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Let $G$ be a finite group, $V$ an irreducible complex representation, $H$ a subgroup.
Let $V^H$ be the subspace of vectors of $V$ invariant under the action of $H$.

Question: Is $dim(V^H) \le 1$ if $H$ is a maximal subgroup of $G$ ?

Remark : If $H = \{ e \}$ then $G = \mathbb{Z}_p$ and $dim(V)=1$.

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  • $\begingroup$ Why would you expect things like this to be true? $\endgroup$
    – Derek Holt
    Feb 21, 2014 at 8:44
  • $\begingroup$ @DerekHolt : my first question on intermediate subgroup has a negative answer (you gave) with $H=1$. Here it's true for $H=1$, so I would like to know if it's true in general. Now thanks to the answer of Jack Schmidt, I know that the counter-examples are a bit rare but exist, so that it's false in general. But, perhaps there is a natural way to augment the question to something with a positive answer or open. $\endgroup$
    – Sebastien Palcoux
    Feb 21, 2014 at 14:45
  • $\begingroup$ It could be fun to compute the ratio of maximal inclusions of groups checking this property (and (or) its dual), at fixed index and up to equivalence. Perhaps it's workable with GAP. $\endgroup$
    – Sebastien Palcoux
    Feb 21, 2014 at 14:46
  • $\begingroup$ @DerekHolt: the original motivation for these questions came from subfactors theory, for testing the question 3 of the post Abelian subfactors, a relevant concept?, and I explain why in this answer. $\endgroup$
    – Sebastien Palcoux
    Feb 24, 2014 at 15:17

2 Answers 2

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$\dim(V^H) = [ \chi_H, 1_H ] = [ \chi, 1_H^G ]$ and it is a bit rare for a primitive permutation character to have a repeated factor, but not too uncommon.

There are several examples amongst simple groups: $G=L_2(11)$, $L_2(13)$, $L_2(17)$, $L_2(19)$, $L_3(3)$, $L_2(23)$, $L_2(25)$, $M_{11}$, $L_2(27)$, $L_2(29)$, $L_2(31)$, $Sz(8)$, $M_{12}$, $J_1$, $A_9$, $L_3(5)$, $J_2$, $L_2(109)$, $L_2(113)$, ..., $A_{10}$, $A_{12}$, $A_{13}$, ${}^2F_4(2)'$ for example.

For $G=L_2(11)$ we can take $H$ to be a Sylow 3-normalizer, and then one of the irreducible representations $W$ of dimension 10 (there are two, but only one works; the one with trace -1 on an element of order 6) works. This one might be a bit easier to check. In this example, one can even find $g$ so that $H \cap H^g = 1$ providing a counterexample to the “dual question.”

For $G=A_{13}$ we can take $H$ to be a Sylow 13-normalizer, and then almost every irreducible representation of $G$ works (all except the ones of degree 1, 12, 65, and 66).

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  • $\begingroup$ Thank you again ! Do you think there exist counter-examples for the principal part and the dual part ? $\endgroup$
    – Sebastien Palcoux
    Feb 21, 2014 at 0:57
  • $\begingroup$ Yes, the L2(11) one works for both. The Sylow 3-normalizer is so small, that things go very badly. You can choose K=1 again. $\endgroup$
    – Jack Schmidt
    Feb 21, 2014 at 1:01
  • $\begingroup$ Is there a condition for a Sylow normalizer to be a maximal subgroup ? For $L_2(11)$, if I'm not mistaken, a 3-Sylow normalizer is isomorphic to $D_{12}$, isn't it ? $\endgroup$
    – Sebastien Palcoux
    Feb 21, 2014 at 2:11
  • $\begingroup$ For $L_2(q)$ and $p$ not dividing $q$, then the Sylow $p$-normalizer is either $D_{q+1}$ or $D_{q-1}$ (depending on which one $p$ divides) and these are always maximal (for $q \geq$ some small constant, maybe 1). Sz(q) has a similar situation. I don't think there is any general criteria. $\endgroup$
    – Jack Schmidt
    Feb 21, 2014 at 2:15
  • $\begingroup$ Also to be clear, as far as I understand, the reason counterexamples exist is purely because $1_H^G$ is huge: it has lots of irreducible components, so many in fact that some have to be repeated. I don't know any general arguments to help out. $\endgroup$
    – Jack Schmidt
    Feb 21, 2014 at 2:17
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Here is the list of counter-examples with $G$ simple and $|G|<30000$ (to be read like $[|G:H|, G,H]$): 

