# 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,

• Higgs Bundles, Gauge Theory maybe? – JJJ Jun 30 '19 at 5:33
• Google D-modules. – Misha Jun 30 '19 at 5:59
• Symmetries are very important in PDEs, and it's not too much of a stretch to say that representation theory is the linearised study of symmetries. If you find an advisor who's interested in both, then you'll be in good shape. Unfortunately, however, I don't think that this is a research-level question. – LSpice Jun 30 '19 at 11:39
• I was a student of Robert Bryant. His work is in PDE and differential geometry, using a lot of representation theory of finite dimensional Lie groups, and very little (but some) numerical methods. As mentioned below, you can look at the book Exterior Differential Systems to see some of the work, but the book doesn't use any representation theory. You will find representation theory in many of his published papers, for example on exotic holonomy groups. – Ben McKay Jun 30 '19 at 15:02
• Following up on @Misha's comment, here's the wikipedia page: en.wikipedia.org/wiki/D-module. It's pretty abstract stuff, though. – Deane Yang Jun 30 '19 at 15:57

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.

• There's an older book by Kolchin on differential algebra that talks about the PDE case more. – arsmath Jul 1 '19 at 17:48

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

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).

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).