An Extended Schur ’ S Lemma and Its Application

The Springer modules have a combinatorial property called " coincidence of dimensions, " i.e., the Springer modules are naturally decomposed into submodules with common dimensions. Morita and Nakajima proved the property by giving modules with common dimensions whose induced modules are isomorphic to the submodules of Springer modules. They proved that the induced modules are isomorphic to the submodules, by showing the coincidence of their characters. Our aim is to construct isomorphisms between the induced modules and their corresponding submodules in a combinatorial manner. For this purpose, we show lemmas, which are equivalent to the classical Schur's lemma in special cases. We also give a procedure to construct isomorphisms, and explicitly construct isomorphisms in the case of the Springer modules corresponding to Young diagrams of two rows.