A PROOF OF THE MACDONALD-MORRIS

ROOT SYSTEM CONJECTURE FOR F_{4}

F.G. GARVAN†

Abstract. We give a proof of the Macdonald-Morris root system conjecture for F_{4} that draws on ideas from Zeilberger’s recent proof of the G_{2}^{∨} case and Kadell’s proof of the q-BC_{n} case. Our proof depends on much computer computation. As in Zeilberger’s proof the problem is reduced to solving a system of linear equations. A FORTRAN program generated the equations which were solved using the computer algebra package MAPLE.

1. Introduction.

The Macdonald-Morris root system conjecture for F_{4} is

(1.1)  C.T. ∏_{1≤i<j≤4} (1 − x_{i}x_{j})^{a}(1 − x_{i}^{−1}x_{j}^{−1})^{a}(1 − \frac{x_{i}}{x_{j}})^{a}(1 − \frac{x_{j}}{x_{i}})^{a}

· ∏_{i=1}^{4}(1 − x_{i}^{2})^{b}(1 − x_{i}^{−2})^{b} ∏_{r_{1},r_{2},r_{3},r_{4}=±1}(1 − x_{1}^{r_{1}}x_{2}^{r_{2}}x_{3}^{r_{3}}x_{4}^{r_{4}})^{b}

= \frac{(6a + 6b)!(4a + 4b)!(2a + 6b)!(4a + 2b)!(2a + 4b)!(4b)!(3a)!}{(5a + 6b)!(3a + 5b)!(3a + 4b)!(3a + 2b)!(2a + 3b)!(a + 3b)!(2a + b)!}

· \frac{(3b)!(2a)!(2b)!}{(a + 2b)!(a + b)!a!a!b!b!b!} = f(a, b) .

Here C.T. means constant term of the Laurent polynomial in x_{1}, x_{1}^{−1}, …, x_{4}, x_{4}^{−1}. We refer to the reader to Macdonald [10] for more general statements of the root system conjectures. The A_{n}, B_{n}, C_{n}, D_{n}, BC_{n} and G_{2} cases of the Macdonald-Morris root system conjectures have been proved. The conjectures have remained open for the F_{4}, E_{6}, E_{7}, E_{8} cases. The best proof of the A_{n} case is due to Good [6]. Macdonald [10] noticed that the BC_{n} case (and hence the B_{n}, C_{n}, and D_{n} cases) follow from Selberg’s [12] integral. The G_{2} case was proved independently by Habsieger [7] and Zeilberger [13]. There are q-analogs of these conjectures.

Typeset by AMS-TEX

1991 Mathematics Subject Classification. Primary: 33A15, Secondary: 33A75, 17B20.

Key words and phrases. Macdonald-Morris root system conjectures, Aomoto’s proof of Selberg’s integral.

†School of Mathematics and Physics, Macquarie University, New South Wales 2109, Australia.

This research was done at the University of Wisconsin, Madison, and later while the author was a postdoctoral member of the Institute of Mathematics and Its Applications, University of Minnesota,  Minneapolis,  Minnesota.

2 F.G. GARVAN†

We refer the reader to Macdonald [10] and Askey [2] for more details. Recently, Zeilberger [14] has proved the G_{2}^{∨} case and Kadell [9] has proved the q-BC_{n} case.

The goal of this paper is to prove the F_{4} case, namely (1.1). Kadell’s paper [9, §2] also contains a new proof of the (q = 1) BC_{n} case of the Macdonald-Morris root system conjecture that avoids integrals. This new proof is analogous to Aomoto’s [1] proof of Selberg’s integral in the following sense: It involves adding extra factors to the Laurent polynomial as opposed to adding extra factors to the integrand of Selberg’s integral and Aomoto’s integration by parts is replaced by the fact that the derivative of a Laurent polynomial has no residue. We extend Kadell’s proof to the F_{4} case but another idea is needed. The extra idea comes from Zeilberger’s [14] proof of the G_{2}^{∨} case of the q-analog of Macdonald-Morris root system conjecture. As in Zeilberger [14] the problem is reduced to finding and solving a system of linear equations whose unknowns are constant terms of certain Laurent polynomials. These equations are generated with the aid of a FORTRAN program. Finally the equations are solved using the computer algebra package MAPLE.

