Affine Hecke Algebras and Quantum Symmetric Pairs.

Author
Fan, Zhaobing [Browse]
Format
Book
Language
English
Εdition
1st ed.
Published/​Created
  • Providence : American Mathematical Society, 2023.
  • ©2023.
Description
1 online resource (108 pages)

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Summary note
"We introduce an affine Schur algebra via the affine Hecke algebra associated to Weyl group of affine type C. We establish multiplication formulas on the affine Hecke algebra and affine Schur algebra. Then we construct monomial bases and canonical bases for the affine Schur algebra. The multiplication formula allows us to establish a stabilization property of the family of affine Schur algebras that leads to the modified version of an algebra Kc n. We show that Kc n is a coideal subalgebra of quantum affine algebra Uppglnq, and Uppglnq,Kc nq forms a quantum symmetric pair. The modified coideal subalgebra is shown to admit monomial and stably canonical bases. We also formulate several variants of the affine Schur algebra and the (modified) coideal subalgebra above, as well as their monomial and canonical bases. This work provides a new and algebraic approach which complements and sheds new light on our previous geometric approach on the subject. In the appendix by four of the authors, new length formulas for the Weyl groups of affine classical types are obtained in a symmetrized fashion"-- Provided by publisher.
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Contents
  • Cover
  • Title page
  • Acknowledgment
  • Notations
  • Chapter 1. Introduction
  • 1.1. History
  • 1.2. The goal
  • 1.3. Main results
  • 1.4. The organization
  • Part 1. Affine Schur algebras
  • Chapter 2. Affine Schur algebras via affine Hecke algebras
  • 2.1. Affine Weyl groups
  • 2.2. Parabolic subgroups and cosets
  • 2.3. Affine Schur algebra via Hecke
  • 2.4. Set-valued matrices
  • 2.5. A bijection
  • 2.6. Computation in affine Schur algebra ^{ }_{ , }
  • 2.7. Isomorphism ^{ , }_{ , }≅ ^{ }_{ , }
  • Chapter 3. Multiplication formula for affine Hecke algebra
  • 3.1. Minimal length representatives
  • 3.2. Multiplication formula for affine Hecke algebra
  • 3.3. An example
  • Chapter 4. Multiplication formula for affine Schur algebra
  • 4.1. A map
  • 4.2. Algebraic combinatorics for ^{ }_{ , }
  • 4.3. Multiplication formula for ^{ }_{ , }
  • 4.4. Special cases of the multiplication formula
  • Chapter 5. Monomial and canonical bases for affine Schur algebra
  • 5.1. Bar involution on ^{ }_{ , }
  • 5.2. A standard basis in ^{ }_{ , }
  • 5.3. Multiplication formula using [ ]
  • 5.4. The canonical basis for ^{ }_{ , }
  • 5.5. A leading term
  • 5.6. A semi-monomial basis
  • 5.7. A monomial basis for ^{ }_{ , }
  • Part 2. Affine quantum symmetric pairs
  • Chapter 6. Stabilization algebra ̇^{ }_{ } arising from affine Schur algebras
  • 6.1. A BLM-type stabilization
  • 6.2. Stabilization of bar involutions
  • 6.3. Multiplication formula for ̇^{ }_{ }
  • 6.4. Monomial and stably canonical bases for ̇^{ }_{ }
  • 6.5. Isomorphism ̇^{ , }_{ }≅ ̇^{ }_{ }
  • Chapter 7. The quantum symmetric pair ( _{ }, ^{ }_{ })
  • 7.1. The algebra _{ } of Type A
  • 7.2. The algebra ^{ }_{ }
  • 7.3. The algebra ^{ }_{ } as a subquotient
  • 7.4. Comultiplication on ^{ }_{ }
  • Chapter 8. Stabilization algebras arising from other Schur algebras.
  • 8.1. Affine Schur algebras of Type
  • 8.2. Monomial and canonical bases for ^{ }_{ , }
  • 8.3. Stabilization algebra of Type
  • 8.4. Stabilization algebra of Type
  • 8.5. Stabilization algebra of Type
  • Appendix A. Length formulas in symmetrized forms by Zhaobing Fan, Chun-Ju Lai, Yiqiang Li and Li Luo
  • A.1. Dimension of generalized Schubert varieties
  • A.2. Length formulas of Weyl groups
  • Bibliography
  • Back Cover.
ISBN
1-4704-7319-4
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