Conjugacy action, induced representations and the Steinberg square for simple groups of Lie type

Gerhard Heide; Jan Saxl; Pham Huu Tiep; Alexandre E. Zalesski

doi:10.1112/plms/pds062

Conjugacy action, induced representations and the Steinberg square for simple groups of Lie type

Gerhard Heide, Jan Saxl, Pham Huu Tiep, Alexandre E. Zalesski

Mathematics

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20 Scopus citations

Abstract

Let G be a finite simple group of Lie type, and let π_G be the permutation representation of G associated with the action of G on itself by conjugation. We prove that every irreducible complex representation of G is a constituent of π_G, unless G=PSU_n(q) and n≥3 is coprime to 2(q+1), where precisely one irreducible representation fails. We also prove that every irreducible representation of G is a constituent of the tensor square St ⊗ St of the Steinberg representation St of G, with the same exceptions as in the previous statement.

Original language	English (US)
Pages (from-to)	908-930
Number of pages	23
Journal	Proceedings of the London Mathematical Society
Volume	106
Issue number	4
DOIs	https://doi.org/10.1112/plms/pds062
State	Published - Apr 2013

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Mathematics(all)

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title = "Conjugacy action, induced representations and the Steinberg square for simple groups of Lie type",

abstract = "Let G be a finite simple group of Lie type, and let πG be the permutation representation of G associated with the action of G on itself by conjugation. We prove that every irreducible complex representation of G is a constituent of πG, unless G=PSUn(q) and n≥3 is coprime to 2(q+1), where precisely one irreducible representation fails. We also prove that every irreducible representation of G is a constituent of the tensor square St ⊗ St of the Steinberg representation St of G, with the same exceptions as in the previous statement.",

author = "Gerhard Heide and Jan Saxl and Tiep, {Pham Huu} and Zalesski, {Alexandre E.}",

note = "Funding Information: Received 19 March 2012; published online 9 November 2012. 2010 Mathematics Subject Classification 20B05, 20C05, 20C33, 20C15, 20D06. The third author is supported by the NSF (grants DMS-0901241 and DMS-1201374).",

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doi = "10.1112/plms/pds062",

language = "English (US)",

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T1 - Conjugacy action, induced representations and the Steinberg square for simple groups of Lie type

AU - Heide, Gerhard

AU - Saxl, Jan

AU - Tiep, Pham Huu

AU - Zalesski, Alexandre E.

N1 - Funding Information: Received 19 March 2012; published online 9 November 2012. 2010 Mathematics Subject Classification 20B05, 20C05, 20C33, 20C15, 20D06. The third author is supported by the NSF (grants DMS-0901241 and DMS-1201374).

PY - 2013/4

Y1 - 2013/4

N2 - Let G be a finite simple group of Lie type, and let πG be the permutation representation of G associated with the action of G on itself by conjugation. We prove that every irreducible complex representation of G is a constituent of πG, unless G=PSUn(q) and n≥3 is coprime to 2(q+1), where precisely one irreducible representation fails. We also prove that every irreducible representation of G is a constituent of the tensor square St ⊗ St of the Steinberg representation St of G, with the same exceptions as in the previous statement.

AB - Let G be a finite simple group of Lie type, and let πG be the permutation representation of G associated with the action of G on itself by conjugation. We prove that every irreducible complex representation of G is a constituent of πG, unless G=PSUn(q) and n≥3 is coprime to 2(q+1), where precisely one irreducible representation fails. We also prove that every irreducible representation of G is a constituent of the tensor square St ⊗ St of the Steinberg representation St of G, with the same exceptions as in the previous statement.

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Conjugacy action, induced representations and the Steinberg square for simple groups of Lie type

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