An irreducible representation of a finite group whose symmetric square is also irreducible

charactersfinite-groupsgaprepresentation-theory

Is there an irreducible representation $V$ (over $\mathbb{C}$) of a finite group $G$ such that $\dim V > 1$ and $\mathrm{Sym}^2V$ is again irreducible? (I believe that there are examples for compact $G$, such as the fundamental representation of $SU(2)$.)

Searching for examples:

Some obvious candidates don't work:

  • $G$ cannot be dihedral, as all dihedral irreps are of dimension 1 or 2;
  • $G$ cannot be $Q_8$ for analogous reasons;
  • $G$ cannot be $S_n$ for $n \le 6$; dimension-counting rules out all irreps except for the two-dimensional irrep of $S_4$, which satisfies $\mathrm{Sym}^2V = 1 \oplus V$;
  • $G$ cannot be $A_n$ for $n \le 6$ for similar dimension-counting reasons.

Obviously the Frobenius-Schur indicator of $V$ cannot be 1, but this is sort of tautological. If $G$ has odd order, then I think the Frobenius-Schur indicator of $V$ must be zero, so perhaps something like the subgroup of $\mathop{PGL}_2(\mathbb{F}_7)$ of order 21 is a good place to look.

A suitable GAP query might also produce examples; unfortunately this is beyond my present ability.

Trying to prove this is impossible:

I also haven't been able to make any headway on showing that there can be no such $G$. The obvious strategy would be to find a nice simplification of $$ \left\langle \chi_{\rho}, \chi_{\mathrm{Sym}^2V} \right \rangle, $$ where presumably the clever choice is to take $\rho = V$. (Taking $\rho = 1$ cannot prove this is impossible, since groups with Frobenius-Schur indicators other than 1 exist.)

Note that if $V$ is not a summand of $\mathrm{Sym}^2V$, we have $$(\dim V)^2 = \sum_{g \in G} \overline{\chi}_V(g)\chi_V(g^2).$$ This feels possibly helpful in special cases; for example using this formula I think I can (via a sort of "complex rearrangement inequality") rule out this case for $G$ of odd order.

It might be possible to use the hook-length formula to rule out most examples in $S_n$. The only candidate irreps $V$ of $S_n$ for $n \le 8$ are the quotient of the standard representation by the trivial representation and the tensor product of this representation with the sign representation. I haven't yet written this down to see if I can make it work; I'd also have to look up a lot about the representations of the symmetric groups to rule out the possibilities that would be left.

Best Answer

Since you mention GAP, here is a way how one could test it using the character table functionality:

# test for property: Arguments: table, character
test:=function(t,c). 
local p;
  if c[1]=1 or ScalarProduct(c,c)<>1 then return false;fi;
  p:=SymmetricParts(t,[c],2)[1];
  return ScalarProduct(p,p)=1;
end;

# test for group -- any irreducible characetr has property?
gptest:=function ( g )
local t;
  t := CharacterTable( g );
  return ForAny( Irr( t ), x->test( t, x );end );
end;

Lets check group up to order 100:

gap> g:=OneSmallGroup(Size,[2..100],gptest,true);
<pc group of size 24 with 4 generators>
gap> StructureDescription(g); # only makes sense for small orders
"SL(2,3)"

and lets check:

gap> t:=CharacterTable(g);
CharacterTable( SL(2,3) )
gap> List(Irr(t),x->x[1]);
[ 1, 1, 1, 2, 2, 2, 3 ]
gap> List(Irr(t),x->test(t,x));
[ false, false, false, true, true, true, false ]

Thus the 2-dimensional representations of $SL_2(3)$ are an example of minimal order. $SL_2(5)$ is a minimal nonsolvable example (also degree 2), and $SL_3(2)$ is an exampel with character degree 3.

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