Rigidity of formal characters of Lie algebras (II)

TL;DR

This paper establishes a characterization of formal characters of Weyl modules for complex simple Lie algebras of types A, B, C, D, providing criteria to construct modules from lower-rank cases.

Contribution

It offers a necessary and sufficient condition for a family of elements to be the formal characters of Weyl modules in classical Lie algebra types.

Findings

Characterization of formal characters for Weyl modules

Criteria for constructing modules from lower-rank modules

Extension of previous results to types B, C, D

Abstract

For a complex simple Lie algebra of type A_l,B_l,C_l or D_l, given a family of elements f_\lambda\ in commutative ring Z[\Lambda], we show that f_\lambda\ is just the formal character of the Weyl module V(\lambda) if f_\lambda\ satisfy several natural conditions. Hence we give a necessary and sufficient condition for constructing a family of g_l-modules from a family of g_{l-1}-modules.