Alternating submodules for partition algebras, rook algebras, and rook-Brauer algebras

Letting n≥2k, the partition algebra CAk≥2(n) has two one-dimensional subrepresentations that correspond in a natural way to alternating and trivial characters of symmetric group Sk. In 2019, Benkart Halverson introduced proved evaluations distinguished bases CAk(n) for nonzero elements regular CAk(n)-submodule corresponds Young symmetrizer ∑σ∈Skσ; 2016, Xiao an explicit formula analogue sign representation rook monoid algebra. this article, we lift Xiao's diagram basis evaluation CAk(n). We prove our lifting, which denote as Altk∈CAk(n), generates module under action multiplication by arbitrary Our Altk gives us cancellation-free other CAk(n)-module, with regard Halverson's lifting ∑σ∈Skσ. then use sign-reversing involution evaluate generators orbit basis, Young's N- P-functions so allow set-partition tableaux arguments, construct Young-type matrix units CA2(n) CA3(n).