After some preliminaries in §2 an idea of the proof is given in §3. The results behind the FORTRAN program that generates the desired equations are given in §§4-6. The proof is completed in §7.

We have been able to verify the results of this paper by another method. Recently we [5] have found a new proof of the G_{2} case of the Macdonald-Morris root system conjecture which is solely in terms of integrals. Our proof was motivated by some conjectures of Askey [2], that have to do with adding roots to the G_{2} case of the Macdonald-Morris root system conjecture, and is analogous to Aomoto’s [1] proof of the Selberg’s integral. We have been able to extend our integral proof of the G_{2} case to the F_{4} case. However, this other proof involves finding equations between certain integrals and converting these into equations involving constant terms. The proof then proceeds as usual by solving a system of linear equations. We have omitted this other proof, finding the approach of working with Laurent polynomials rather than with integrals more straightforward.

In §8 we give some other results that involve adding extra factors to the F_{4} case of the Macdonald-Morris root system conjecture. The results are analogous to Askey’s [2] conjectures for G_{2}. Although many of these results can be written as products of factorials we are unable to generalize them to all root systems. These other results may indicate that a simpler proof of the F_{4} case is possible.

All computer programs used in this paper are available from the author on request. Some preliminary calculations were done using REDUCE at the University of Wisconsin, Madison. The final FORTRAN and MAPLE programs were run on an APOLLO DN-5800 at the I.M.A., University of Minnesota, Minneapolis.

2. Some Preliminaries.

In this section we prove some properties of the root system F_{4} that will be needed later. We assume that the reader is familiar with the basics of root systems and their Weyl groups. See Bourbaki [3], Carter [4] and Humphreys [8] for treatments of root systems and Weyl groups.

A PROOF OF THE MACDONALD-MORRIS ROOT SYSTEM CONJECTURE FOR F_{4} 3

Let {e_{1}, e_{2}, e_{3}, e_{4}} be the standard basis of ℝ^{4}. The roots of F_{4} are usually written as

±e_{i} (1 ≤ i ≤ 4),  ±e_{i} ± e_{j}  (1 ≤ i < j ≤ 4), (2.1)
\frac{1}{2} (±e_{1} ± e_{2} ± e_{3} ± e_{4}).

See, for example Bourbaki [3, p. 272]. We call this set of roots Φ^{(1)}(F_{4}). It is clear that the long roots of F_{4} are isomorphic to D_{4}. In this paper we shall use two other ways of writing the roots of F_{4}.

Firstly, we rewrite the roots of F_{4} in a way that makes it clear that the short roots of F_{4} are isomorphic to D_{4}. Let Φ^{(2)}(F_{4}) be the set of vectors:

±2e_{i} (1 ≤ i ≤ 4),  ±e_{i} ± e_{j}  (1 ≤ i < j ≤ 4), (2.2)
(±e_{1} ± e_{2} ± e_{3} ± e_{4}).

Φ^{(2)}(F_{4}) and Φ^{(1)}(F_{4}) are isomorphic as root systems. The isomorphism is given by

(2.3)  A : Φ^{(1)}(F_{4}) ⟶ Φ^{(2)}(F_{4})

where A is the transformation with matrix

(2.4)  ( \begin{matrix}0 & 1 & 0 & 1 \\ 1 & 0 & 1 & 0 \\ 1 & 0 & −1 & 0 \\ 0 & −1 & 0 & 1 \end{matrix} )

with respect to the standard basis of ℝ^{4}. This is a root system isomorphism since A^{t}A = 2I. As an immediate consequence we have

Lemma 2.5. Short roots of F_{4} ≅ D_{4}.

