Jucys–Murphy elements of partition algebras for the rook monoid.

Kudryavtseva and Mazorchuk exhibited Schur–Weyl duality between the rook monoid algebra ℂ R n and the subalgebra ℂ I k of the partition algebra ℂ A k (n) acting on (ℂ n) ⊗ k . In this paper, we consider a subalgebra ℂ I k + 1 2 of ℂ I k + 1 such that there is Schur–Weyl duality between the actions of ℂ R n − 1 and ℂ I k + 1 2 on (ℂ n) ⊗ k . This paper studies the representation theory of partition algebras ℂ I k and ℂ I k + 1 2 for rook monoids inductively by considering the multiplicity free tower ℂ I 1 ⊂ ℂ I 3 2 ⊂ ℂ I 2 ⊂ ⋯ ⊂ ℂ I k ⊂ ℂ I k + 1 2 ⊂ ⋯. Furthermore, this inductive approach is established as a spectral approach by describing the Jucys–Murphy elements and their actions on the canonical Gelfand–Tsetlin bases, determined by the aforementioned multiplicity free tower, of irreducible representations of ℂ I k and ℂ I k + 1 2 . Also, we describe the Jucys–Murphy elements of ℂ R n which play a central role in the demonstration of the actions of Jucys–Murphy elements of ℂ I k and ℂ I k + 1 2 . [ABSTRACT FROM AUTHOR]