of these actions are explicit in
the sense that relative to the bases obtained, the entries for the
representing matrices of certain Lie algebra generators are rational
coefficients we assign in pairs to the lattice edges. In Theorem 4.4
we show that if such coefficients can be assigned to the edges, then
the assignment is unique up to products; we conclude that the
associated weight bases enjoy certain uniqueness and extremal
properties (the ``solitary'' and ``edge-minimal'' properties
respectively). Our proof of this result is uniform and combinatorial
in that it depends only on certain properties possessed by all
semistandard posets. For certain families of semistandard lattices
some of these results were obtained in previous papers; in Proposition
5.6 we explicitly construct new weight bases for a certain family of
rank two symplectic representations. These results are used to help
obtain in Theorem 5.1 the classification of those semistandard
lattices which are supporting graphs.
\bye