By: David Bressoud @dbressoud

The q-Dyson theorem that was the subject of my August Launchings was not an isolated result. It sits within a larger context that was a very active area during the 1980s and has continued to expand through the decades. This is some of the most beautiful mathematics that I have known. While I did not make significant contributions to its development, I cannot resist using this opportunity to introduce it to what I hope is a wider audience. It has its origins in the Weyl groups of Lie algebra and representation theory. I begin by recasting the Vandermonde determinant in the language of the root system A_n (starting with Figure 1).

A root system is any finite collection of non-zero vectors, called roots, with the following properties:

1.     They span the space.

2.     The only scalar multiples of a root are itself and its negative.

3.     Reflection across the hyperplane perpendicular to any one of the roots permutes the roots.

4.     For each root r, the difference between any root and its reflection across the hyperplane perpendicular to the root r is an integral multiple of r.

A root system is irreducible if it cannot be decomposed into two mutually orthogonal root systems. In addition to A_n, there are three other infinite families of irreducible root systems, B_n, C_n, and D_n (Figure 5) as well as root systems that only exist in particular dimensions: E_6, E_7, E_8, F_4, and G_2. In each case, the subscript designates the dimension of the space that is spanned. Figure 6 shows two-dimensional projections of E6, E7, and E8. The Weyl group is the group of symmetries generated by reflexions in hyperplanes perpendicular to the roots. And the sign (sgn) of an element of the Weyl group is +1 if the element can be constructed from an even number of reflexions in hyperplanes perpendicular to the roots, –1 if it requires an odd number. Thus, for A_2, the sign of each reflexion is –1 while the sign of each rotation is +1.

Problem. Verify that G_2, defined in Figure 7 and shown in Figure 8, satisfies the requirements of a root system.

The Weyl Denominator Formula, part of his character formula for complex semisimple Lie algebras, is simply the statement that A_n in equation (1) can be replaced by any root system.

In 1972, Ian Macdonald extended the Weyl Denominator  Formula to include the affine Lie algebras that had been defined by Kac and Moody (Figures 9 and 10).

These identities all involve the full affine extension, with powers of q extending from negative infinity to positive infinity. Macdonald explored what happens with truncated products, as in the q-Dyson identity, and made several conjectures which were further extended and elaborated by Walter Garfield “Chip” Morris in his doctoral thesis of 1982, written under the direction of Dick Askey (Figure 12). Chip’s thesis is unusual in that while it proves some theorems, the real contribution is the wealth of conjectures he provided. As I have stated before, coming up with great conjectures is the most important part of mathematics research. Chip had a love of both mathematics and dance. He told me that he pursued mathematics to see how far he could take that passion, but once he had taken it all the way to a PhD he decided that dance was where he really belonged. He went on to become owner and Director of the Acton School of Ballet in Massachusetts as well as co-founder and Artistic Director of the Commonwealth Ballet Company. Chip never published any of his mathematical results, but his thesis is well know. MathSciNet lists 45 research papers that reference his thesis. That must be some kind of a record.

References

Macdonald, I.G. (1982). Some conjectures for root systems and finite reflection groups. SIAM J. Math. Analysis 13, 988–1007.

Morris, W.G. II. (1982). Constant term identities for finite and affine root systems: conjectures and theorems. Ph.D. dissertation, University of Wisconsin-Madison.

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