[ 55, PSL(2,11), Group([ (1,10)(2,9)(3,4)(5,6)(7,11)(8,12), (1,6,8,2,3,11)(4,9,12,5,10,7) ]) ], 
[ 78, PSL(2,13), Group([ (1,14)(4,12)(5,6)(7,10)(8,13)(9,11), (2,8)(3,12)(4,5)(6,10)(9,14)(11,13) ]) ], 
[ 91, PSL(2,13), Group([ (1,7)(3,12)(4,14)(5,9)(6,13)(10,11), (1,9)(2,10)(3,12)(4,7)(6,11)(8,13) ]) ], 
[ 91, PSL(2,13), Group([ (1,3,8)(2,7,12)(4,5,14)(10,11,13), (1,7)(2,8)(4,9)(5,14)(6,11)(10,13) ]) ], 
[ 136, PSL(2,17), Group([ (1,3,13,2,17,4,8,12,15)(5,14,10,9,7,6,18,11,16),(1,4)(2,13)(3,17)(6,7)(8,15)(9,18)(10,11)(14,16) ]) ], 
[ 153, PSL(2,17), Group([ (1,10)(2,18)(3,12)(4,17)(6,14)(8,11)(9,16)(13,15), (2,11,3,12,8,18,15,13)(4,6,7,14,17,16,5,9) ]) ], 
[ 171, PSL(2,19), Group([ (1,19)(2,11)(3,20)(4,12)(5,17)(6,15)(7,13)(8,18)(9,10)(14,16), (1,15)(2,6)(3,4)(5,11)(7,20)(8,12)(9,13)(10,16)(14,19)(17,18) ]) ], 
[ 190, PSL(2,19), Group([ (1,4)(2,7)(3,15)(5,9)(6,20)(8,14)(10,17)(11,13)(12,19)(16,18), (1,8)(2,7)(3,4)(5,15)(6,13)(9,20)(10,18)(11,12)(14,16)(17,19) ]) ], 
[ 234, PSL(3,3), Group([ (1,5,3)(2,6,9)(4,7,13)(8,11,10), (1,3,12,5)(2,8,10,6)(4,7)(9,11) ]) ], 
[ 253, PSL(2,23), Group([ (1,7,9)(2,21,23)(3,19,16)(4,10,20)(5,14,11)(6,18,17)(8,15,22)(12,24,13), (1,4,23,18)(2,20,15,12)(3,17,21,13)(5,7,6,16)(8,14,19,24)(9,11,22,10) ]) ], 
[ 253, PSL(2,23), Group([ (1,10,24)(2,11,3)(4,15,13)(5,23,6)(7,12,8)(9,22,20)(14,17,19)(16,18,21), (1,14,12,2)(3,6,22,10)(4,20,23,16)(5,11,7,18)(8,19,15,21)(9,13,17,24) ]) ], 
[ 253, PSL(2,23), Group([ (1,16)(2,17)(3,9)(4,14)(5,10)(6,23)(7,19)(8,24)(11,20)(12,15)(13,22)(18,21), (1,9)(2,19)(3,14)(4,6)(5,13)(7,16)(8,18)(10,20)(11,17)(12,21)(15,23)(22,24) ]) ], 
[ 276, PSL(2,23), Group([ (1,6)(2,16)(3,23)(4,13)(5,15)(7,17)(8,14)(9,18)(10,24)(11,22)(12,20)(19,21), (1,9)(2,24)(3,4)(5,11)(6,22)(7,8)(10,15)(12,20)(13,21)(14,16)(17,19)(18,23) ]) ], 
[ 300, PSL(2,25), Group([ (1,14,22,19,7,2,9,8,3,11,15,13,12)(4,5,26,21,16,25,10,6,23,17,24,18,20), (2,3)(4,5)(7,11)(8,9)(10,23)(12,14)(13,22)(15,19)(16,24)(17,25)(18,21)(20,26) ]) ], 
[ 325, PSL(2,25), Group([ (1,17)(2,10)(3,16)(4,21)(5,12)(6,23)(7,14)(8,22)(11,15)(13,20)(19,25)(24,26), (1,7)(2,18)(3,17)(4,5)(6,23)(8,19)(9,14)(10,11)(12,20)(13,26)(15,16)(22,24) ]) ], 
[ 165, M11, Group([ (1,8,5,4,9,7,3,11)(2,6), (1,7)(3,5)(6,10)(8,9) ]) ], 
[ 351, PSL(2,27), Group([ (1,14)(2,10)(3,13)(4,18)(5,7)(6,21)(8,16)(9,22)(11,19)(12,24)(15,26)(17,27)(20,23)(25,28), (1,26)(2,6)(3,19)(4,23)(5,14)(7,24)(8,25)(9,16)(10,12)(11,15)(13,28)(17,21)(18,27)(20,22) ]) ],
[ 378, PSL(2,27), Group([ (1,24,25,16,11,4,5,9,26,6,8,18,21)(2,7,10,20,22,27,19,23,12,17,13,28,3), (1,23)(2,9)(3,26)(4,10)(5,7)(6,28)(8,13)(11,20)(12,21)(14,15)(16,22)(17,18)(19,24)(25,27) ]) ], 
