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The representation theory of the symmetric groups, and the closely related cyclotomic Hecke algebras of type A, was transformed when Brundan and Kleshchev's discovered of a grading on these algebras following work of Khovanov-Lauda and Rouquier. The graded theory is harder than the "classical" approach to this subject but it reveals new features of the representation theory which we could not see before.
The underlying problems are still the same - we want to compute the (graded) dimensions and the (graded) decomposition numbers of these algebras - but there is now more structure to work with. There are indications that this new perspective may furnish us with the tools to finally answer these questions.
This theory is intimately connected with the representation theory of affine Hecke algebras and quantum groups; there are also ramifications for the representation theory of the symmetric groups and finite reductive groups.
Other active interests include:
Quiver Hecke algebras and quiver Schur algebras
Cyclotomic Hecke algebras, complex reflection groups and their braid groups.
The q-Schur algebras and cyclotomic q-Schur algebras.
Combinatorics of symmetric groups and Hecke algebras.
The theory of (graded) cellular algebras.
Affine Hecke algebras.
Quantum groups, canonical bases, and crystal graphs.
Kazhdan-Lusztig polynomials and cell representations.
Coxeter groups and groups of Lie type, and their representation theory.
The representation theory of the symmetric groups, and the closely related cyclotomic Hecke algebras of type A, was transformed when Brundan and Kleshchev's discovered of a grading on these algebras following work of Khovanov-Lauda and Rouquier. The graded theory is harder than the "classical" approach to this subject but it reveals new features of the representation theory which we could not see before.
The underlying problems are still the same - we want to compute the (graded) dimensions and the (graded) decomposition numbers of these algebras - but there is now more structure to work with. There are indications that this new perspective may furnish us with the tools to finally answer these questions.
This theory is intimately connected with the representation theory of affine Hecke algebras and quantum groups; there are also ramifications for the representation theory of the symmetric groups and finite reductive groups.
Other active interests include:
Quiver Hecke algebras and quiver Schur algebras
Cyclotomic Hecke algebras, complex reflection groups and their braid groups.
The q-Schur algebras and cyclotomic q-Schur algebras.
Combinatorics of symmetric groups and Hecke algebras.
The theory of (graded) cellular algebras.
Affine Hecke algebras.
Quantum groups, canonical bases, and crystal graphs.
Kazhdan-Lusztig polynomials and cell representations.
Coxeter groups and groups of Lie type, and their representation theory.
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Mathematische Annalenpp.1-2, (2024)
arXiv (Cornell University) (2023)
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Representation Theory of The American Mathematical Societyno. 15 (2023): 508-573
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arxiv(2023)
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REPRESENTATION THEORY (2023): 508-573
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JOURNAL OF THE LONDON MATHEMATICAL SOCIETY-SECOND SERIESno. 3 (2023): 1002-1044
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