94 research outputs found

    On the Classification of Weakly Integral Modular Categories

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    We classify all modular categories of dimension 4m4m, where mm is an odd square-free integer, and all ranks 66 and 77 weakly integral modular categories. This completes the classification of weakly integral modular categories through rank 77. Our results imply that all integral modular categories of rank at most 77 are pointed (that is, every simple object has dimension 11). All strictly weakly integral (weakly integral but non-integral) modular categories of ranks 66 and 77 have dimension 4m4m, with mm an odd square free integer, so their classification is an application of our main result. The classification of rank 77 integral modular categories is facilitated by an analysis of two actions on modular categories: the Galois group of the field generated by the entries of the SS-matrix and the group of isomorphism classes of invertible simple objects. The interplay of these two actions is of independent interest, and we derive some valuable arithmetic consequences from their actions.Comment: Version 2: fixed missing metadata, version 3: corrected incomplete introduction and added theorem numbers, version 4: 32 pages, significant cosmetic revisions, version 5: final revision pre-submissio

    Endotrivial complexes

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    Let GG be a finite group, pp a prime, and kk a field of characteristic pp. We introduce the notion of an endotrivial complex of pp-permutation kGkG-modules, and study the corresponding group of endotrivial complexes, Ek(G)\mathcal{E}_k(G). Such complexes are shown to induce splendid Rickard autoequivalences of kGkG. The elements of Ek(G)\mathcal{E}_k(G) are determined uniquely by integral invariants arising from the Brauer construction and a degree 1 character G→k×G \to k^\times. Using ideas from Bouc's theory of biset functors, we provide a canonical decomposition of Ek(G)\mathcal{E}_k(G), and as an application, determine complete descriptions of Ek(G)\mathcal{E}_k(G) for abelian groups and pp-groups of normal pp-rank 1. We investigate the image of Ek(G)\mathcal{E}_k(G) in the orthogonal unit group of the trivial source ring O(T(kG))O(T(kG)) induced via the Lefschetz invariant map, and using recent results of Boltje and Carman, we determine a Frobenius stability condition an orthogonal unit must satisfy to lift to an endotrivial complex.Comment: Minor changes for version 2: fixed numerous typos and revised some proofs and statements for clarity and correctness. 36 pages, comments welcome

    Quaternary Affine-Invariant Codes

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    This thesis concerned with extended cyclic codes. The objective of this thesis is to give a full description of binary and quaternary affine-invariant codes of small dimensions. Extended cyclic codes are studied using group ring methods. Affine-invariant codes are described by their defining sets. Results are presented by enumeration of defining sets. Full description of affine-invariant codes is given for small dimension

    Unitary representations of rational Cherednik algebras and Hecke algebras

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    Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2010.Cataloged from PDF version of thesis.Includes bibliographical references (p. 49-50).We begin the study of unitary representations in the lowest weight category of rational Cherednik algebras of complex reflection groups. We provide the complete classification of unitary representations for the symmetric group, the dihedral group, as well as some additional partial results. We also study the unitary representations of Hecke algebras of complex reflection groups and provide a complete classification in the case of the symmetric group. We conclude that the KZ functor defined in [16] preserves unitarity in type A. Finally, we formulate a few conjectures concerning the classification of unitary representations for other types and the preservation of unitarity by the KZ functor and the restriction functors defined in [2].by Emanuel Stoica.Ph.D
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