Unipotent Hecke algebras: the structure, representation theory, and combinatorics

Last updated: 26 March 2015

Contents

Abstract
Acknowledgements 1 Introduction 2 Preliminaries 2.1 Hecke algebras 2.1.1 Modules and characters 2.1.2 Weight space decompositions 2.1.3 Characters in a group algebra 2.1.4 Hecke algebras 2.2 Chevalley Groups 2.2.1 Lie algebra set-up 2.2.2 >From a Chevalley basis of ${𝔤}_{s}$ to a Chevalley group 2.3 Some combinatorics of the symmetric group 2.3.1 A pictorial version of the symmetric group ${S}_{n}$ 2.3.2 Compositions, partitions, and tableaux 2.3.3 Symmetric functions 2.3.4 RSK correspondence 3 Unipotent Hecke algebras 3.1 The algebras 3.1.1 Important subgroups of a Chevalley group 3.1.2 Unipotent Hecke algebras 3.1.3 The choices $\mu$ and $\psi$ 3.1.4 A natural basis 3.2 Parabolic subalgebras of ${ℋ}_{\mu }$ 3.2.1 Weight space decompositions for ${ℋ}_{\mu }\text{-modules}$ 3.3 Multiplication of basis elements 3.3.1 Chevalley group relations 3.3.2 Local Hecke algebra relations 3.3.3 Global Hecke algebra relations 4 A basis with multiplication in the $G={GL}_{n}\left({𝔽}_{q}\right)$ case 4.1 Unipotent Hecke algebras 4.1.1 The group ${GL}_{n}\left({𝔽}_{q}\right)$ 4.1.2 A pictorial version of ${GL}_{n}\left({𝔽}_{q}\right)$ 4.1.3 The unipotent Hecke algebra ${ℋ}_{\mu }$ 4.2 An indexing for the standard basis of ${ℋ}_{\mu }$ 4.3 Multiplication in ${ℋ}_{\mu }$ 4.3.1 Pictorial versions of ${e}_{\mu }v{e}_{\mu }$ 4.3.2 Relations for multiplying basis elements 4.3.3 Computing ${\phi }_{k}$ via painting, paths and sinks 4.3.4 A multiplication algorithm 5 Representation theory in the $G={GL}_{n}\left({𝔽}_{q}\right)$ case 5.1 The representation theory of ${ℋ}_{\mu }$ 5.2 A generalization of the RSK correspondence 5.3 Zelevinsky's decomposition of ${\text{Ind}}_{U}^{G}\left({\psi }_{\mu }\right)$ 5.3.1 Preliminaries to the proof (1) 5.3.2 The decomposition of ${\text{Ind}}_{U}^{G}\left({\psi }_{\left(n\right)}\right)$ (2) 5.3.3 Decomposition of ${\text{Ind}}_{U}^{G}\left({\psi }_{\mu }\right)$ (3) 5.4 A weight space decomposition of ${ℋ}_{\mu }\text{-modules}$ 6 The representation theory of the Yokonuma algebra 6.1 General type 6.1.1 A reduction theorem 6.1.2 The algebras ${𝒯}_{\gamma }$ 6.2 The $G={GL}_{n}\left({𝔽}_{q}\right)$ case 6.2.1 The Yokonuma algebra and the Iwahori-Hecke algebra 6.2.2 The representation theory of ${𝒯}_{\nu }$ 6.2.3 The irreducible modules of ${ℋ}_{1}$ A Commutation Relations
Bibliography

Notes and References

This is an excerpt of the PhD thesis Unipotent Hecke algebras: the structure, representation theory, and combinatorics by F. Nathaniel Edgar Thiem.

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