Affine quantum Schur--Weyl duality

Abstract

Let mathpzc K be an arbitrary commutative ring containing an invertible element ε. Let ℋ_vartriangle(r)_{mathpzc K} be the extended affine Hecke algebra of type A with Hecke parameter ε, let Ω_{mathpzc K}^{⊗ r} be the affine tensor space, and let 𝒮_vartriangle(n,r)_{mathpzc K} be the corresponding affine quantum Schur algebra. We first prove that the natural right action of ℋ_vartriangle(r)_{mathpzc K} on Ω_{mathpzc K}^{⊗ r} is always faithful. Assume further that mathpzc K is a field of characteristic $0$ and that ε is not a root of unity. We prove that, for any n≥ 2, the natural algebra homomorphism ξ_r:ℋ_vartriangle(r)_{mathpzc K}→End_{𝒮_vartriangle(n,r)_{mathpzc K}}(Ω_{mathpzc K}^{⊗ r})^op is an isomorphism. This proves Conjecture~3.8.8 of \cite{DDF}. As an application, we prove the conjecture formulated in \cite[5.2.4]{DDF} concerning the center of the affine quantum Schur algebra. We also prove that 𝒮_vartriangle(n,r)_{mathpzc K} is left and right Noetherian whenever mathpzc K is a Noetherian commutative ring, which verify a conjecture in \cite[Rem. 1.7]{DY}.

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BibTeX RIS

BibTeX

@misc{fu2026affine,
  title = {{Affine quantum Schur--Weyl duality}},
  author = {Qiang Fu and Jun Hu},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.19608},
  eprint = {2609.19608},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.19608},
}

RIS

TY  - UNPB
TI  - Affine quantum Schur--Weyl duality
AU  - Qiang Fu
AU  - Jun Hu
PY  - 2026
DA  - 2026/09/17
PB  - arXiv
DO  - 10.48550/arxiv.2609.19608
UR  - https://arxiv.org/abs/2609.19608
AB  - Let mathpzc K be an arbitrary commutative ring containing an invertible element ε. Let ℋ_ vartriangle (r)_mathpzc K be the extended affine Hecke algebra of type A with Hecke parameter ε, let Ω_mathpzc K^⊗ r be the affine tensor space, and let 𝒮_ vartriangle (n,r)_mathpzc K be the corresponding affine quantum Schur algebra. We first prove that the natural right action of ℋ_ vartriangle (r)_mathpzc K on Ω_mathpzc K^⊗ r is always faithful. Assume further that mathpzc K is a field of characteristic $0$ and that ε is not a root of unity. We prove that, for any n≥ 2, the natural algebra homomorphism ξ_r:ℋ_ vartriangle (r)_mathpzc K→End_𝒮_ vartriangle (n,r)_mathpzc K(Ω_mathpzc K^⊗ r)^op is an isomorphism. This proves Conjecture~3.8.8 of \cite{DDF}. As an application, we prove the conjecture formulated in \cite[5.2.4]{DDF} concerning the center of the affine quantum Schur algebra. We also prove that 𝒮_ vartriangle (n,r)_mathpzc K is left and right Noetherian whenever mathpzc K is a Noetherian commutative ring, which verify a conjecture in \cite[Rem. 1.7]{DY}.
ER  -

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