Irreducible tensor product representations in finite simple groups

$$\DeclareMathOperator\GL{GL}\DeclareMathOperator\PSL{PSL}\DeclareMathOperator\PSp{PSp}\DeclareMathOperator\PSU{PSU}$$Background:

A representation $$\rho: G \to \GL(V)$$ of a group $$G$$ on a (complex) vector space $$V$$ is said to be irreducible if there are no nontrivial invariant subspaces under the action of $$G$$. A tensor product of two irreducible representations $$\rho_i: G \to \GL(V_i)$$, $$i=1,2$$, is a representation on the tensor product space $$V_1 \otimes V_2$$, defined by $$\rho_1(g) \otimes \rho_2(g)$$ for $$g \in G$$.

In some cases, this tensor product representation $$\rho_1 \otimes \rho_2$$ can itself be irreducible.

Examples:

• If $$V_1$$ or $$V_2$$ is one-dimensional, then $$\rho_1 \otimes \rho_2$$ is irreducible.

• Let $$G$$ be the product $$G_1 \times G_2$$. If $$\rho_i$$ is an irreducible representation of $$G_i$$, then the representation $$\rho_1 \boxtimes \rho_2 = (\rho_1 \boxtimes 1) \otimes (1 \boxtimes \rho_2)$$ is irreducible for $$G_1 \times G_2$$.

To avoid such straightforward examples, let us assume that $$G$$ is a non-abelian finite simple group and $$\rho_i \neq 1$$. Surprisingly, such phenomenon is still possible in this case, but quite rare.

It occurs just on $$10$$ (among $$204$$) such groups of order less than $$2.7\times 10^8$$ (see the computations in Appendix), namely,
$$\PSp(4,3), \ M_{12}, \ A_9, \ \PSp(6,2), \ M_{23}, \ \PSU(5,2), \ 2F(4,2)', \ O_+(8,2), \ ^3D(4,2), \ M_{24}.$$

More specifically, for the alternating groups $$A_n$$, with $$5 \le n \le 21$$, this phenomenon occurs just for $$n=9,16$$, with $$(\dim(V_1),\dim(V_2))=(8,21), (15,12012)$$, respectively.

For the sporadic groups, it occurs for $$9$$ ones (among $$26$$), namely,
$$M_{12}, \ M_{23}, \ M_{24}, \ Co_3, \ Co_2, \ Th, \ Co_1, \ F_{3+}, \ M.$$ Warning: This last list differs from (the complement of) the list in this review mentioned by Nick Gill in this comment.

Question: For which non-abelian finite simple groups this phenomenon occurs? more specifically, for which alternating groups $$A_n$$? if and only if $$n$$ is a square? and $$\dim(V_1) = n-1$$?

My laptop was not able to check up to $$n=25$$.

Computations

For general non-abelian finite simple groups (we know that we can avoid $$\PSL(2,q)$$, see here):

gap> simpnames:= AllCharacterTableNames( IsSimple, true, IsAbelian, false, IsDuplicateTable, false : OrderedBy:= Size);
gap> for nam in simpnames do if Size(nam)<3 or List([1..3],i->nam[i]) <> "L2(" then ct:=CharacterTable(nam); Print(nam," ",Size(ct)," ",c,"\n"); Phenomenon(ct); fi; od;
...
U4(2), M12, A9, S6(2), M23, U5(2), 2F4(2)', O8+(2), 3D4(2), M24, ... the list should be incomplete after, because AllCharacterTableNames contains finitely many simple groups


For the alternating group $$A_n$$, for $$5 \le n \le 21$$:

gap> for n in [5..21] do Print(n,"\n");; ct:=CharacterTable("Alternating", n); Phenomenon(ct);; od;
...
A9, A16


gap> spornames:= AllCharacterTableNames(IsSporadicSimple, true, IsDuplicateTable, false : OrderedBy:= Size);;
gap> for nam in spornames do ct:=CharacterTable(nam); Print(nam," ",Size(ct),"\n"); Phenomenon(ct); od;
M12, M23, M24, Co3, Co2, Th, Co1, F3+, M


Script

LoadPackage("ctbllib"); LoadPackage("atlasrep");
Phenomenon:=function(ct)
local irr,r,L,i,j,k;
irr:=Irr(ct);
r:=Size(irr);
for i in [2..r] do
for j in [i..r] do
if IsIrreducible(irr[i]*irr[j]) then
Print([i,j],"\n");;
fi;
od;
od;
end;;

