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

Letting n & GE; 2k, the partition algebra CAk & GE;2(n) has two one-dimensional subrepresentations that correspond in a natural way to the alternating and trivial characters of the symmetric group Sk. In 2019, Benkart and Halverson introduced and proved evaluations in the two distinguished bases of CAk(n) for nonzero elements in the one-dimensional regular CAk(n)-submodule that corresponds to the Young symmetrizer E & sigma;& ISIN;Sk & sigma;; in 2016, Xiao proved an explicit formula for the analogue of the sign representation for the rook monoid algebra. In this article, we lift Xiao's formula to a diagram basis evaluation in the partition algebra CAk(n). We prove that our diagram basis evaluation for this lifting, which we denote as Altk & ISIN; CAk(n), generates a one-dimensional module under the action of multiplication by arbitrary elements in CAk(n). Our explicit formula for Altk gives us a cancellation-free formula for the other one-dimensional regular CAk(n)-module, with regard to Benkart and Halverson's lifting of E & sigma;& ISIN;Sk & sigma;. We then use a sign-reversing involution to evaluate our one-dimensional generators in the orbit basis, and we use our explicit formula for Altk to lift Young's N-and P-functions so as to allow set-partition tableaux as arguments, and we use this lifting to construct Young-type matrix units for CA2(n) and CA3(n).& COPY; 2023 Elsevier B.V. All rights reserved.