Detecting elements in a group using characters.

Question

It is well known that the trace of the regular representation $\rho$ of a group $G$ 'detects' the identity element of the group. More precisely, we have

$$ Tr(\rho)(g) = \begin{cases}|G|&\mbox{if }g=e\\ 0&\mbox{otherwise.}\end{cases} $$

Now, if $G$ is Abelian, this can be used to detect arbitrary elements in $G$ as

\begin{equation}\label{Equation 1} Tr(\rho)(h^{-1}g) = \begin{cases}|G|&\mbox{if }g=h\\ 0&\mbox{otherwise.}\end{cases} \end{equation}

The key here is that every irreducible representation of $G$ (in this case) is one dimensional so that $Tr(\rho)(h^{-1}g) = \rho(h^{-1}g)$ can be written as a linear combination of irreducible representations of $G$.

My question is the following.

Is there an analogous way to detect arbitrary elements when $G$ is non Abelian

EDIT: Let me add some context to why I am asking this question. The motivation comes from Artin $L$ functions (over $\mathbb{Q}$ for simplicity). Consider the Riemann zeta function $\zeta(s)$ and suppose, for a fixed $a,N\in \mathbb{N}$ with $GCD(a,N)=1$, I am interested in the function $$ L(s):=\sum_{m\equiv a\mod N} \frac{1}{m^s}. $$ Then it is well known (and in fact a consequence of the first of the two equations above) that

$$ L(s) = \frac{1}{\varphi(N)}\sum_\chi \chi(a)^{-1}L(s,\chi) $$ where $L(s,\chi)$ is the Dirichlet $L$ function of the character $\chi$ modulo $N$, given by

$$ L(s,\chi) = \sum_{n=1}^\infty \frac{\chi(n)}{n^s}, $$ and the summation runs over all characters of $(\mathbb{Z}/N\mathbb{Z})^\times$.

Something very similar can be done for other number fields as well (although I have not done the exact calculations, I think it should be possible to do the same for other Dirichlet/Hecke $L$ functions too). Now I am interested in replacing $\zeta(s)$ with Artin $L$ functions of representations of higher dimensions and see if something similar is possible. I hope this makes my question clear.

Answer 1

If I understand your question correctly, the point is that characters are class functions, so in a non-abelian group you will not be able to distinguish two distinct elements that are conjugate.

Answer 2

Irrespective of the abelianess of $G$, the regular character over the complex numbers for finite groups at least, is defined in a similar way: $$\rho=\underset{\chi \in Irr(G)}\sum \chi(1)\chi.$$ And $\rho(g)=|G|$ if $g=1$, $\rho(g)=0$, if $g \neq 1$. Hence similarly, this character can “detect” arbitrary elements.

Edit (April 8th 2021)

As you probably know, Richard Dedekind studied the quotients of what are now called Dedekind zeta functions and proved that the quotient $\zeta_M(s)/\zeta(s)$ is entire for every pure cubic extension $M/\mathbb{Q}$. This result suggests that the quotient $\zeta_M(s)/\zeta_k(s)$ of Dedekind zeta functions is entire whenever $M/k$ is an extension of number fields, and this is now called the Dedekind conjecture.

Somewhat differently, for any $G = Gal(K/k)$, Artin conjectured that every $L$-function $L(s, \chi, K/k)$ with $\chi \in Irr(G)-\{1_G\}$, extends to an entire function. Artin Reciprocity Law implies this conjecture is valid if $\chi$ is monomial, that is, induced from a linear character of some subgroup of $G$. From this and his famous induction theorem, Brauer further showed that each $L(s, \chi, K/k)$ extends to a meromorphic function over $\mathbb{C}$. Aramata and Brauer independently proved the Dedekind conjecture for every Galois extension of number fields. Furthermore, if $M$ is contained in a solvable normal closure of $k$, Uchida and Robert van der Waall established (1975) the Dedekind conjecture in this case. In general, the Dedekind conjecture would follow from Artin’s conjecture. However, both conjectures are still open.

I hope this provides you with more background and reading.