Relationship between the characters and the elementary symmetric sums of weights

Abstract

Let [phi] be a root system, [delta] = {alpha[subscript 1], ..., alpha[subscript n]} a base of [phi], [lambda] the weight lattice of [phi], lambda[subscript 1], ..., lambda[subscript n] fundamental weights, W the Weyl group of [phi], Z[lambda] the group ring of lambda over Z and zeta[lambda][superscript w] the set of elements in Z[lambda] which are invariant under W. For a weight [micro], we define the elementary symmetric sum S(e[superscript micro]) of [micro], the elementary alternating sum A(e[superscript micro]) of [micro] and the character X[subscript micro] of [micro] as follows: S(e[superscript micro]) = [sigma[subscript beta W[subscript micro]] e[superscript beta], A(e[superscript micro]) = [sigma[subscript w W]] det(w)e[superscript w(micro)] และ X[subscript micro] = A(e[superscript micro + delta])/A(e[superscript delta]) respectively, where [delta] is the half sum of all positive roots. Let S = {S(e[superscript lambda[subscript i]]) : 1 [is less than or equal to] i [is less than or equal to] n} and X = {X[lambda[subscript i]] : 1 [is less than or equal to] i [is less than or equal to] n} be the set of elementary symmetric sums of fundamental weights and set of characters of fundamental weights, respectively. It is well-known that both S and X are bases for Z-module Z[lambda][superscript W]. In this research, we are interested in finding relations between elements in the sets S and X in the case of root systems whose Dynkin diagrams are A[subscript n], B[subscript n], C[subscript n], D[subscript n] and G[subscript 2] for appropriate integers n