Young-Type Matrix Units for Non-Propagating Partition Algebra Submodules

We construct bases of Young-type matrix units for all irreducible, non-propagating representations for partition algebras. In contrast to Halverson and Ram’s application of Bourbaki’s basic construction, our matrix units are not constructed recursively from smaller-order partition algebras, and our explicit construction for non-propagating matrix units is by direct analogy with Young’s classical construction for symmetric group algebra matrix units. Our matrix unit bases, which we construct using the semimodularity of partition lattices and define via an analogue of Young’s idempotent γ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma $$\end{document}-elements, cannot be obtained from Enyang’s seminormal form for individual isomorphic copies of the irreducible representations of partition algebras.