[ 819, PSL(2,27), Group([ (1,28)(2,10)(3,23)(4,6)(5,25)(7,21)(8,13)(9,15)(11,14)(12,20)(16,19)(17,26)(18,22)(24,27), (1,8,2)(3,27,5)(4,25,19)(6,7,23)(9,10,17)(11,14,12)(13,22,26)(15,18,28)(16,24,21) ]) ] 
[ 406, PSL(2,29), Group([ (1,7)(3,22)(4,9)(5,16)(6,21)(10,26)(11,12)(13,19)(14,17)(15,23)(18,25)(20,30)(24,28)(27,29), (1,22)(2,16)(3,24)(4,21)(5,19)(6,17)(7,14)(8,27)(9,29)(11,30)(12,18)(13,23)(15,25)(20,26) ]) ], 
[ 435, PSL(2,29), Group([ (1,11)(2,28)(3,14)(4,23)(5,27)(6,12)(7,16)(8,25)(9,22)(10,18)(15,20)(17,30)(19,24)(21,29), (1,15)(2,28)(3,4)(5,22)(7,27)(8,16)(10,21)(11,12)(13,29)(14,30)(17,19)(18,23)(20,25)(24,26) ]) ], 
[ 465, PSL(2,31), Group([ (1,24)(2,17)(3,4)(5,29)(6,12)(7,8)(9,19)(10,28)(11,18)(13,31)(14,16)(15,20)(21,26)(22,32)(23,30)(25,27), (1,4)(2,6)(3,15)(5,20)(7,25)(8,22)(9,31)(10,11)(12,23)(13,26)(14,27)(16,21)(17,28)(18,32)(19,24)(29,30) ]) ], 
[ 496, PSL(2,31), Group([ (1,27)(2,17)(3,29)(4,10)(5,14)(6,24)(7,22)(8,20)(9,30)(11,15)(12,28)(13,21)(16,32)(18,19)(23,31)(25,26), (1,26)(2,25)(3,23)(4,7)(5,22)(6,31)(8,24)(9,30)(10,32)(11,19)(12,21)(13,27)(14,17)(15,20)(16,29)(18,28) ]) ], 
[ 620, PSL(2,31), Group([ (1,26,25,10)(2,30,15,14)(3,9,12,13)(4,18,22,6)(5,32,20,8)(7,27,28,31)(11,23,19,24)(16,21,17,29), (1,2)(3,4)(5,32)(6,31)(7,30)(8,29)(9,28)(10,27)(11,26)(12,25)(13,24)(14,23)(15,22)(16,21)(17,20)(18,19) ]) ], 
[ 620, PSL(2,31), Group([ (1,21,10)(2,4,23)(5,12,15)(6,24,26)(7,19,25)(8,18,20)(9,29,32)(13,17,28)(14,30,22)(16,27,31), (1,17)(2,22)(3,8)(4,21)(5,18)(6,31)(7,13)(9,10)(11,15)(12,20)(14,16)(19,27)(23,25)(24,28)(26,32)(29,30) ]) ], 
[ 666, PSL(2,37), Group([ (1,21)(2,9)(3,14)(4,23)(5,31)(6,22)(7,26)(8,24)(11,34)(12,18)(13,28)(15,33)(16,27)(17,38)(19,32)(25,35)(29,30)(36,37), (1,32)(3,35)(4,20)(5,27)(6,13)(7,15)(8,34)(9,12)(10,25)(11,16)(14,31)(17,19)(18,29)(21,22)(23,36)(24,30)(26,38)(28,37) ]) ], 
[ 703, PSL(2,37),Group([ (1,6,9,33,12,13,32,15,16,31,19,22,2,21,17,14,11,7)(3,37,28,35,24,4,29,36,27,25,34,8,10,5,23,18,20,30), (1,32)(3,35)(4,20)(5,27)(6,13)(7,15)(8,34)(9,12)(10,25)(11,16)(14,31)(17,19)(18,29)(21,22)(23,36)(24,30)(26,38)(28,37) ]) ], 
[ 2109, PSL(2,37), Group([ (1,3)(2,31)(4,9)(5,34)(6,27)(7,28)(8,19)(10,22)(11,14)(12,24)(13,21)(15,26)(16,32)(17,35)(18,38)(20,23)(25,30)(29,36), (1,12,13)(2,37,38)(3,36,5)(6,35,23)(7,22,28)(8,27,15)(9,14,11)(16,26,20)(17,19,32)(18,31,33)(21,25,29)(24,34,30) ]) ], 
[ 560, Sz(8), <permutation group of size 52 with 2 generators> ], 
[ 1456, Sz(8), <permutation group of size 20 with 2 generators> ], 
[ 2080, Sz(8), <permutation group of size 14 with 2 generators> ]
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