Imaginary Schur-Weyl Duality.
Kleshchev, Alexander.
Imaginary Schur-Weyl Duality. - 1st ed. - 1 online resource (108 pages) - Memoirs of the American Mathematical Society ; v.245 . - Memoirs of the American Mathematical Society .
Cover -- Title page -- Chapter 1. Introduction -- 1.1. Convex preorders and cuspidal systems -- 1.2. Imaginary representations -- 1.3. Imaginary Schur-Weyl duality -- 1.4. Imaginary Howe and Ringel dualities -- 1.5. Gelfand-Graev words and representations -- 1.6. Example: type ₁⁽¹⁾ -- 1.7. Structure of the paper -- 1.8. Acknowledgements -- Chapter 2. Preliminaries -- 2.1. Partitions and compositions -- 2.2. Coset representatives -- 2.3. Schur algebras -- 2.4. Representation theory of Schur algebras -- 2.5. Induction and restriction for Schur algebras -- 2.6. Schur functors -- Chapter 3. Khovanov-Lauda-Rouquier algebras -- 3.1. Lie theoretic notation -- 3.2. The definition and first properties -- 3.3. Basic representation theory of _ -- 3.4. Induction, coinduction, and duality for KLR algebras -- 3.5. Crystal operators and extremal words -- 3.6. Mackey Theorem -- 3.7. Convex preorders and root partitions -- 3.8. Cuspidal systems and standard modules -- 3.9. Colored imaginary tensor spaces -- Chapter 4. Imaginary Schur-Weyl duality -- 4.1. Imaginary tensor space and its parabolic analogue -- 4.2. Action of \Si_ on _ -- 4.3. Imaginary Schur algebras -- 4.4. Characteristic zero theory -- 4.5. Imaginary induction and restriction -- Chapter 5. Imaginary Howe duality -- 5.1. Gelfand-Graev modules -- 5.2. Imaginary symmetric, divided, and exterior powers -- 5.3. Parabolic analogues -- 5.4. Schur algebras as endomorphism algebras -- 5.5. Projective generator for imaginary Schur algebra -- Chapter 6. Morita equaivalence -- 6.1. Morita equivalence functors -- 6.2. Induction and Morita equivalence -- 6.3. Alternative definitions of standard modules -- 6.4. Base change -- 6.5. Ringel duality and double centralizer properties -- Chapter 7. On formal characters of imaginary modules -- 7.1. Weight multiplicities in \La^}( _). 7.2. Gelfand-Graev words and shuffles -- 7.3. Gelfand-Graev fragment of the formal character of \Dede(\la) -- 7.4. Imaginary Jacobi-Trudy formula -- Chapter 8. Imaginary tensor space for non-simply-laced types -- 8.1. Minuscule representations for non-simply-laced types -- 8.2. The endomorphism ᵣ: _→ _ for non-simply-laced types -- Bibliography -- Back Cover.
The authors study imaginary representations of the Khovanov-Lauda-Rouquier algebras of affine Lie type. Irreducible modules for such algebras arise as simple heads of standard modules. In order to define standard modules one needs to have a cuspidal system for a fixed convex preorder. A cuspidal system consists of irreducible cuspidal modules-one for each real positive root for the corresponding affine root system _l^, as well as irreducible imaginary modules-one for each l-multiplication. The authors study imaginary modules by means of "imaginary Schur-Weyl duality" and introduce an imaginary analogue of tensor space and the imaginary Schur algebra. They construct a projective generator for the imaginary Schur algebra, which yields a Morita equivalence between the imaginary and the classical Schur algebra, and construct imaginary analogues of Gelfand-Graev representations, Ringel duality and the Jacobi-Trudy formula.
9781470436032
Duality theory (Mathematics).
Electronic books.
QA564.K547 2017
512.48199999999997
Imaginary Schur-Weyl Duality. - 1st ed. - 1 online resource (108 pages) - Memoirs of the American Mathematical Society ; v.245 . - Memoirs of the American Mathematical Society .
Cover -- Title page -- Chapter 1. Introduction -- 1.1. Convex preorders and cuspidal systems -- 1.2. Imaginary representations -- 1.3. Imaginary Schur-Weyl duality -- 1.4. Imaginary Howe and Ringel dualities -- 1.5. Gelfand-Graev words and representations -- 1.6. Example: type ₁⁽¹⁾ -- 1.7. Structure of the paper -- 1.8. Acknowledgements -- Chapter 2. Preliminaries -- 2.1. Partitions and compositions -- 2.2. Coset representatives -- 2.3. Schur algebras -- 2.4. Representation theory of Schur algebras -- 2.5. Induction and restriction for Schur algebras -- 2.6. Schur functors -- Chapter 3. Khovanov-Lauda-Rouquier algebras -- 3.1. Lie theoretic notation -- 3.2. The definition and first properties -- 3.3. Basic representation theory of _ -- 3.4. Induction, coinduction, and duality for KLR algebras -- 3.5. Crystal operators and extremal words -- 3.6. Mackey Theorem -- 3.7. Convex preorders and root partitions -- 3.8. Cuspidal systems and standard modules -- 3.9. Colored imaginary tensor spaces -- Chapter 4. Imaginary Schur-Weyl duality -- 4.1. Imaginary tensor space and its parabolic analogue -- 4.2. Action of \Si_ on _ -- 4.3. Imaginary Schur algebras -- 4.4. Characteristic zero theory -- 4.5. Imaginary induction and restriction -- Chapter 5. Imaginary Howe duality -- 5.1. Gelfand-Graev modules -- 5.2. Imaginary symmetric, divided, and exterior powers -- 5.3. Parabolic analogues -- 5.4. Schur algebras as endomorphism algebras -- 5.5. Projective generator for imaginary Schur algebra -- Chapter 6. Morita equaivalence -- 6.1. Morita equivalence functors -- 6.2. Induction and Morita equivalence -- 6.3. Alternative definitions of standard modules -- 6.4. Base change -- 6.5. Ringel duality and double centralizer properties -- Chapter 7. On formal characters of imaginary modules -- 7.1. Weight multiplicities in \La^}( _). 7.2. Gelfand-Graev words and shuffles -- 7.3. Gelfand-Graev fragment of the formal character of \Dede(\la) -- 7.4. Imaginary Jacobi-Trudy formula -- Chapter 8. Imaginary tensor space for non-simply-laced types -- 8.1. Minuscule representations for non-simply-laced types -- 8.2. The endomorphism ᵣ: _→ _ for non-simply-laced types -- Bibliography -- Back Cover.
The authors study imaginary representations of the Khovanov-Lauda-Rouquier algebras of affine Lie type. Irreducible modules for such algebras arise as simple heads of standard modules. In order to define standard modules one needs to have a cuspidal system for a fixed convex preorder. A cuspidal system consists of irreducible cuspidal modules-one for each real positive root for the corresponding affine root system _l^, as well as irreducible imaginary modules-one for each l-multiplication. The authors study imaginary modules by means of "imaginary Schur-Weyl duality" and introduce an imaginary analogue of tensor space and the imaginary Schur algebra. They construct a projective generator for the imaginary Schur algebra, which yields a Morita equivalence between the imaginary and the classical Schur algebra, and construct imaginary analogues of Gelfand-Graev representations, Ringel duality and the Jacobi-Trudy formula.
9781470436032
Duality theory (Mathematics).
Electronic books.
QA564.K547 2017
512.48199999999997