Secondly, we may write the roots of F_{4} (as given in (2.2)) as ℤ-linear combinations of α_{1}, α_{2}, α_{3}, α_{4} where

(2.6)  α_{1} = e_{1} − e_{2}, α_{2} = e_{3} + e_{4}, α_{3} = e_{3} − e_{4}, α_{4} = e_{2} − e_{3}.

These ℤ-linear combination are given Appendix A. The α_{i} come from the Dynkin diagram for D_{4} which is given below in Figure 1.

Figure 1 Dynkin diagram for D_{4}

From the symmetry of the Dynkin diagram we see that any permutation of α_{1}, α_{2}, α_{3} leaves the root system of D_{4} invariant. It is interesting to note that any such permutation also leaves the root system of F_{4} invariant. Let π ∈ S_{3} and

suppose α = ∑_{i=1}^{4} k_{i}α_{i} ∈ Φ^{(2)}(F_{4}) where k_{i} ∈ ℤ (1 ≤ i ≤ 4). Then we define

(2.7)  πα = k_{1}α_{π(1)} + k_{2} α_{π(2)} + k_{3}α_{π(3)} + k_{4}α_{4}.

We have

Lemma 2.8. For π ∈ S_{3}, π(Φ^{(2)}(F_{4})) = Φ^{(2)}(F_{4}).

For a root system R we denote its Weyl group by W (R). We need a nice way to code the elements of W (F_{4}). By Bourbaki [3, p. 257] W (D_{4}) consists of all signed permutations with an even number of signs, that act on the coordinates e_{1}, e_{2}, e_{3}, e_{4}. Let H denote the set of all signed permutations that act on the coordinates e_{1}, e_{2}, e_{3}, e_{4}. For α ∈ R we denote by w_{α} the reflection through the hyperplane orthogonal to α. Since w_{2e_{i}} ∈ W (F_{4}) (1 ≤ i ≤ 4) we have

(2.9)  W (D_{4}) ⊂ H ⊂ W (F_{4}).

We introduce some notation to describe the elements of H. We denote the permutations on the coordinates e_{1}, e_{2}, e_{3}, e_{4} using the usual cycle notation. We define the sign changes as follows: For 1 ≤ i ≤ 4 let s_{i} denote the transformation given by

(2.10)  s_{i} : ℝ^{4} ⟶ ℝ^{4}  e_{i} ⟼ −e_{i}  and  e_{j} ⟼ e_{j} (j ≠ i).

For example,

s_{1}(34) (e_{1} + e_{2} + 2e_{3} − e_{4}) = −e_{1} + e_{2} − e_{3} + 2e_{4}.

Lemma 2.11. Every w ∈ W (F_{4}) can be written

w = (τσ)^{k} h

where k = 0, 1, 2 h ∈ H and τ = w_{2e_{4}} and σ = w_{e_{1}−e_{2}−e_{3}−e_{4}} .

Proof. | H | = 2^{4}.4! = 2^{7}.3 | W (F_{4}) | = 2^{7}.3^{2} (Bourbaki [3, p.273 (X)]).

The result follows since τσ ∉ H and (τσ)^{3} = I. □

Let,

(2.12) \overline{C} = \{(x_{1}, x_{2}, x_{3}, x_{4}) ∈ ℝ : x_{1} ≥ x_{2} ≥ x_{3} ≥ x_{4} ≥ 0,
x_{2} + x_{3} + x_{4} ≥ x_{1}, x_{1} + x_{4} ≥ x_{2} + x_{3}}.

Lemma 2.13. For every x_{∼} ∈ ℝ^{4} there is a unique v_{∼} ∈ \overline{C} such that x_{∼} can be transformed into v_{∼} by some element of W (F_{4}).

