Alternating submodules for partition algebras, rook algebras, and rook-Brauer algebras
Résumé
Letting $n \geq 2k$, the partition algebra $\mathbb{C}A_{k \geq 2}(n)$ has two one-dimensional subrepresentations that correspond in a natural way to the alternating and trivial characters of the symmetric group $S_{k}$. In 2019, Benkart and Halverson introduced and proved evaluations in the two distinguished bases of $\mathbb{C}A_k(n)$ for nonzero elements in the one-dimensional regular $\mathbb{C}A_k(n)$-submodule that corresponds to the Young symmetrizer $\sum_{\sigma \in S_k} \sigma$; 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 $\mathbb{C}A_{k}(n)$. We prove that our diagram basis evaluation for this lifting, which we denote as $\textsf{Alt}_{k} \in \mathbb{C}A_k(n)$, generates a one-dimensional module under the action of multiplication by arbitrary elements in $\mathbb{C}A_k(n)$. Our explicit formula for $\textsf{Alt}_{k}$ gives us a cancellation-free formula for the other one-dimensional regular $\mathbb{C}A_k(n)$-module, with regard to Benkart and Halverson's lifting of $\sum_{\sigma \in S_k} \sigma$. We then use a sign-reversing involution to evaluate our one-dimensional generators in the orbit basis, and we use our explicit formula for $\textsf{Alt}_{k}$ 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 $\mathbb{C}A_2(n)$ and $\mathbb{C}A_3(n)$.
Domaines
Mathématiques [math]Origine | Fichiers produits par l'(les) auteur(s) |
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