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This is a somewhat imprecise question, as I am not sure how exactly how to formalise how to do mathematics "without" a certain key tool, but hopefully the intent of the question will still be clear.

Let $G$ be a finite group. Traditionally, a character $\chi: G \to {\bf C}$ on $G$ is defined as being the trace of a finite-dimensional unitary representation $\rho: G \to U(V)$ of $G$, and then representation-theoretic tools (including Schur's lemma) can then be used to derive the basic results of character theory, including the following assertions:

  1. The irreducible characters form an orthonormal basis of the space $L^2(G)^G$ of class functions, and all other characters are natural number combinations of the irreducible characters.
  2. The space of characters form a semiring with identity; in particular, for three irreducible characters $\chi_1,\chi_2,\chi_3$, the structure constants $\langle \chi_1 \chi_2, \chi_3 \rangle$ are natural numbers.
  3. For any character $\chi$, $\chi(1)$ is a positive integer.
  4. For any character $\chi$, $\chi(g^{-1})=\overline{\chi(g)}$ for all $g$.
  5. For any irreducible character $\chi$, the convolution operation $f \mapsto f * \chi(1) \chi$ is a minimal idempotent in $L^2(G)$, and one has the Fourier inversion formula $f = \sum_\chi f * \chi(1) \chi$ for all class functions $f$. Furthermore, the image $I_\chi$ of the convolution operation $f \mapsto f * \chi(1) \chi$ in $L^2(G)$ (or ${\bf C} G$) is an irreducible $G \times G$ representation.
  6. If $\chi$ is an irreducible character of $G$, and $\eta$ is an irreducible character of a subgroup $H$ of $G$, then the structure constant $\langle \chi, \operatorname{Ind}^G_H \eta \rangle_{L^2(G)^G} = \langle \operatorname{Res}^H_G \chi, \eta \rangle_{L^2(H)^H}$ is a natural number.

Note that representation theory does not make an explicit appearance in the above character theory facts (induction and restriction of characters can be done at a purely character-theoretic level without explicit reference to representations), other than as one of the conclusions to Fact 5.

Note also that one can define characters directly, without mention of representations. The space $L^2(G)^G$ of class functions is a finite-dimensional commutative algebra under the convolution operation, and one can then locate the minimal idempotents $f$ of this algebra; $f(1)$ will then be positive, so one can define an "irreducible" character $\chi$ associated to this idempotent by the formula $f = \chi(1) \chi$ with $\chi(1) := f(1)^{1/2}$ being the positive square root of $f(1)$. One can then define an arbitrary character to be a natural number combination of the irreducible characters. This definition of character automatically gives Facts 1, 4, and 5 above (the $G \times G$-irreducibility of $I_\chi$ can be obtained by computing that the dimension of $\operatorname{Hom}^{G \times G}(I_\chi,I_\chi)$ is one), and if one is allowed to use Schur's lemma (or the Artin-Wedderburn theorem), one can also show that this definition is equivalent to the usual representation-theoretic definition of a character which then gives the remaining Facts 2, 3, and 6.

My (imprecise) question is whether one can still recover Facts 2, 3, and 6 from this non-representation-theoretic definition of a character if one is "not allowed" to use Schur's lemma or the Artin-Wedderburn theorem. Now, I do not know how to rigorously formalise the concept of not being allowed to use a particular mathematical result; my first attempt was to phrase the problem with the complex numbers replaced by the field of definition of the characters $\chi$ (or the cyclotomic field of order $|G|$), so that the underlying field is not algebraically closed and so Schur's lemma or Artin-Wedderburn do not directly apply. But this is hardly any constraint at all since one can immediately pass to the algebraic closure of these fields in order to bring Schur or Artin-Wedderburn back into play. Another option is to take a constructivist (or maybe reverse mathematics) point of view, and only permit one's mathematical reasoning to work with representations as long as they can be constructed from characters using explicit, "functorial", representation-theoretic constructions (e.g. tensor sum, tensor product, orthogonal complement, isotypic component, Schur functors, induction, or restriction) from concrete representations (e.g. trivial representation, regular representation, or quasiregular representation), but prohibit any argument that requires one to make "arbitrary" or "non-functorial" choices on representations (and in particular, to split an isotypic representation or module into irreducibles). However, I do not know how to formalise this sort of mathematical reasoning (perhaps one has to introduce a suitable topos?). It has also been suggested to me (by Allen Knutson) that perhaps the correct setting for this framework is that of quantum groups over roots of unity rather than classical groups, but I am not familiar enough with quantum groups to formalise this suggestion.