Proof. From Bourbaki [3, p. 272 (II)] the following vectors form a base for F_{4}:

(2.14)  e_{2} − e_{3}, e_{3} − e_{4}, e_{4}, \frac{1}{2} (e_{1} − e_{2} − e_{3} − e_{4}),

with the roots written in the usual way (i.e: as elements of Φ^{(1)}(F_{4})). In our representation of F_{4} (i.e: as elements of Φ^{(2)}(F_{4})) this corresponds to

(2.15)  e_{1} − e_{2} + e_{3} − e_{4}, −e_{1} + e_{2} − e_{3} − e_{4}, e_{1} + e_{4}, −e_{1} + e_{3}.

By applying the relevant signed permutation we find that the following vectors also form a base for F_{4}:

(2.16)  −e_{1} + e_{2} + e_{3} + e_{4}, e_{1} − e_{2} − e_{3} + e_{4}, e_{3} − e_{4}, −e_{3} + e_{2},

by Carter [4, Theorem 2.2.4]. The closure of the chamber, C, corresponding to this base is given in (2.12). The result follows from Carter [4, Proposition 2.3.4]. □

Let,

(2.17)  funch : ℝ^{4} ⟶ \overline{C}  funch (x_{∼}) = v_{∼}

with x_{∼}, v_{∼} as in Lemma (2.13) above. For α = ∑_{i=1}^{4} k_{i}e_{i} (k_{i} ∈ ℤ) we let

(2.18)  x^{α} = ∏_{i=1}^{4} x_{i}^{k_{i}}.

The elements w of the Weyl group act on monomials by

(2.19)  w(x^{α}) = x^{w(α)},

and by linearity on Laurent polynomials that are linear combinations of the x^{α}. Let

(2.20)  F (x_{∼}; a, b) = ∏_{α∈Φ^{(2)}(F_{4})} (1 − x^{α})^{k_{α}}

where  k_{α} = \{ \begin{matrix}a, & if α is a short root, \\ b, & if α is a long root. \end{matrix}

We note that F (x_{∼}; a, b) is the Laurent polynomial on the left hand side of (1.1).

Lemma 2.21. For w ∈ W (F_{4}),

C.T. x^{α} F = C.T. x^{w(α)}F

where F is defined in (2.20).

Proof. The result follows from the fact that F is symmetric with respect to the Weyl group and w does not change the constant term. □

3. The idea of the proof.

Let,

(3.1)  f′(a, b) = C.T.F (x_{∼}; a, b)

where F is defined in (2.20). Our goal is to prove that f′(a, b) = f(a, b) for all a, b ≥ 0. The idea is to proceed by induction on a. That is, we want to prove that

(3.2)  \frac{f′(a + 1, b)}{f′(a, b)} = \frac{f(a + 1, b)}{f(a, b)} .

This will be enough because the case a = 0 is already known since

(3.3)  Long roots of F_{4} ≅ D_{4}.

The flavor of our proof is similar to Zeilberger’s [14] proof of the G_{2}^{∨} case of the q-version of the Macdonald-Morris root system conjecture. Let L be the lattice generated by α (α ∈ F_{4}). Now,

f′(a + 1, b) = C.T. F (x_{∼}; a + 1, b) (3.4)

= C.T. ∏_{α∈ short F_{4}} (1 − x^{α}) F (x_{∼}; a, b)

= C.T. ∏_{α∈D_{4}} (1 − x^{α})F (x_{∼}; a, b) (by Lemma (2.5))

= C.T. ∑_{α∈L′} a_{α}x^{α}F (x_{∼}; a, b) (for some L′ ⊂ L)

= C.T. ∑_{α∈L′} a_{α}x^{funch(α)} F (x_{∼}; a, b) (by Lemma (2.13), (2.17) and Lemma (2.21))

= C.T. ∑_{α∈S} a′_{α}x^{α} F (x_{∼}; a, b)

for some finite subset S of \overline{C}, which is defined in (2.12). ∏_{α∈D_{4}}(1−x^{α}) was multiplied

at utilizing a FORTRAN program, funch(α) was calculated for each monomial x^{α} that arose in each stage of the multiplication. The 37 vectors that arose, v(i) (1 ≤ i ≤ 37), are listed in Appendix B. S = {v(i) : 1 ≤ i ≤ 37} and

(3.5)  ∑_{α∈S} a′_{α}x^{α} = 192x^{v(1)} − 768x^{v(2)} + ⋯ + 192x^{v(37)}.

