Application of representation theory I often read that one can use representation theory in the field of quantum physics or for the analysis of symmetries in physics or chemistry. Unfortunately I couldn't find a concrete example for this. I would be very happy if someone could describe me a concrete application of linear representation theory of finite groups (modular and arbitrary theory over rings). Is this theory e.g. useful for the Schrödinger equation?
 A: Your question seems to ask more for a reference. There are many good papers for this that I read through while looking at this for my undergraduate thesis. Here are a few good references to get you started at why the two are connected:
Representation Theory, Symmetry, and Quantum Mechanics
Quantum Mechanics and Representation Theory
and this which is not as related but easier to read.
A: Scientists tend to be more concerned with inspecting maps from a group to vector space automorphisms rather than properties of groups themselves.
Here are some concrete examples for you:
For simple-enough molecules, representations of dihedral groups are a good place to start. They can describe discrete mirror/rotational transformations that leave regular polygons invariant.
Example. An example for a representation of the group $$D_4:=\langle x,y; x^4=1=y^2, y^{-1}xy=x^{-1}\rangle$$ is the group homomorphism $\rho:D_4\rightarrow GL(2,\mathbb{C})$. The group itself is the group of symmetries of the square in the Cartesian plane (keeping the center fixed). To find a representation $\rho$, you can write some generic matrices with complex entries and brute force it or use some simple geometric intuition to recall what types of matrices give you mirror/rotation transformations, then apply the right ones.
I’m no expert on chemistry, but both the topology and symmetries of molecules are very important to their functions and properties.
As far as quantum theory, the basic example is actually not a normal (faithful) representation, but a projective one onto ray space (see projective Hilbert space). The intuition is quantum states are only required to be the same up to a global $U(1,\mathbb{C})$ phase factor. The group associativity induces a 2-cocycle condition which imposes a restriction on the particular form of phase difference the two states may have. The canonical reference for this is Weinberg’s Quantum Theory of Fields, Volume 1, $\S 2.7$. So yes it applies to quantum mechanics and the Schrodinger equation, but representation theory tends to appear much more in other facets of quantum theory such as gauge theory and quantum field theory. 
A lot of credit for incorporating representation theory into quantum mechanics and gauge theory was Hermann Weyl and his book on Groups and Quantum Mechanics is very worth checking out.
A: I Googled representation theory in quantum mechanics and found:


*

*Quantum Field Theory and Representation Theory
To quote the source: In 1928 Weyl published a book called "Theory of Groups and Quantum
Mechanics", which had alternate chapters of group theory and quantum mechanics.
I think that you will find the rest of the the source most enlightening. 
A: One simple application would be to estimate if a given electron distribution (wave or state function) in a molecule of a given symmetry (group) may have a non-zero dipole moment. In the usual approximation one takes the positions of the atoms as given parameters in $\Bbb R^3$. The functional equation that describes all possible structures of electron distributions  ($\Psi$) with determined energies ($E$) is the Schrödinger equation, its an eigenvalue equation $$\hat H \Psi = E \Psi$$ where $\hat H$ is the so called Hamilton operator. 
Now the important bit is that $\hat H$ belongs to the same symmetry group as the distribution of nuclei. Thus it commutes with all symmetry operations from the group and thus all possible electronic state functions $\Psi$ are can be represented by irreducible representations (symmetry species) of this group. 
Now in order to have a non-zero dipole moment a certain integral must be non-zero. This integral $$\int_{space} \Psi \hat{D}\Psi d\tau$$ where $\hat{D}$ is the dipole operator. The dipole operator is represented in $\Bbb R^3$ (all molecular point groups can be faithfully represented in $O(3)$) like the basis $x, y$ or $z$.
Now one can check if the tensor product of the irreducible representations of $\Psi$, $\Psi$ and $\hat{D}$ decomposes into a direct sum that contains the trivial representation. If so the molecule may have a non-zero dipole moment.
In general group representation theory in quantum mechanics is often used to see if certain integrals are required to be zero or not in exactly that way. Those integrals are Hilbert space inner products that describe transitions between possible quantum states of the systems while Operators that act on the functions represent measurements of physical observable. 
