Is there a connection between representation theory and PDEs? As a PhD student, if I want to do something algebraic / linear-algebraic such as representation theory as well as do PDEs, in both the theoretical and numerical aspects of PDEs, would this combination be compatible and / or useful? Is it feasible?
I'd be grateful for an online resource to look into.
Thanks,
 A: Peter Olver has an interesting book on Symmetry and PDEs.  Another area to consider (that is particularly important for geometric PDEs) are exterior differential systems.  Here are some notes on the subject by Robert Bryant (who sometimes posts here).
A: The book "D-Modules, Perverse Sheaves, and Representation Theory " by   Ryoshi Hotta, Kiyoshi Takeuchi and Toshiyuki Tanisaki  is the perfect source for this topic. The introduction gives a very nice (and elementary) explanation how representation theory of D-modules and symstems of partial differential equations are related. I just give a very nice excerpt from the introduciton of the book.
Let $X$ be an open subset of $\mathbb{C^n}$ and $\mathcal{O}$ the commutative ring of complex analytic functions defined on $X$. 
Let $D$ be the set of partial differential operators with coefficients in $\mathcal{O}$, whose elements are thus of the form $\sum\limits_{i_1,...,i_n}^{\infty}{f_{i_1,...,i_n} (\frac{\delta}{\delta x_1})^{i_1} ... (\frac{\delta}{\delta x_n})^{i_n}} $.
Let $P \in D$ and consider the partial differential equation $Pu=0$ and $M$ the D-module $M=D/DP$.
We then have $Hom_D(M,\mathcal{O}) \cong \{f \in \mathcal{O} | Pf=0 \}$.
This shows that the set of analytic solution of $Pu=0$ is isomorphic to a $Hom$-space, which are the natural objects of study of representation theory (representation theory can be summarized more or less as the study of representations of rings and their Hom-spaces).
A: This goes back to the beginning of the subject of unitary representations of locally compact noncompact groups. Wigner was looking for all possible generalizations of the Dirac equation to higher spin, and developing the representation theory of the Poincaré group is how he obtained his results (Bargmann did this independently, so they published together). See here:
https://www.pnas.org/content/34/5/211
A: There is e.g. a book 
Differential Galois Theory by M. van der Put and M. F. Singer, where in Appendix D one can find things on the PDE case (the book is mostly about ODEs).
In mathematical physics there are topics such as KZ equations
which are related to linear representations of braid groups.
