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. 1
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