On tensor spaces for rook monoid algebras

Abstract

Abstract. Let m, n ∈ N, and V be a m-dimensional vector space over a field F of characteristic 0. Let U = F ⊕ V and Rn be the rook monoid. In this paper, we construct a certain quasi-idempotent in the annihilator of U ⊗n in F Rn, which comes from some one-dimensional two-sided ideal of rook monoid algebra. We show that the two-sided ideal generated by this element is indeed the whole annihilator of U ⊗n in F Rn.

The Annihilator of Tensor Space in the -Rook Monoid Algebra

“…It is surprising to some extent that in both the symplectic and orthogonal cases and their quantised versions, the annihilator of n-tensor space in a specialised Brauer algebra or BMW algebra is generated by an explicitly described quasi-idempotent. Motivated by these results, we have found that the annihilator of tensor space U ⊗n in a rook monoid algebra (the case q = 1 in the present paper) is also generated by a quasi-idempotent [18]. We shall construct a quasi-idempotent Φ m+1 (see Section 3) in Ker ϕ and prove the following result.…”

“…The classical case (q = 1). In this subsection, we recall the main results of [18] for later use. Let R n be the set of all n × n matrices that contain at most one entry equal to 1 in each row and column and zeros elsewhere.…”

“…For any positive integer k ≤ n, the natural map s i → s i , p j → p j for all 1 ≤ i ≤ k − 1 and 1 ≤ j ≤ k extends to an algebra embedding from CR k into CR n . In [18,Section 4], when m < n, we defined a quasi-idempotent…”

“…By Propositions 2.5 and 2.4, to construct the generators of Ker(ϕ), we only need to construct a q-analogue of Y m+1 . In other words, we need to construct an element Φ m+1 ∈ R m+1 (q) having the one-dimensional sign representation of R m+1 (q) (see [18,Section 3]), that is,…”

“…Note that one of the main differences between q-rook monoid algebras and the Hecke algebras, Brauer algebras and BMW algebras is that the q-rook monoid algebra R n (q) generally cannot be realised as a diagram algebra except in the case of q = 1 (see [5,Remark 4.4]). Therefore our proof of Theorem 1.2 differs from that in [8,11,18] and we will view R n (q) as a module of the Hecke algebra of a symmetric group.…”