The rook partition algebra

Abstract

The rook partition algebra RP k (x) is a generically semisimple algebra that arises from looking at what commutes with the action of the symmetric group S n on U ⊗k , where U is the direct sum of the natural representation and the trivial representation of S n . We give a combinatorial description of this algebra, construct its irreducible representations, and exhibit a Murnaghan-Nakayama formula to compute certain character values.

Schur–Weyl dualities for symmetric inverse semigroups

“…In the latter paper it was shown that the representation V of GL(n) can be slightly modified such that the centralizing object obtained on the right-hand side is the symmetric inverse semigroup IS n , introduced in [24] and also known as the rook monoid, see [20]. This idea of modification of V was recently used in [6] to obtain a Schur-Weyl duality between S n and a generalization of the partition algebra, called in [6] the rook partition algebra.…”

Rook version of colored partition algebras

“…Hence we have P k (n) ⊆ P k+ 1 2 (n). The rook partition algebra P k+ 1 2 (n) has been realized as a subalgebra of the partition algebra P k+1 (n) as the span of all partition diagrams in which the last two vertices (k +1th and 2(k +1)th) are in a same class, see for example [2,3,9].…”

“…The rook (or half) partition algebras have been introduced by Martin and Rollet [9], also studied by Halverson and Ram [3] and Grood [2] with different notations. We will use the notation P k+ 1 2 (x) for the half partition algebra.…”

The Hopf algebra of uniform block permutations

“…Note that σ is invertible and id A is idempotent, by (13). It follows that P n is a factorizable inverse monoid [6, Section 2], [20, Chapter 2.2].…”

“…This paper explores one aspect of the relationship between Schur-Weyl duality, diagram algebras, and inverse semigroups whose general study was started by Solomon [27] and continued by several authors [13][14][15]19]. We thank the referees for bringing up these and other references to our attention.…”