2025/09/17 by Volodymyr Mazorchuk, Mazorchuk, Volodymyr, Shraddha Srivastava +1
Mathematics · #Advanced Combinatorial Mathematics #Algebraic structures and combinatorial models #Combinatorics (math.CO) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Representation Theory (math.RT)
paper · pdf · doi:10.48550/arxiv.2509.14044
openalex publication_date 2025/09/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let Pk(δ), where k is a positive integer and δ some complex parameter, be the classical partition algebra over the complex numbers. In the case when δ=n, it is well-known that the algebra Pk(δ) is the centralizer of the symmetric group Sn acting on the k-fold tensor space of the natural representation of Sn, for n≥ 2k. The algebra Pk(δ) is semi-simple for generic values of δ. In this paper, we show that semi-simple partition algebras appear as the centralizer algebras for certain representations of the rook monoids given by an iterative restriction-induction of the trivial representation. Along the way, we also give a decomposition of this iterative representation of the rook monoid into various tensor spaces and show that the corresponding dimensions are given by generalized Bell numbers.