A modified Brauer algebra as centralizer algebra of the unitary group

We introduce a new family of superalgebras − → B r,s for r, s ≥ 0 such that r + s > 0, which we call the walled Brauer superalgebras, and prove the mixed Scur-Weyl-Sergeev duality for queer Lie superalgebras. More precisely, let q(n) be the queer Lie superalgebra, V = C n|n the natural representation of q(n) and W the dual of V. We prove that, if n ≥ r + s, the superalgebra − → B r,s is isomorphic to the supercentralizer algebra End q(n) (V ⊗r ⊗W ⊗s ) op of the q(n)-action on the mixed tensor space V ⊗r ⊗W ⊗s . As an ingredient for the proof of our main result, we construct a new diagrammatic realization − → D k of the Sergeev superalgebra Ser k . Finally, we give a presentation of − → B r,s in terms of generators and relations.2000 Mathematics Subject Classification. 17B10, 05E10. Key words and phrases. mixed Schur-Weyl-Sergeev duality, queer Lie superalgebras, walled Brauer superalgebras.

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“…The relations (1)-(4) were verified in the proof of [5,Theorem 4.10]. The relation (5.4) implies e r−1,r+2 e r,r+1 = e r,r+1 e r−1,r+2 .…”

Section: Proof Of Stepmentioning

confidence: 71%

“…Remark 2.3. There have been a lot of works done on diagram algebras with marked edges or marked vertices (see, for example, [2,5,10,11]). This work is different from those in that we deal with superalgebras.…”

Section: Diagrammatic Realization Of Sergeev Superalgebrasmentioning

confidence: 99%

Mixed Schur–Weyl–Sergeev duality for queer Lie superalgebras

Jung

Kang

2014

Journal of Algebra

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“…The category D T is generated as a monoidal category by the two elements (10) 5.3. The general definition.…”

Section: Diagram Categoriesmentioning

confidence: 99%

“…Furthermore C Br is generated as a category by these two subcategories. It follows that C Br is generated as a monoidal category by the morphisms ( 8) and (10).…”

Section: The Brauer Categorymentioning

confidence: 99%

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A combinatorial approach to classical representation theory

Rubey¹,

Westbury²

2014

Preprint

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A fundamental problem from invariant theory is to describe, given a representation V of a group G, the algebra End G (⊗ r V ) of multilinear functions invariant under the action of the group. According to Weyl's classic, a first main (later: 'fundamental') theorem of invariant theory provides a finite spanning set for this algebra, whereas a a second main theorem describes the linear relations between those basic invariants.Here we use diagrammatic methods to carry Weyl's programme a step further, providing explicit bases for the subspace of ⊗ r V invariant under the action of G, that are additionally preserved by the action of the long cycle of the symmetric group S r .The representations we study are essentially those that occur in Weyl's book, namely the defining representations of the symplectic groups, the defining representations of the symmetric groups considered as linear representations, and the adjoint representations of the linear groups.In particular, we present a transparent, combinatorial proof of a second fundamental theorem for the defining representation of the symplectic groups Sp(2n). Our formulation is completely explicit and provides a very precise link to (n+1)-noncrossing perfect matchings, going beyond a dimension count. Extending our argument to the k-th symmetric powers of these representations, the combinatorial objects involved turn out to be (n + 1)-noncrossing k-regular graphs. As corollaries we obtain instances of the cyclic sieving phenomenon for these objects and the natural rotation action.In general, we are able to derive branching rules for the diagram algebras corresponding to the representations in a uniform way. Moreover, we compute the Frobenius characteristics of modules of the diagram algebras restricted to the action of the symmetric group. Via a general theorem, we also obtain the isotypic decomposition of ⊗ r V when n is large enough in comparison to r.

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A modified Brauer algebra as centralizer algebra of the unitary group

Cited by 2 publications

References 20 publications

Mixed Schur–Weyl–Sergeev duality for queer Lie superalgebras

Mixed Schur–Weyl–Sergeev duality for queer Lie superalgebras

A combinatorial approach to classical representation theory

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