• Irreducible tensor products do exist for alternating group $A_n$ where $n$ is a square. Namely take $V_1$ to be representation of dimension $n-1$ (just as you suggested) and $V_2$ to be representation corresponding to square Young diagram of size $k\times k$ (since the diagram is invariant under transposition, the corresponding representation of the symmetric group splits into 2 non-isomorphic representations of the alternating group; choose any of them to be $V_2$). Then tensor product $V_1\otimes V_2$ is isomorphic to representation corresponding to the diagram $(k+1,k,\ldots,k,k-1)$. Commented May 26 at 0:25
• Here is an example: for $k=5$ representation $V_1$ is of dimension 24 and representation $V_2$ is of dimension $350574510$ (I used wonderful Young Diagram Calculator integral-domain.org/lwilliams/Applets/Math/YoungDiagrams.php ) Commented May 26 at 0:32
• Magaard and Tiep have studied this problem in several papers. For instance, zbmath.org/0992.20009 Commented May 26 at 14:22
• The problem has been solved for the alternating groups by Bessenrodt and Kleshchev: zbmath.org/1009.20013 Commented May 26 at 14:25
• You get more examples if you work with quasimple groups instead of simple ones ( though the product characters might not be faithful, but contain (some of) the centre in their kernel. For example, the degree 4 irreducible characters of $A_{5}$ is a product of two characters of degree $2$ of ${\rm SL}(2,5).$ Commented May 26 at 18:33

This is not an answer, but it reminded me of a beautiful general argument of R. Brauer (which I first saw in a paper of David Wales, (credited to Brauer, though I am not sure whether Brauer ever published it in his own right)). This result may be marginally relevant to the question.

If we have an irreducible character $$\chi$$ of a finite group $$G$$ such that $$\chi \overline{\chi} = 1 + \theta$$ where $$\theta$$ is a (necessarily non-trivial and real-valued) irreducible character, then either $$\theta$$ is rational valued, or else there is a Galois conjugation $$\sigma$$ such that $$\chi \chi^{\sigma}$$ is irreducible.

For suppose that $$\theta$$ is not rational valued. Then there is a Galois conjugation $$\sigma$$ (commuting with complex conjugation) such that $$\theta^{\sigma} \neq \theta.$$

Then $$\chi \chi^{\sigma} \overline{\chi \chi^{\sigma}} = 1 + \theta + \theta^{\sigma} + \theta \theta^{\sigma}.$$

Since $$\theta = \overline{\theta}\neq \theta^{\sigma},$$ the irreducible character $$\theta^{\sigma}$$ is different from $$\overline{\theta}.$$ Hence the trivial character only occurs once in $$\chi \chi^{\sigma} \overline{\chi \chi^{\sigma}}$$ and $$\chi \chi^{\sigma}$$ is irreducible.

Two examples where this occurs in practice are when $$G ={\rm SL}(2,5)$$ and $$\chi$$ is an irreducible character of degree $$2$$. Then $$\chi \overline{\chi} = 1 +\theta,$$ where $$\theta$$ is irreducible of degree three, but is not rational valued. Then $$\chi \chi^{\sigma}$$ is irreducible of degree $$4$$ where $$\sigma$$ is the Galois automorphism which squares odd order roots of unity and fixes $$2$$-power roots of unity. A similar example occurs for $$G$$ a triple cover of $$A_{6}$$, with $$\chi$$ irreducible of degree $$3$$ (with $$\sigma$$ cubing roots of unity of order prime to $$3$$ anf fixing roots of unity of $$3$$-power order).

• There is an odd typo in the second line of the second paragraph. I'm guessing it ought to read "necessarily non-trivial irreducible character", but I'm not sure. Commented May 26 at 22:49
• @DonuArapura : Yes, that was what I intended, thanks. Commented May 26 at 23:17

Also not an answer, just a hint on how the GAP code can be made a bit faster, for future reference: GAP actually "knows" the character tables for the alternating groups. For some reason I don't know this is not used here, but we can ask GAP directly for that character table, bypassing all computations in groups, to get a faster computation, like so:

LoadPackage("ctbllib");
Phenomenon:=function(n)
local ct, irr,r,L,i,j,k;
ct:=CharacterTable("Alternating", n);
irr:=Irr(ct);
r:=Size(irr);
for i in [2..r] do
for j in [i..r] do
if IsIrreducible(irr[i]*irr[j]) then
Print([i,j],"\n");;
fi;
od;
od;
end;;

for n in [5..20] do Print(n,"\n");; Phenomenon(n);; od;

• Thanks! I expect your hint also applies to any finite simple groups of not too large an order. Commented May 27 at 8:51
• Yes, the GAP packages atlasrep and ctbllib have character tables for the sporadic finite simple groups and several of the Lie types of not-too-big degree. Commented May 27 at 9:31
• The BrowseAtlasInfo() function can be used to, well, browse a list of available tables. That can be helpful in finding out the right table for a group (e.g. the table for the O'Nan group is "ON") Commented May 27 at 9:36