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Kazhdan-Lusztig bases of parabolic Hecke algebras and applications to Schur-Weyl duality

Jeremie Guilhot, Loic Poulain d'Andecy

Abstract

With an eye to applications to type A and Schur-Weyl duality, we study Kazhdan-Lusztig bases for a general parabolic Hecke algebra. Parabolic Hecke algebras are idempotent subalgebras of Hecke algebras corresponding to parabolic subgroups, and for type A they coincide with the fused Hecke algebras appearing in a generalisation of the Schur-Weyl duality with the quantum group of GL(N). In this paper we investigate two different Kazhdan-Lusztig bases for parabolic Hecke algebras, together with the associated cells and the corresponding representations. We quickly specialise to type A, for which we describe the cells in terms of the RSK correspondence generalising thus the well-known description for the symmetric group. As a first application we recover the classification of irreducible representations of parabolic Hecke algebras of type A and provide a new construction of these representations. Next we turn to the Schur-Weyl duality and describe the kernel in terms of one the basis studied precedently. Moreover, we formulate some conjectures about a generator of these kernels in terms of Kazhdan-Lusztig basis elements, give some evidence and prove these conjectures in some special cases.

Kazhdan-Lusztig bases of parabolic Hecke algebras and applications to Schur-Weyl duality

Abstract

With an eye to applications to type A and Schur-Weyl duality, we study Kazhdan-Lusztig bases for a general parabolic Hecke algebra. Parabolic Hecke algebras are idempotent subalgebras of Hecke algebras corresponding to parabolic subgroups, and for type A they coincide with the fused Hecke algebras appearing in a generalisation of the Schur-Weyl duality with the quantum group of GL(N). In this paper we investigate two different Kazhdan-Lusztig bases for parabolic Hecke algebras, together with the associated cells and the corresponding representations. We quickly specialise to type A, for which we describe the cells in terms of the RSK correspondence generalising thus the well-known description for the symmetric group. As a first application we recover the classification of irreducible representations of parabolic Hecke algebras of type A and provide a new construction of these representations. Next we turn to the Schur-Weyl duality and describe the kernel in terms of one the basis studied precedently. Moreover, we formulate some conjectures about a generator of these kernels in terms of Kazhdan-Lusztig basis elements, give some evidence and prove these conjectures in some special cases.
Paper Structure (71 sections, 15 theorems, 190 equations)

This paper contains 71 sections, 15 theorems, 190 equations.

Table of Contents

  1. Introduction
  2. Content of the paper.
  3. Organisation.
  4. Acknowledgements.
  5. Preliminaries and notations on Hecke algebras
  6. Definition and standard basis
  7. Kazhdan--Lusztig bases
  8. Cells and representations
  9. Orders and cells.
  10. Ideals and representations.
  11. The particular case of type A
  12. Generators and relations.
  13. The Robinson--Schensted correspondence for permutations in $S_n$.
  14. Cells and RS correspondence.
  15. Cells and representations.
  16. ...and 56 more sections

Key Result

Proposition 3.2

Let $\mathcal{D},\mathcal{D}'\in W_J\backslash W/ W_J$. We have:

Theorems & Definitions (55)

  • Example 3.1
  • Proposition 3.2
  • proof
  • Definition 3.3
  • Remark 3.4
  • Proposition 4.1
  • proof
  • Example 4.2
  • Proposition 4.3
  • proof
  • ...and 45 more