2025/07/18 by Campbell, John M.
#05E10 #Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.2507.14313
Wilcox has considered a twisted semigroup algebra structure on the partition algebra ℂAk(n), but it appears that there has not previously been any known basis that gives ℂAk(n) the structure of a "non-twisted" semigroup algebra or a monoid algebra. This motivates the following problem, for the non-degenerate case whereby n ∈ ℂ ∖ \ 0, 1, …, 2 k - 2 \ so that ℂAk(n) is semisimple. How could a basis Mk = M of ℂAk(n) be constructed so that M is closed under the multiplicative operation on ℂAk(n), in such a way so that M is a monoid under this operation, and how could a product rule for elements in M be defined in an explicit and combinatorial way in terms of partition diagrams? We construct a basis M of the desired form using Halverson and Ram's matrix unit construction for partition algebras, Benkart and Halverson's bijection between vacillating tableaux and set-partition tableaux, an analogue given by Colmenarejo et al. for partition diagrams of the RSK correspondence, and a variant of a result due to Hewitt and Zuckerman characterizing finite-dimensional semisimple algebras that are isomorphic to semigroup algebras.