Leaving aside the question of how to properly formalise the question, one can make some intriguing partial progress towards Facts 2, 3, 6 without invoking Schur or Artin-Wedderburn. The isotypic representation $I_\chi$ has character $\chi(1) \chi$ by construction, so in particular this shows that $\chi(1)^2$ is a positive integer which is in some sense the "square" of Fact 3. By considering the $\chi_3\otimes \overline{\chi_3}$-isotypic component of $I_{\chi_1} \otimes I_{\chi_2}$, viewed as $G \times G$ representations, one can similarly show that the square $|\langle \chi_1 \chi_2, \chi_3 \rangle|^2$ of the structure constant $\langle \chi_1 \chi_2, \chi_3 \rangle$ is a natural number, and by viewing these representations instead as $G$-representations one also gets that $\chi_1(1)\chi_2(1) \chi_3(1) \langle \chi_1 \chi_2, \chi_3 \rangle$ is a natural number. These two facts don't quite establish Fact 2, but they do at least show that Fact 3 implies Fact 2. Finally, for Fact 6, the Frobenius reciprocity $\langle \chi, \operatorname{Ind}^G_H \eta \rangle_{L^2(G)^G} = \langle \operatorname{Res}^H_G \chi, \eta \rangle_{L^2(H)^H}$ is an easy algebraic identity if one defines induction and restriction in character-theoretic terms, and the same sort of arguments as before show that $|\langle \chi, \operatorname{Ind}^G_H \eta \rangle_{L^2(G)^G}|^2$ and $\chi(1) \eta(1) \langle \chi, \operatorname{Ind}^G_H \eta \rangle_{L^2(G)^G}$ are natural numbers, so again Fact 3 will imply Fact 6. (Conversely, it is not difficult to deduce Fact 3 from either Fact 2 or Fact 6.)

So it all seems to boil down to Fact 3, or equivalently that the dimension of any minimal ideal of $L^2(G)$ (or ${\mathbf C} G$, if you prefer) is a perfect square. This is immediate from the Artin-Wedderburn theorem, and also follows easily from Schur's lemma (applied to an irreducible representation of this ideal) but I have been unable to demonstrate this fact without such a representation-theoretic (or module-theoretic) tool.

Characters do give a partial substitute for Schur's lemma, namely that the dimension of the space $\operatorname{Hom}^G(V,W)$ of $G$-morphisms between two representations $V,W$ is equal to $\langle \chi_V, \chi_W \rangle_{L^2(G)^G}$, where $\chi_V, \chi_W$ are the characters associated to $V, W$. This gives Schur's lemma when the characters $\chi_V, \chi_W$ are irreducible in the sense defined above (defined through minimal idempotents and square roots rather than through irreducible representations). At the level of $G \times G$-representations, the isotypic components $I_\chi$ are irreducible (in both the character-theoretic and representation-theoretic senses) and so one has a satisfactory theory at this level (which is what is giving the "square" of Facts 2, 3, and 6) but at the level of $G$-representations there are these annoying multiplicities of $\chi(1)$ which do not seem to be removable without the ability to reduce to subrepresentations for which Schur's lemma applies.

Somehow the enemy is a sort of phantom scenario in which a group-like object $G$ has an irreducible (but somehow not directly observable) "ghost representation" of an irrational dimension such as (say) $\sqrt{24}$, creating an isotypic component $I_\chi$ whose dimension ($24$, in this case) is not a perfect square. This is clearly an absurd situation, but one which appears consistent with the weakened version of representation theory discussed above, in which Schur's lemma, the Artin-Wedderburn theorem, or "non-constructive" representations are not available. But I do not know if this is a genuine limitation to this sort of theory (e.g. can one cook up a quantum group with such irrational representations?), or whether I am simply missing some clever argument.

(My motivation for this question, by the way, is to explore substitutes for character-theoretic or representation-theoretic methods in finite group theory, for instance to find alternate proofs of Frobenius's theorem on Frobenius groups, which currently relies crucially on Fact 6.)

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Schur's Lemma is really old, I think. Burnside used it in 1905 almost with no justification, for instance... A completely off-hand suggestion: Deligne has defined a category that is supposed to behave like the representations of the symmetric group $S_t$ where $t$ is not a positive integer. Maybe those could be useful? –  Denis Chaperon de Lauzières May 21 '13 at 17:57
I think these "ghost representations" do exist; the literature on fusion categories might be a place to find them? –  Qiaochu Yuan May 21 '13 at 18:36
Frobenius defined characters and proved theorems about them before there was a definition of representation of a group. See ams.org/notices/199803/lam.pdf, esp. (6.4) and (7.4). He used the group determinant in place of representations (e.g., a character is irreducible when it corresponds to an irreducible factor of the group determinant). –  KConrad May 21 '13 at 19:00
one might consider a generalization of fact $4$, that if $p$ does not divide the order of $G$, then $\chi(g^p)= Frob_p(\chi(g))$ for $Frob_p$ a Frobenius lift in $\mathbb Q(\chi(g))$. Or simply the fact that $\chi(g)$ is a cyclotomic integer. –  Will Sawin May 21 '13 at 19:08
To be more precise, my suggestion was that your framework might be allowing in quantum groups at roots of unity, and their representations' "quantum dimensions", alongside the things you actually want. –  Allen Knutson May 21 '13 at 19:20
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2 Answers

