94 research outputs found
On the Classification of Weakly Integral Modular Categories
We classify all modular categories of dimension , where is an odd
square-free integer, and all ranks and weakly integral modular
categories. This completes the classification of weakly integral modular
categories through rank . Our results imply that all integral modular
categories of rank at most are pointed (that is, every simple object has
dimension ). All strictly weakly integral (weakly integral but non-integral)
modular categories of ranks and have dimension , with an odd
square free integer, so their classification is an application of our main
result. The classification of rank 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 -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
Let be a finite group, a prime, and a field of characteristic
. We introduce the notion of an endotrivial complex of -permutation
-modules, and study the corresponding group of endotrivial complexes,
. Such complexes are shown to induce splendid Rickard
autoequivalences of . The elements of are determined
uniquely by integral invariants arising from the Brauer construction and a
degree 1 character . Using ideas from Bouc's theory of biset
functors, we provide a canonical decomposition of , and as an
application, determine complete descriptions of for abelian
groups and -groups of normal -rank 1. We investigate the image of
in the orthogonal unit group of the trivial source ring
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
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
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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