The finite-dimensional irreducible representations of finite-dimensional complex simple Lie
algebras have been studied extensively during the last 100 years by mathematicians and physi-
cists alike. These investigations have led to numerous formulas for computing their dimensions,
weights, and weight multiplicities. Although these formulas are generally tedious to perform by
hand, they lend themselves nicely to computer calculations. However even then, the computa-
tions can be done only for relatively small rank algebras because of limitations due to storage
space. In this work we consider the problem of determining information about representations
as the rank grows large, in fact, tends to infinity. Here we show, in a sense to be made more
precise later, that the set of dominant weights stabilizes as the rank goes to infinity and the
multiplicities become polynomials in the rank. In addition, we give effective, easily computable
algorithms for determining the set of dominant weights and illustrate how to calculate their
multiplicity polynomials. As we comment below, many of these questions have been addressed
previously using group characters. The proofs presented in this work are of a different, more
constructive nature and so reveal detailed information concerning the internal structure of the
modules as they establish the results.
To discuss some of our major objectives, we assume that Q is a finite-dimensional simple
Lie algebra over the complex numbers with Cartan subalgebra 7i. Let CJI, ...,u;r be a basis of
fundamental weights for the dual space H* of H. If A £ W* and A = X^i=i ^Wi w n e r e the At
are nonnegative integers, then A is a dominant weight of Q. In this work we identify A with
the infinite tuple A = (Ai,... , A
,0,0,...) where A,- £ Z + U {0}. Under this identification, if
A = (Ai,...,A
,0,0,... ) where Xp ^ 0, then A may be viewed as a dominant weight for each
simple Lie algebra having rank r p.
The finite-dimensional irreducible (/-modules are in one-to-one correspondence with the
set of dominant weights of Q. If A is a dominant weight of Q, we denote the irreducible Q-
module having highest weight A by V(X,Q). Then ^(A,^ ) = ®flen*V(X,g)fl where V(X,Q)ll =
{v £ V(X,G) | hv = fi(h)v for all h £ H}. Whenever V(A,£)M ^ (0), p is a weight of
V(X,G). The set of weights of V(X,Q) is invariant under the action of the Weyl group W((?) of
Q. Moreover, dimV(X^Q)^ = dimV(X^Q)w^ for all w £ W(Q). Since each weight is conjugate
under the Weyl group to a dominant weight, it suffices to determine the set 3(A, Q) of dominant
weights of V(X, Q) and to compute the multiplicities, dim V(X, 0)^ as \i ranges over the dominant
weights in $(A,£/).
Since our primary concern in this work is what happens as the rank grows large, we restrict
our attention to the four infinite families of simple Lie algebras: A
, 2?r, C
, Dr . Thus, we
assume that Q = Xr where X = A, B, C, Z), and that A = (Ai,... , Ap, 0,0,...) is a dominant
weight of Xr. We let rj(Xr) = r for X = A or C; r - 1 for X = B\ and r - 2 for X = D. We
offer constructive proofs of the following results concerning the module V(A,X
), its dominant
weights $(A,X
) , and their multiplicities:
(i) (Theorem 4.1) If M = Xx + 2 A2 + .. . + p Ap, then $(A,X
) = $(A,X
) for all r
satisfying rf(Xr) M.
(ii) (Theorem 4.1) $(A, Ar) C $(A, Cr) = $(A, Dr) C $(A, Br) for all r M + 2.
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