
TL;DR
This paper constructs a natural basis for the partition algebra that makes it a monoid algebra, using combinatorial and algebraic tools, clarifying its structure in the semisimple case.
Contribution
It introduces a basis for the partition algebra as a monoid algebra, unifying combinatorial and algebraic approaches to its structure.
Findings
Constructed a basis of the partition algebra as a monoid algebra.
Provided an explicit combinatorial product rule for the basis elements.
Connected the algebra's structure to semigroup algebra characterizations.
Abstract
Wilcox has considered a twisted semigroup algebra structure on the partition algebra , but it appears that there has not previously been any known basis that gives the structure of a "non-twisted" semigroup algebra or a monoid algebra. This motivates the following problem, for the non-degenerate case whereby so that is semisimple. How could a basis of be constructed so that is closed under the multiplicative operation on , in such a way so that is a monoid under this operation, and how could a product rule for elements in be defined in an explicit and combinatorial way in terms of partition diagrams? We construct a basis of the desired form using Halverson and Ram's matrix unit construction for partition…
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