By M. Aizenman (Chief Editor)
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Extra info for Communications in Mathematical Physics - Volume 216
Choose any total order compatible with this partial order, so we have λ = λ1 > λ2 > λ3 > . . At the first step we expand m+ in all directions. Then we expand in all directions all monomials of weight λ1 obtained at the first step. Then we expand in all directions all monomials of weight λ2 obtained at the previous steps, and so on. Since the monomials obtained in the expansion of a monomial of Uq g-weight µ have weights less than µ, the result does not depend on the choice of the total order.
1. , it does not contain any factors Ai,c ). Proof. 1. 1 for fundamental representations Vωi (a). 1 will be true for any tensor product of the fundamental representations. 4, any irreducible representation V can be represented as a quotient of a submodule of a tensor product W of fundamental representations, which is generated by the highest weight vector. Therefore each monomial in a q-character of V is also a monomial in the q-character of W . In addition, the highest weight monomials of the q-characters of V and W coincide.
3(3), the module V has highest weight λ = i∈I deg Pi · ωi , which has multiplicity 1. 4 implies that χq (V ) contains a unique monomial of weight λ. This monomial equals ni i∈I k=1 Yi,a (i) . 13) k We call it the highest weight monomial of V . All other monomials in χq (V ) have lower weight than λ. , if it is a product of Yi,a ’s in positive powers only). The highest weight monomial is dominant, but in general the highest weight monomial is not the only dominant monomial occurring in χq (V ). 5 that the only dominant monomial contained in the q-character of a fundamental representation Vωi (a) is its highest weight monomial Yi,a .
Communications in Mathematical Physics - Volume 216 by M. Aizenman (Chief Editor)