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Decompositions of small tensor powers and Larsen's conjecture

Author(s): Robert M. Guralnick; Pham Huu Tiep
Journal: Represent. Theory 9 (2005), 138-208.
MSC (2000): Primary 20C15, 20C20, 20C33, 20C34, 20G05, 20G40
Posted: February 2, 2005
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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 composition 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
Email: guralnic@math.usc.edu

Pham Huu Tiep
Affiliation: Department of Mathematics, University of Florida, Gainesville, Florida 32611
Email: tiep@math.ufl.edu

DOI: 10.1090/S1088-4165-05-00192-5
PII: S 1088-4165(05)00192-5
Received by editor(s): March 31, 2003
Received by editor(s) in revised form: December 8 and December 15, 2004
Posted: 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 of article: Copyright 2005, American Mathematical Society

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