Tabloids and Weighted Sums of Characters of Certain Modules of the Symmetric Groups

Abstract

We consider certain modules of the symmetric groups whose basis elements are called tabloids. Some of these modules are isomorphic to subspaces of the cohomology rings of subvarieties of flag varieties as modules of the symmetric groups. We give a combinatorial description for some weighted sums of their characters, i.e., we introduce combinatorial objects called $(ρ,\Ll)$-tabloids and rewrite weighted sums of characters as the numbers of these combinatorial objects. We also consider the meaning of these combinatorial objects, i.e., we construct a correspondence between $(ρ,\Ll)$-tabloids and tabloids whose images are eigenvectors of the action of an element of cycle type $ρ$ in quotient modules.