Affine walled Brauer-Clifford superalgebras

Abstract

In this paper, a notion of affine walled Brauer-Clifford superalgebras BC aff r,t is introduced over an arbitrary integral domain R containing 2 −1 . These superalgebras can be considered as affinization of walled Brauer superalgebras in [9]. By constructing infinite many homomorphisms from BC aff r,t to a class of level two walled Brauer-Clifford superagebras over C, we prove that BC aff r,t is free over R with infinite rank. We explain that any finite dimensional irreducible BC aff r,t -module over an algebraically closed field F of characteristic not 2 factors through a cyclotomic quotient of BC aff r,t , called a cyclotomic (or level k) walled Brauer-Clifford superalgebra BC k,r,t . Using a previous method on cyclotomic walled Brauer algebras in [15], we prove that BC k,r,t is free over R with super rank (k r+t 2 r+t−1 (r + t)!, k r+t 2 r+t−1 (r + t)!) if and only if it is admissible in the sense of Definition 6.4. Finally, we prove that the degenerate affine (resp., cyclotomic) walled Brauer-Clifford superalgebras defined by are isomorphic to our affine (resp., cyclotomic) walled Brauer-Clifford superalgebras. Contents 1. Introduction 1 2. Walled Brauer-Clifford superalgebras 3 3. Affine walled Brauer-Clifford superalgebras 10 4. A basis of BC 2,r,t with special parameters 13 5. Homomorphisms between BC r,t and BC 2,r+k,t+k 18 6. A basis of the cyclotomic Brauer-Clifford superalgebra 25 7. Isomorphisms between affine (resp., cyclotomic) Brauer-Clifford algebras and Comes-Kujawa's affine (resp., cyclotomic) algebras 30 References 33

A proof of Comes-Kujawa's conjecture

“…The aim of this section is to establish connections between two cyclotomic walled Brauer-Clifford superalgebras defined in [8,9]. The level two cases has been dealt with in [9] under the assumption that f (t) = t 2 − u with u = 0 over the complex field.…”

“…We start by recalling the notion of cyclotomic walled Brauer-Clifford superalgebras in [9]. Let Σ r be the symmetric group in r letters.…”

“…It results in an induced surjective superalgebra homomorphism φf : BC f r,t → End OBC f κ (δ) (↓ t ↑ r ). It is easy to check that φf sends a regular monomial of BC f r,t (see [9,Definition 3.15]) to a normally ordered dotted oriented Brauer-Clifford diagram, and moreover, the images of two regular monomials are not equivalent. By [9,Corollary 3.16] and Theorem 3.12, we see that φf sends a basis of BC f r,t to a basis of End OBC f κ (δ) (↓ t ↑ r ), forcing φf to be an isomorphism.…”

“…Let AOBC κ (δ) and OBC f κ (δ) be the category obtained from AOBC κ and OBC f κ by specializing ∆ k at δ k , where δ = (δ k ) k∈Z ≥0 . It is proven in [8,9] that there is a κ-superalgebra homomorphism…”

“…In particular, ϕ sends each regular monomial of BC aff r,t (δ) in [9, Definition 3.15] to a unique equivalence class of normally ordered dotted oriented Brauer-Clifford diagram (without bubbles). The tiny difference between the isomorphisms established in [8,9] is that we number the leftmost strand as the first one in [9] while Comes and Kujawa number the rightmost as the first strand (see, e.g, (3.4)). Moreover, we use right tensor ideal in [9] while they use left tensor ideal for the definition of OBC f κ .…”