up vote 15 down vote accepted

I spent a long time writing an answer to this question, but MO did not believe I was a human being ( I did mis-spell one of the test words, but everyone deserves a second chance, I think ), so it seems to have disappeared. I am not sure I have the energy to do it again right now, but here (in precis, though not precise) are three points I thought worth making/suggesting:

  1. The group determinant has been mentioned: in a 1991 Proc AMS paper, Formanek and Sibley proved that the group determinant determines the group. Perhaps you could use the analogue of the group determinant to tease out the properties that an algebraic structure $G$, whose "formal character theory" satisfies the properties you can get without Schur and Wedderburn, would have. Such a "group determinant" would not a priori have the property that the multiplicity of an irreducible factor equals its degree.

  2. It is possible to get quite far into the structure of the group algebra just using the symmetric algebra structure of the complex group algebra of $G$ induced by the linear form $t$ with $t(\sum_{g \in G} a_{g}g ) = a_{1}.$ Since $t$ vanishes on nilpotent elements, it follows that no non-zero right ideal of $\mathbb{C}G$ consists of nilpotent elements and that for each minimal (two-sided) ideal $A$ of $\mathbb{C}G$, $Z(A)$ is $1$-dimensional. You might argue that this is using representation theory, since it is not a priori immediately obvious that $t$ vanishes on nilpotent elements until one notes that (up to the multiple $|G|$ ) $t$ is the trace afforded by the regular representation.

  3. Frobenius's theorem, and other normal complement theorems of a similar nature suggest that there might be an analogue (under certain hypotheses) of the transfer homomorphism but when the target group is not necessarily Abelian. That theorem can be proved in the case that $H$ is solvable by using the usual transfer homomorphism, and what the theorem says in the end (in the general case) is that the identity homomorphism from $H$ to itself extends to a homomorphism from $G$ onto $H.$ If one tries a transfer-type proof, it looks as though it almost would work, except that the order of products matters when the target group is non-Abelian. Nevertheless, the Theorem does in the end say that the homomorphism you would like to define by "transfer" is well-defined after all.

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@Geoff: It's easy for things to get lost in transit, so I'd recommend composing a text document separately if your answer (or question) gets lengthy and hard to reproduce. This also allows better for proofreading (such as "teh" --> "the"). Then just paste it in, but keep a saved copy temporarily. In the present case the question is unusually complicated, so an answer also has to be. (By the way, are you actually a human being...?) –  Jim Humphreys May 27 '13 at 23:51
Thanks Jim. I've eventually learned to keep saving text, but the human/computer question threw me, and I didn't expect it just to kick me off- I thought I would get returned to the text I was composing at worst. I was a human being yesterday- today, who knows? –  Geoff Robinson May 28 '13 at 7:15
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It might be useful here to think about the Morava $K$-theory rings $K(n)^*(BG)$ (for finite groups $G$). For the trivial group you get the graded ring $k=\mathbb{Z}/p[v_n,v_n^{-1}]$, where $|v_n|=2p^n-2$; for general groups you get a finitely generated graded module over $k$ (and all such modules are free). There is a ring structure, induction and restriction maps, an inner product and so on, making $K(n)^*(BG)$ closely analogous to $R(G)$. However, one can find examples where $K(n)^*(BG)$ has no basis that is permuted by $Aut(G)$, which means that we have nothing analogous to irreducible characters. One way to make your question more precise would be to restrict attention to methods that also work in this context.

If you don't like the fact that $k$ has characteristic $p>0$, there are naturally occurring lifted versions where the trivial group gives you the ring of $p$-adic integers, or a formal power series ring over the $p$-adics or something a bit larger than that. If we let $E$ denote one of these variants, it works out that there is a kind of character theory due to Hopkins, Kuhn and Ravenel. Instead of conjugacy classes of elements of $G$, you need to consider the set $C_n$ of conjugacy classes of $n$-tuples of mutually commuting elements of $p$-power order. There is then a certain ring $L$ that is an algebraic extension of $\mathbb{Q}\otimes E^*(\text{point})$, and a natural isomorphism $L\otimes E^*(BG)\to Map(C_n,L)$, analogous to the description of $\mathbb{C}\otimes R(G)$ by class functions.

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