# Some interesting and elementary topics with connections to the representation theory?

I'm going to give a talk to talented high school seniors (for nearly 1.25-1.75 hours, maybe a little bit longer). They know some abstract algebra (groups, rings, modules...), linear algebra (including the Jordan normal form, nilpotent operators, etc.), a real (and basics of complex) calculus, some elementary topology (basics from the general topology, fundamental groups and maybe something about manifolds or vector fields), basics of algebraic geometry and a lot of combinatorial stuff (including the graph theory and generating functions).

My main interest is the representation theory, so I'd like to discuss an algebraic topic connected with this branch of mathematics. It'll be cool if this topic contains some beautiful combinatorial constructions. Despite this, some level of abstraction is required... I think that it mustn't also be too famous (so the standard things like representations of symmetric groups (even in Vershik-Okounkov approach) or the basics of Lie theory aren't acceptable). Ideally, it has to be new to me...

Can someone give me a piece of advice about which topic can be chosen in this situation?

I thought about the cluster algebras (maybe in the flavor of the first pages of the paper ?)... But are there any elementary applications of them? (And, by the way, are there any classical monographs about this subject?)

UPD: There is a question which looks like similar to this on MO. Namely, Fun applications of representations of finite groups . But, of course, it's very different. The reason is that I'm more interested in representations of more complicated than finite groups structures like, for example, quivers or Lie algebras. So the other question isn't relevant: its topic is too narrow.

• may be something along the lines of polya enumeration, like polya polynomial and its relation to symmetric polynomials, including schur polynomials. – vidyarthi Oct 6 '19 at 8:31
• also may be something about using representation theory in graph coloring – vidyarthi Oct 6 '19 at 8:33
• Possible duplicate of Fun applications of representations of finite groups – Francois Ziegler Oct 6 '19 at 9:31
• Me and everyone else who won’t repeat what was there. Thank you :-) – Francois Ziegler Oct 6 '19 at 9:59
• @kotlinski: Ah, you're pre-emptively avoiding overlap with others' offerings; that makes sense. I take it these are students that have spent some time in undergraduate classes, but I'm still a bit skeptical about how much they are willing to care about the things representation theorists care about. Thus I would personally focus on the combinatorial structures underneath than on their algebraic origins. Say, promotion of standard tableaux is a game anyone can play and reason about, while Kazhdan-Lusztig polynomials are an acquired taste among a rather narrow class of mathematicians. ... – darij grinberg Oct 8 '19 at 18:55

Here three suggestions:

-An applications of representation theory to chemistry: "The Representation-Theory for Buckminsterfullerene" by Gordon James: https://www.sciencedirect.com/science/article/pii/S0021869384712130?via%3Dihub

-For a mix with probability theory the book Harmonic Analysis on Finite Groups: Representation Theory, Gelfand Pairs and Markov Chains by Tullio Ceccherini-Silberstein, Fabio Scarabotti and Filippo Tolli is great and gives insight to the works of Persi Diaconis.

-The applications of the representation theory of the symmetric group in Hurwitz theory, see for example the book Riemann Surfaces and Algebraic Curves: A First Course in Hurwitz Theory by Renzo Cavalieri and Eric Miles .

For cluster algebras there is a forthcoming book by Sergey Fomin, Lauren Williams and Andrei Zelevinsky : https://arxiv.org/abs/1608.05735 .

I gave a 4 minute(!) talk for an audience of high-school students about my research. I talked about the cyclic sieving phenomenon, which has a close connection with representation theory.

Most people have an intuitive idea about what symmetries are, and what objects are "more symmetric", so this concept is not that difficult.

Here is a link to the slides (swedish).

• A great material! Thank you! – kotlinski Oct 6 '19 at 14:30

Since you're mentioning the Jordan decomposition and Lie algebras, the connection between the representation theory of $$\mathfrak{sl}_2$$ (explicitly: Jacobson-Morozov, classification of irreducible representations, primitives), the Jordan decomposition of nilpotent operators and the weight filtration associated with a nilpotent operator comes to my mind. To which extent this is suitable heavily depends on the audience's background in $$\mathfrak{sl}_2$$-representations and linear algebra. Disclaimer: quite a few people I know consider this useless/ridiculous overkill. Moreover, I can't offer any combinatorics here and the representation theory is not at all deep or even very interesting. Nonetheless, I like it a lot, so here it goes:

