Earlier I mentioned that the theory of Young tableaux is the source of one of my favorite proofs. Today I’d like to present, again from the theory of Young tableaux, one of my favorite pairs of proofs.
A standard Young tableau is a chain in Young’s lattice; equivalently, it is a sequence of Young diagrams, each of which has exactly one more box than the previous. One can succinctly write down such a chain by taking the last Young diagram in the chain and recording the “path” traced out by the rest of the chain: write down a 1 in the box corresponding to the first Young diagram, a 2 in the box added by the next Young diagram, and so forth. In other words, a standard Young tableau is a Young tableau filled with positive integers which is strictly increasing across rows and columns.
Given a Young diagram , let denote the number of standard Young tableaux of shape ; equivalently, one can define to be the poset of Young diagrams fitting inside , and then is equal to the number of maximal chains of .
Finally, one last bit of notation. If is a Young diagram for a partition of , we will write either or . Then we have the following beautiful result.
The representation-theoretic proof, which I believe is the method by which the theorem was first discovered, is this: every Young diagram has an associated Specht module that defines an irreducible representation of , and the dimension of this representation turns out to be precisely ; in fact it is possible to write down the basis explicitly. Since the number of conjugacy classes in is precisely the number of partitions of , these are all the irreducible representations. For example, the single row diagram corresponds to the trivial representation, while the single column diagram corresponds to the sign representation. In any case, standard results in representation theory then guarantee that for a finite group with irreducible representations , it is true that
and then the result follows as a corollary.
In the spirit of bijective combinatorics, however, the appropriate combinatorial interpretation of this identity is that there is a bijection between permutations and pairs of standard Young tableaux of the same shape. The interesting question is whether this bijection can be concisely described.
This is the content of the second proof: exactly such an explicit bijection is provided by the Robinson-Schensted-(Knuth) algorithm. Stanley provides a good explanation of the algorithm as well as a proof of a stronger result concerning so-called Hasse walks in Young’s lattice using operator methods, but it is not so much the details of the algorithm I want to talk about so much as its existence.
The fact that there can exist two proofs of the same result that are so different is fascinating to me. The network of ideas in this area bring together combinatorics, representation theory, invariant theory and symmetric function theory, and via Schur-Weyl duality relate to the representations of the linear groups as well, an important subject to physics and other branches of mathematics that one would think to be far removed from combinatorics.
Even without knowing all this, there are many questions raised by the existence of these two proofs. For example, how much information about a representation can be read off from the corresponding Young diagram? A lot, as it turns out: for example, if a representation of is given by a Young diagram , then its induced representation in is given by a direct sum over the representations corresponding to Young diagrams obtained by adding one box to , i.e. all elements covering in Young’s lattice. Similarly the restricted representation in is a direct sum over the representations corresponding to Young diagrams obtained by removing a box from . For a very readable introduction to these ideas, consult Yufei Zhao’s article in the Harvard College Mathematics Review.
Perhaps the only downside of the subject is that as it’s usually presented the relationship between Young tableaux and representations of the symmetric group is not deduced, but rather claimed (at least I have never seen anyone explain in intuitive terms why this should be.) A purely deductive approach would presumably carry over to other Coxeter groups as well, and indeed this seems to be the goal of this paper by Okounkov and Vershik.
Incidentally, it’s only very recently that I’ve come to learn that most research doesn’t consist of constructing new theories and solving open problems; a lot of it consists of compressing old theories into more manageable chunks or solving old problems in new, shorter ways. It seems to me that academic culture tends to undervalue this to some extent.