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Is it true that for any finite nontrivial group G, there exist two inequivalent irreducible representations of G over the complex numbers that have the same degree.

If so, is there an easy proof? If not, what is the smallest counterexample?

Note: Any counterexample group must be perfect, because if the abelianization is nontrivial, we get multiple irreducible representations of degree one. [EDIT: Further, as Colin Reid notes in the comment, a minimal counterexample must be a simple (non-abelian) group]. This whittles down our search considerably. The general expressions for the degrees of irreducible representations for the families of simple groups that I've checked suggests that there is plenty of repetition of degrees in these cases.

edited Oct 30 2011 at 2:32

asked Oct 30 2011 at 1:37

Vipul Naik
4,945●13●44

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A minimal counterexample would have to be simple, so if you can prove it for the simple groups, you are done. – Colin Reid Oct 30 2011 at 1:45

Thanks for pointing it out! I forgot to say it explicitly, but that's the direction I was going with "checking the families of simple groups." I've edited the original question to make that clear. – Vipul Naik Oct 30 2011 at 2:33

This is a nice question. I wonder if it's possible to give a proof that doesn't rely on the classification of simple groups. – Faisal Oct 30 2011 at 2:44

Faisal · Accepted Answer · 2011-10-30 04:57:08Z UTC

It seems that the answer is yes. A MathSciNet search brought up the paper

Y. Berkovich, D. Chillag, and M. Herzog, Finite groups in which the degrees of the nonlinear irreducible characters are distinct, Proc. Amer. Math. Soc. 115 (1992), 955–959.

In it you can find a characterization of groups whose nonlinear irreducible characters have distinct degrees. In particular, such a group can't be perfect (see Lemma 1), and so will always have multiple linear characters as was noted in the OP. The proof, however, relies on the classification of finite simple groups, so is not "easy".

Addendum: I took a closer look at the related literature and happened across the following interesting result, which I figured was worth sharing. (It can also be used to give an affirmative answer to the original question.)

Theorem. Let $G$ be a nontrivial finite group. If the character table of $G$ has a column or row containing distinct rational entries, then $G$ must be isomorphic to either $S_2$ or $S_3$ .

The reference is

M. Bianchi, D. Chillag, A. Gillio, Finite groups with many values in a column or a row of the character table, Publ. Math. Debrecen 69 (2006), no. 3, 281–290.

The result from the classification of finite simple groups used in the Berkovich–Chillag–Herzog paper is also used here (in very much the same spirit).

Does every finite nontrivial group have two distinct irreducible representations over the complex numbers of equal degree?

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