Affine Schur--Weyl theory for loop Semigroups and polynomial representations

Abstract

Classical Schur--Weyl duality plays a fundamental role in the representation theory of general linear and symmetric groups. In this paper, we develop an affine Schur--Weyl theory for the Laurent polynomial loop semigroup. Let Ĝ_K(n) denote the Laurent polynomial loop semigroup inside the loop group GL_n(K((t))) over a field K, and let Ŝ_K(n,r) be the affine Schur algebra. The natural action of Ĝ_K(n) on the affine tensor space Ω_K^{⊗ r} induces an algebra homomorphism KĜ_K(n)→ Ŝ_K(n,r). We prove that this homomorphism is surjective for every field K satisfying |K|>r. Furthermore, we establish the double centralizer property for affine Schur algebras over unique factorization domains for all n≥2. For n≥2 and any field K with |K|>r, this yields the corresponding centralizer property for Ĝ_K(n). In contrast, for n=1 and r≥2, the corresponding natural homomorphism is not surjective over any field. For n≥ 2, we use the double centralizer theorem to determine the center of the affine Schur algebra and, when K is a field with |K|>r, show that ζ̂_r,K maps the center of KĜ_K(n) onto it. Over an algebraically closed field K of arbitrary characteristic, we prove that every finite-dimensional irreducible Ŝ_K(n,r)-module is a tensor product of evaluation modules. Moreover, we obtain explicit character formulas expressing these irreducible characters as products of characters of irreducible classical Schur algebra modules. Finally, we study polynomial representations of Ĝ_K(n) over algebraically closed fields K, and classify its finite-dimensional irreducible polynomial representations. For finite fields 𝔽_q with q>r, this yields a classification in the defining-characteristic setting.

Schur tests and estimates for positive semigroups on non commutative L_p
Clément Coine, Mélina Laforge, Éric Ricard
arXiv17 Sep 2026PreprintarXiv licenceFull textAnalysis & PDEsarXiv
Schur--Weyl duality for coideal subalgebras of quantum affine superalgebras of type A
Weideng Cui
arXiv23 Oct 2026PreprintCC0Full textAlgebra & number theoryarXiv
Affine quantum Schur--Weyl duality
Qiang Fu, Jun Hu
arXiv17 Sep 2026PreprintarXiv licenceFull textAlgebra & number theoryarXiv
Covariant representations of actions of inverse semigroups: a new approach to the reduced and essential crossed-product C*-algebras
Astrid an Huef, Ilija Tolich
arXiv22 Sep 2026PreprintarXiv licenceFull textAnalysis & PDEsarXiv
The classes of bivariate Schur and Herglotz matrix-valued rational functions: realizations, symmetrizations, and related determinantal representations
Radomił Baran, Piotr Pikul, Hugo J. Woerdeman +1
arXiv2 Sep 2026PreprintarXiv licenceFull textAnalysis & PDEsarXiv

Cite

BibTeX RIS

BibTeX

@misc{fu2026affine,
  title = {{Affine Schur--Weyl theory for loop Semigroups and polynomial representations}},
  author = {Qiang Fu},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.11282},
  eprint = {2610.11282},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.11282},
}

RIS

TY  - UNPB
TI  - Affine Schur--Weyl theory for loop Semigroups and polynomial representations
AU  - Qiang Fu
PY  - 2026
DA  - 2026/10/08
PB  - arXiv
DO  - 10.48550/arxiv.2610.11282
UR  - https://arxiv.org/abs/2610.11282
AB  - Classical Schur--Weyl duality plays a fundamental role in the representation theory of general linear and symmetric groups. In this paper, we develop an affine Schur--Weyl theory for the Laurent polynomial loop semigroup. Let Ĝ_K(n) denote the Laurent polynomial loop semigroup inside the loop group GL_n(K((t))) over a field K, and let Ŝ_K(n,r) be the affine Schur algebra. The natural action of Ĝ_K(n) on the affine tensor space Ω_K^⊗ r induces an algebra homomorphism KĜ_K(n)→ Ŝ_K(n,r). We prove that this homomorphism is surjective for every field K satisfying |K|>r. Furthermore, we establish the double centralizer property for affine Schur algebras over unique factorization domains for all n≥2. For n≥2 and any field K with |K|>r, this yields the corresponding centralizer property for Ĝ_K(n). In contrast, for n=1 and r≥2, the corresponding natural homomorphism is not surjective over any field. For n≥ 2, we use the double centralizer theorem to determine the center of the affine Schur algebra and, when K is a field with |K|>r, show that ζ̂_r,K maps the center of KĜ_K(n) onto it. Over an algebraically closed field K of arbitrary characteristic, we prove that every finite-dimensional irreducible Ŝ_K(n,r)-module is a tensor product of evaluation modules. Moreover, we obtain explicit character formulas expressing these irreducible characters as products of characters of irreducible classical Schur algebra modules. Finally, we study polynomial representations of Ĝ_K(n) over algebraically closed fields K, and classify its finite-dimensional irreducible polynomial representations. For finite fields 𝔽_q with q>r, this yields a classification in the defining-characteristic setting.
ER  -

Abstract

Related

Cite