Presenting Schur superalgebras

Turkey, Houssein El; Kujawa, Jonathan R.

doi:10.2140/pjm.2013.262.285

Cited by 21 publications

(33 citation statements)

References 25 publications

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“…Remark 4.4. Both proofs in [14] and [8] for the fact that Y spans U υ,Z /I r ∩ U υ,Z and Y spans U υ,Z use a PBW type basis involving all root vectors. Thus, almost all the commutation formulas in Propositions 2.3 and 2.4 have to be used in a lengthy case-by-case argument.…”

Section: 3) and Relations Formentioning

confidence: 99%

“…[M ] ! (a = b, M ≥ 1) in U υ,Z , are given in [14]. They continue to hold in the specialisation to an arbitrary commutative ring R via υ → q ∈ R:…”

Section: Introductionmentioning

confidence: 98%

“…The other case can be done similarly. We are now ready to give a presentation for S q,R (m|n, r); compare the presentation over Q(υ) in [14]. Recall the map η r,R in (3.2.5) and Remark 2.5 for a presentation of U q,R (m|n).…”

Section: Introductionmentioning

confidence: 99%

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Polynomial super representations of $$U_{q}^{\mathrm{res}}(\mathfrak {gl}_{m|n})$$ at roots of unity

Zhou

2019

J Algebr Comb

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As a homomorphic image of the hyperalgebra U q,R (m|n) associated with the quantum linear supergroup U υ (gl m|n ), we first give a presentation for the q-Schur superalgebra S q,R (m|n, r) over a commutative ring R. We then develop a criterion for polynomial supermodules of U q,F (m|n) over a filed F and use this to determine a classification of polynomial irreducible supermodules at roots of unity. This also gives classifications of irreducible S q,F (m|n, r)-supermodules for all r. As an application when m = n ≥ r and motivated by the beautiful work [3] in the classical (non-quantum) case, we provide a new proof for the Mullineux conjecture related to the irreducible modules over the Hecke algebra H q 2 ,F (S r ); see [2] for a proof without using the super theory.

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Section: 3) and Relations Formentioning

confidence: 99%

“…[M ] ! (a = b, M ≥ 1) in U υ,Z , are given in [14]. They continue to hold in the specialisation to an arbitrary commutative ring R via υ → q ∈ R:…”

Section: Introductionmentioning

confidence: 98%

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Polynomial super representations of $$U_{q}^{\mathrm{res}}(\mathfrak {gl}_{m|n})$$ at roots of unity

Zhou

2019

J Algebr Comb

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“…The analogue of the Schur-Weyl duality consists in the following assertions (see [28] for the case M = 0, and see [29,30,39] for the general case).…”

Section: Jimbo-schur-weyl Duality and Its Z/2z-graded Analoguementioning

confidence: 99%

Fusion Formulas and Fusion Procedure for the Yang-Baxter Equation

d'Andecy

2017

Algebr Represent Theor

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We use the fusion formulas of the symmetric group and of the Hecke algebra to construct solutions of the Yang-Baxter equation on irreducible representations of gl N , gl N|M , U q (gl N ) and U q (gl N|M ). The solutions are obtained via the fusion procedure for the Yang-Baxter equation, which is reviewed in a general setting. Distinguished invariant subspaces on which the fused solutions act are also studied in the general setting, and expressed, in general, with the help of a fusion function. Only then, the general construction is specialised to the four situations mentioned above. In each of these four cases, we show how the distinguished invariant subspaces are identified as irreducible representations, using the relevant fusion formula combined with the relevant Schur-Weyl duality.

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“…[10], while the latter has been introduced in [26,14] in the context of Schur-Weyl-Sergeev duality and investigated in [14] in terms of a Drinfeld-Jimbo type presentation (cf. [15]).…”

Section: Introductionmentioning

confidence: 99%

The Queer -Schur Superalgebra

Wan

2018

J. Aust. Math. Soc.

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As a natural generalisation of q-Schur algebras associated with the Hecke algebra H r,R (of the symmetric group), we introduce the queer q-Schur superalgebra associated with the Hecke-Clifford superalgebra H c r,R , which, by definition, is the endomorphism algebra of the induced H c r,R -module from certain q-permutation modules over H r,R . We will describe certain integral bases for these superalgebras in terms of matrices and will establish the base change property for them. We will also identify the queer q-Schur superalgebras with the quantum queer Schur superalgebras investigated in the context of quantum queer supergroups and provide a constructible classification of their simple polynomial representations over a certain extension of the field C(v) of complex rational functions.We dedicate the paper to the memory of Professor James A. Green

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Presenting Schur superalgebras

Cited by 21 publications

References 25 publications

Polynomial super representations of $$U_{q}^{\mathrm{res}}(\mathfrak {gl}_{m|n})$$ at roots of unity

Polynomial super representations of $$U_{q}^{\mathrm{res}}(\mathfrak {gl}_{m|n})$$ at roots of unity

Fusion Formulas and Fusion Procedure for the Yang-Baxter Equation

The Queer -Schur Superalgebra

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