The central concept is the weight filtration associated with a nilpotent endomorphism $$N\colon V\to V$$ of a finite-dimensional $$k$$-vector space $$V$$, $$W_{\bullet} = W_{\bullet}(N)$$: $$0 = W_{-n-1}\subset W_{-n}\subset\dots\subset W_{n-1}\subset W_n = V$$ which is, up to isomorphism, uniquely determined by the following two properties:

1. $$N$$ decreases the degree by two, i.e., for each integer $$i$$, $$NW_{i}\subset W_{i-2}$$, and
2. for each integer $$i$$, the map $$N^i\colon W_i\to W_{-i}$$ induces an isomorphism $$\overline{N}^i\colon \mathrm{Gr}_i(W_{\bullet})\to\mathrm{Gr}_{-i}(W_{\bullet})$$.

This is a classical thing in Hodge theory; the reference I'd like to mention here is Deligne, La Conjecture de Weil. II, IHES 1980, (1.6) Autour de Jacobson-Morosov. A web search will find you some more references.

Example 1 (Relation with the Jordan normal form) — Let $$N$$ have a Jordan decomposition with basis given by elements $$v_1,v_2,\dots,v_k$$ and their images under $$N$$, $$N^2$$, etc. Then we get a splitting of the filtration as follows: Define the primitive pieces $$P_i\subset W_i$$ as the span of those $$v_j$$ such that $$N^{i}v_j\not=0$$, but $$N^{i+1}v_j=0$$. Then $$\mathrm{Gr}_i(W_{\bullet}) \cong P_i\oplus NP_{i+2}\oplus N^2P_{i+4}\oplus \dots$$ via the obvious maps.

Abstractly, however, the primitive pieces as subspaces of the graded pieces are independent of the chosen basis, as $$P_i = \ker(\overline N^{i+1}\colon \mathrm{Gr}_i(W_{\bullet})\to\mathrm{Gr}_{-i-2}(W_{\bullet}))$$. Thus, reading the above backwards, the Jordan normal form itself, without the basis, is already encoded in the (dimensions of the) primitive pieces. Moreover, basic information about the normal form can be read off the filtration without looking at the primitive pieces.

Choosing a Jordan basis usually (with the standard algorithm) means choosing some a particular vectors, calculating their images under $$N$$, repeatedly, and then making some further choices depending on the prior choices, and so on. But if you know the weight filtration, choosing a Jordan basis boils down to the (conceptually easier) choice of a splitting and then simply choosing bases of the primitive pieces.

Example 2 (Relation with $$\mathfrak{sl}_2$$) — By the Jacobson-Morozov Theorem, $$N$$ is part of an $$\mathfrak{sl}_2$$-triple $$(M,H,N)$$. Here, $$H$$ determines (and is determined by) a splitting of the filtration, since the graded pieces are the eigenspaces of $$H$$. Moreover, from the grading and $$N$$, one can inductively construct $$M$$. (This yields a proof of Jacobson-Morozov.) Moreover, there is an obvious relation between the highest weight vectors (or the decomposition into irreducible representations) and the primitive pieces.

A nice concrete example is the exterior algebra of a complex symplectic vector space with nilpotent endomorphism given by wedge product with the fundamental form (the Lefschetz operator). But to get the representation one usually considers, one should better take the dual Lefschetz operator, which is given by contraction with the fundamental form. In either case, a natural splitting is given by the degree and the stuff about primitive parts (“Lefschetz decomposition”) is of great importance in the (local) theory of Kähler manifolds.

Let me conclude with a remark: One can spend quite some time talking about the weight filtration and the Jordan normal form without ever mentioning $$\mathfrak{sl}_2$$. Perhaps it is an option to solely mention that there is some $$\mathfrak{sl}_2$$-representation theory lurking in the background. But that would feel quite unsatisfactory, I’m afraid.

One example of an elementary application of cluster algebras is the proof that the Somos-4 and Somos-5 sequences, which are defined by a simple recursion, are integral. This is so because the entries of these sequences are cluster variables with the initial cluster variables specialized to 1, so it follows from the Laurent phenomenon. See Fomin and Zelevinsky's paper The Laurent Phenomenon.

One wouldn't have time to prove the Laurent phenomenon, but one can demonstrate it, and it is quite striking.