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Representation Theory

Published by the American Mathematical Society, the Representation Theory (ERT) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-4165

The 2020 MCQ for Representation Theory is 0.7.

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Decompositions of small tensor powers and Larsen’s conjecture
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by Robert M. Guralnick and Pham Huu Tiep PDF
Represent. Theory 9 (2005), 138-208 Request permission

Abstract:

We classify all pairs $(G,V)$ with $G$ a closed subgroup in a classical group $\mathcal G$ with natural module $V$ over $\mathbb C$, such that $\mathcal G$ and $G$ have the same position factors on $V^{\otimes k}$ for a fixed $k\in \{2,3,4\}$. In particular, we prove Larsen’s conjecture stating that for $\dim (V)>6$ and $k=4$ there are no such $G$ aside from those containing the derived subgroup of $\mathcal G$. We also find all the examples where this fails for $\dim (V)\le 6$. As a consequence of our results, we obtain a short proof of a related conjecture of Katz. These conjectures are used in Katz’s recent works on monodromy groups attached to Lefschetz pencils and to character sums over finite fields. Modular versions of these conjectures are also studied, with a particular application to random generation in finite groups of Lie type.
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Additional Information
  • Robert M. Guralnick
  • Affiliation: Department of Mathematics, University of Southern California, Los Angeles, California 90089-1113
  • MR Author ID: 78455
  • Email: guralnic@math.usc.edu
  • Pham Huu Tiep
  • Affiliation: Department of Mathematics, University of Florida, Gainesville, Florida 32611
  • MR Author ID: 230310
  • Email: tiep@math.ufl.edu
  • Received by editor(s): March 31, 2003
  • Received by editor(s) in revised form: December 8, 2021, and December 15, 2004
  • Published electronically: February 2, 2005
  • Additional Notes: The authors gratefully acknowledge the support of the NSF (grants DMS-0236185 and DMS-0070647), and of the NSA (grant H98230-04-0066)
  • © Copyright 2005 American Mathematical Society
  • Journal: Represent. Theory 9 (2005), 138-208
  • MSC (2000): Primary 20C15, 20C20, 20C33, 20C34, 20G05, 20G40
  • DOI: https://doi.org/10.1090/S1088-4165-05-00192-5
  • MathSciNet review: 2123127