Introduction

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

r

,0,0,...) where A,- £ Z + U {0}. Under this identification, if

A = (Ai,...,A

p

,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

r

, 2?r, C

r

, 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

r

), its dominant

weights $(A,X

r

) , and their multiplicities:

(i) (Theorem 4.1) If M = Xx + 2 A2 + .. . + p Ap, then $(A,X

r

) = $(A,X

M

) 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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