Good effective versions of theorems of Artin and Brauer

Question

The theorem of Artin and Brauer of the title are the famous theorem in the theory of representation of finite groups.

For example, Artin's theorem is the statement that for every character $\chi$ of a finite group $G$, there are a sequence of cyclic subgroups $H_1,\dots,H_r$ (possibly with repetition), one-dimensional characters $\chi_i$ of $H_i$ for $i=1,\dots,r$, signs $\epsilon_i = \pm 1$ for $i=1,\dots,r$ and an integer $d \geq 1$, such that $$(1)\ \ \ \ \ \chi = \frac{1}{d} \sum_{i=1}^r \varepsilon_i \ Ind_{H_i}^G \chi_i.$$ Brauer's theorem states similarly that if we weaken the assumption that the $H_i$ are cyclic, assuming just that they are elementary, then such a writing (1) exists with $d=1$.

I'd like to know if there is a version of these theorems with an explicit control of the complexity of the writing (1) in term of $\chi(1)$ and perhaps of $|G|$. More specifically, if all the $\epsilon_i$ were $+1$, then one one would have $\frac{1}{d} \sum_i [G:H_i] = \chi(1)$ . In general of course, the $\epsilon_i$ can be $+1$ or $-1$, and $\frac{1}{d} \sum_{i=1}^r [G:H_i]$ will be larger that $\chi(1)$ but

Do you know a version a version of Artin or Braueur with an explicit bound on $\frac{1}{d} \sum_{i=1}^r [G:H_i]$ in terms of $\chi(1)$ and $|G|$, or a place where can I find some ?

In the case of the Brauer's theorem, one has $d=1$, so the question is simply to get a bound on $\sum_{i=1}^r [G:H_i]$. That would also directly give a bound on $r$, the number of subgroups $H_i$ involved, and I would be also interested in a version with such a bound on $r$. In the case of Artin's theorem, there is the further and orthogonal question of finding a bound on $d$, but that is not what primarily interests me here.

It seems pretty clear to me that the usual proofs (e.g. In Serre's book on representation of finite groups) of Artin and Braueur are effective, so give an upper bound as asked but a huge one. I am looking for something better, or even the best possible bound if it is known.

ADDED: the completely explicit form of Artin's theorem mentioned in Denis's answer is the following:

$$ \chi = \sum_C \alpha_C [G:C]^{-1} Ind_C^G 1,$$ the sum being on cyclic subgroup $C$ of $G$, and $$\alpha_C = \sum_{C \subset B} \mu([B:C]) \chi(b)$$ the sum being now and cyclic subgroups $B$ containing $C$ and $b$ being any generator of $B$. This is a nice-looking formula, but the complexity of this formula, in the sense of my question, is big. Indeed, this complexity $\gamma$ is the sum over $C$ (cyclic subgroup of $C$) of $[G:C]$ times $[G:C]^{-1} |\alpha_C \chi(b)|$, that is this complexity is $$ \gamma = \sum_C |\alpha_C \chi(b)| \leq \dim \rho \sum_C |\alpha_C|.$$ Now a trivial lower bound for $\sum_C |\alpha_C|$ in the case $G=(\mathbb Z /p\mathbb Z)^n$ is $(p^n-1)/(p-1)$, since this already the value of $\alpha_{\{1\}}$, i.e. the number of subgroup of order $p$ of $G$. This shows that the complexity of this formula is not an order of magnitude better than $|G| \dim \rho$. For my application this is grossly insufficient. I would expect something in $\dim \rho \log|G|$.

Answer 1

Maybe Snaith's book "Explicit Brauer induction: with applications to algebra and number theory" (Cambridge Univ. Press) could be helpful? I don't really know much about it except for its existence, but looking in Google books shows, e.g., a completely explicit form of Artin's Theorem in his Theorem 2.1.3.

Answer 2

Work of Snaith, and of Robert Boltje, on Explicit Brauer induction should be helpful here. Their results are essentially equivalent, but Boltje shows that there is a unique explicit Brauer induction formula which commutes with restriction, while Snaith obtains a unique explicit form of Brauer's induction theorem which commutes with induction. These explicit forms are different in general. Snaith's work is more topological, which Boltje's work is more algebraic, working with a nice generalization of the Burnside ring (where monomial representations replace permutation representations). However, the formulae are indeed complicated, and may not meet the criteria you impose.

Later edit: Note also that in Artin's induction theorem, each maximal cyclic subgroup of $G$ must occur as one of the $H_{i}$, and J.A. Green showed in around 2005 that similarly in Brauer's theorem, you need to use a conjugate of every maximal Brauer elementary subgroup. This seems to indicate that the sums you are considering are unavoidably large.