Jucys–Murphy elements and a presentation for partition algebras

Abstract

We give a new presentation for the partition algebras. This presentation was discovered in the course of establishing an inductive formula for the partition algebra Jucys-Murphy elements defined by Halverson and Ram [European J. Combin. 26 (2005), 869-921]. Using Schur-Weyl duality we show that our recursive formula and the original definition of Jucys-Murphy elements given by Halverson and Ram are equivalent. The new presentation and inductive formula for the partition algebra Jucys-Murphy elements given in this paper are used to construct the seminormal representations for the partition algebras in a separate paper.In §2 we recall the presentation of the partition algebras given by Halverson and Ram [HR] and East [Ea]. In §3 we state a recursion defining a family of operators (L i , L i+ 1 2 : i = 0, 1, . . .) (1.2) and establish that the operators (1.2) form a commuting family with properties analogous to the Jucys-Murphy elements that arise in the representation theory of the symmetric group. Simultaneously, we establish some basic commutativity results for certain operators denotedwhich arose in the recursive definition of the Jucys-Murphy elements (1.2). In §4 we show that the elements of (1.3) are involutions which are related to the Coxeter generators for the symmetric group in a precise way. Using the relation between the involutions (1.3) and the Coxeter generators for the symmetric group, and the properties established in §3, we derive a new presentation for the partition algebras. In §5 we give formulae for the actions of the Jucys-Murphy elements (1.2) and the involutions (1.3) on tensor space. Using Schur-Weyl duality, we demonstrate that the recursive definition of Jucys-Murphy elements given in §3 is equivalent to the definition of Jucys-Murphy elements given by Halverson and Ram [HR].

A seminormal form for partition algebras

“…, σ 2i = s i−1 s i σ 2i−2 s i s i−1 +e 2i−2 L i−1 s i e 2i−2 s i + s i e 2i−2 L i−1 s i e 2i−2 −e 2i−2 L i−1 s i−1 e 2i e 2i−1 e 2i−2 − s i e 2i−2 e 2i−1 e 2i s i−1 L i−1 e 2i−2 s i . (2.6) It is shown in [6,Theorem 5.5] that the family {L i | i 1} determined by the recursions above coincides with the family of JM elements defined by Halverson and Ram [11]. For k ∈ Z >0 , we let L k denote the JM subalgebra of A k generated by L 1 , .…”

“…Remark 2.1. In [6], the elements σ 2i and σ 2i+1 are respectively denoted by σ i+ 1 2 and σ i+1 , while e 2i and e 2i+1 are denoted by p i+ 1 2 and p i+1 respectively.…”

Defining an Affine Partition Algebra

“…These elements were originally defined diagrammatically by Halverson and Ram in [HR05]. They were later given a recursive definition by Enyang in [Eny12]. For this recursive definition, Enyang introduced new elements σ i which resemble the Coxeter generators s i .…”

“…For this recursive definition, Enyang introduced new elements σ i which resemble the Coxeter generators s i . We recall this recursive definition, and a new presentation of the partition algebra given in [Eny12] in terms of the generators e i and σ i . The following definition is the one given in Section 2.3 of [Eny13].…”

“…We started this work by trying to use the presentation for the partition algebra given by T. Halverson and A. Ram in [HR05] but were unable to define an algebra with the expected properties in this way. So we instead used the more recent presentation given by J. Enyang in [Eny12] (which uses a new set of generators) to define the affine partition algebra A aff 2k . We prove that it satisfies many properties analogous to those for other affine diagram algebras.…”