Opinion 5: The perils of `Algebraic' Combinatorics:

By: Doron Zeilberger

Written: Aug. 11, 1995.

It is the same old story: The underdogs become topdogs, and then they are even worse than the old topdogs.

In a captivating talk (and article) read in the Garsia Taoramina conference, Gian-Carlo Rota narrated the charming story (fairy-tale(?)) of how the ugly duckling of combinatorics became a beautiful swan, getting a new, gentrified name, in the process: `Algebraic Combinatorics'. Let's hope that it would live happily ever after, but if it is not to reach the ultimate fate of its previous oppressors (the Bourbaki-style, once mainstream, mathematicians), with their extreme abstract nonsense that borders with decadence, its practicioners should keep their feet on the ground, and not despise, `pure' combinatorialists, who are unable (or unwilling!) to connect their work with representation theory, sheaves, etc. etc.

Many of the successes of `fancy' mathematics are due to sociological and linguistic reasons. They are really high-school-algebra arguments in disguise.

Once you strip the fancy verbiage off, what remains is a bare (and much prettier, in my eyes) argument in high-school mathematics, or at most, in Freshman linear-algebra and (formal!) Calculus. For example, I am (almost) sure that Wiles's proof would be expressible in simple language, and if not, there would be a much nicer proof that would.

A good example is Cerednick's recent proof of Macdonald's constant term identities (and its generalizations to the inner product of Macdonald's polynomials). The `conceptualists' regard it as a triumph for their camp, since Cerednick's language is theirs. But a careful reading shows that it is easily transformable to elementary mathematics. The central idea was to use the amazing Dunkl operators, which are NOTHING BUT (and that's a big compliment!) than differential operators intertwined with symmetrization operations.

Another beautiful example of the versatile Dunkl operators was given recently by Luc Lapointe and Luc Vinet, who used Dunkl operators to prove the long-standing conjecture (for which several algebraic combinatorialists found equivalent statements, but were unable to prove) that the Jack polynomials are polynomials in alpha (with integer coeffs). I am willing to bet that their proof would generalize to prove that the Macdonald polynomials (for all root systems) are polynomials in p, and q, and I am willing to bet (a lesser amount) that it would eventually also give the non-negativity of the Kostka polynomials.

Another victory for down-to-earth physics, which is a branch of high-school math. Once known, their proof could in fact be made Shaloshable (One has to compute commutators of differential-symmetry operators, which for any fixed number of variables is obviously routine, and for a variable number of variables, could be made so).

To see the Lapointe-Vinet proof in action see my Maple program LUC.