publication . Article . Preprint . 2019

The Rectangular Representation of the Double Affine Hecke Algebra via Elliptic Schur–Weyl Duality

Jordan, David; Vazirani, Monica;
Open Access
  • Published: 21 Feb 2019 Journal: International Mathematics Research Notices (issn: 1073-7928, eissn: 1687-0247, Copyright policy)
  • Publisher: Oxford University Press (OUP)
  • Country: United Kingdom
Abstract
Given a module $M$ for the algebra $\mathcal{D}_{\mathtt{q}}(G)$ of quantum differential operators on $G$, and a positive integer $n$, we may equip the space $F_n^G(M)$ of invariant tensors in $V^{\otimes n}\otimes M$, with an action of the double affine Hecke algebra of type $A_{n-1}$. Here $G= SL_N$ or $GL_N$, and $V$ is the $N$-dimensional defining representation of $G$. In this paper we take $M$ to be the basic $\mathcal{D}_{\mathtt{q}}(G)$-module, i.e. the quantized coordinate algebra $M= \mathcal{O}_{\mathtt{q}}(G)$. We describe a weight basis for $F_n^G(\mathcal{O}_{\mathtt{q}}(G))$ combinatorially in terms of walks in the type $A$ weight lattice, and sta...
Subjects
arXiv: Mathematics::Representation Theory
free text keywords: General Mathematics, Pure mathematics, Double affine Hecke algebra, Schur–Weyl duality, Mathematics, Mathematical analysis, Mathematics - Representation Theory, Mathematics - Combinatorics, Mathematics - Quantum Algebra
Funded by
EC| QuantGeomLangTFT
Project
QuantGeomLangTFT
The Quantum Geometric Langlands Topological Field Theory
  • Funder: European Commission (EC)
  • Project Code: 637618
  • Funding stream: H2020 | ERC | ERC-STG
,
NSF| Mathematical Sciences Research Institute (MSRI)
Project
  • Funder: National Science Foundation (NSF)
  • Project Code: 1440140
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Per(T ab(u)) 16 DAVID JORDAN AND MONICA VAZIRANI We can associate to SLn the quotient Sn = Sbn/hπni. Then we can think of the image, π, of π as the Dynkin diagram automorphism or the generator of the cyclic group ΛslN /Q.

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