The complete list of coefficients in (3.5) is given in Appendix B. Let,

(3.6)  an(i) = C.T. x^{v(i)}F (x_{∼}; a, b)  (1 ≤ i ≤ 37).

The problem is to get an(i) (2 ≤ i ≤ 37) in terms of an(1) = C.T. x^{v(1)}F (x_{∼}; a, b) = C.T. F (x_{∼}; a, b) = f′(a, b). Once we have done this (3.2) should follow from (3.4) and (3.5).

Hence we need to find 36 independent equations in the unknowns an(i) (1 ≤ i ≤ 37). Our goal is to write a FORTRAN program that will generate equations. The input of this program is a 4-tuple k_{∼} = (k_{1}, k_{2}, k_{3}, k_{4}) ∈ ℕ^{4} and the output will be either a homogeneous linear equation in the an(i) (1 ≤ i ≤ 37) or an error message, which says such an equation is not possible. It will turn out that when k_{∼} = v(i) (2 ≤ i ≤ 37) that the corresponding outputs will be the required 36 independent equations in the an(i) (1 ≤ i ≤ 37). See §6 for more details. We describe how we came by this program in 4 steps:

STEP 1: Use

(3.7)  C.T. x_{1} \frac{∂}{∂x_{1}} x_{1}^{k_{1}}x_{2}^{k_{2}}x_{3}^{k_{3}}x_{4}^{k_{4}} F (x_{∼}; a, b) = 0 .

As noted before this idea was used by Kadell [9] in his proof of the q-BC_{n} case. This gives rise to an equation involving constant terms of rational functions in x_{1}, x_{2}, x_{3}, x_{4} times F.

STEP 2: Use the Weyl group to reduce the number of types of terms arising in STEP 1.

STEP 3: Use the Weyl group to write the constant terms that arise in STEP 2 as constant terms of Laurent polynomials times F.

STEP 4: Use the Weyl group to write the constant terms that arise in STEP 3 in terms of the an(i) (1 ≤ i ≤ 37), if possible.

In this way, for certain k_{∼}, (3.7) can be written as a linear homogeneous equation in the an(i) (1 ≤ i ≤ 37).

4. Steps 1 and 2: An Equation involving constant terms with denominators.

In this section we describe the first two steps, mentioned in §3, that are needed in turning the equation (3.7) into an equation involving the an(i). (3.7) is

0 = C.T.\{(k_{1} + a ∑_{j=2}^{4} (\frac{−x_{1}x_{j}}{1 − x_{1}x_{j}} + \frac{x_{1}^{−1}x_{j}^{−1}}{1 − x_{1}^{−1}x_{j}^{−1}} − \frac{x_{1}/x_{j}}{1 − x_{1}/x_{j}} + \frac{x_{j}/x_{1}}{1 − x_{j}/x_{1}} ) (4.1)

+ b(\frac{−2x_{1}^{2}}{1 − x_{1}^{2}} + \frac{2x_{1}^{−2}}{1 − x_{1}^{−2}} )

+ b ∑_{r_{2},r_{3},r_{4}=±1}(\frac{−x_{1}x_{2}^{r_{2}}x_{3}^{r_{3}}x_{4}^{r_{4}}}{1 − x_{1}x_{2}^{r_{2}}x_{3}^{r_{3}}x_{4}^{r_{4}}} + \frac{x_{1}^{−1}x_{2}^{r_{2}}x_{3}^{r_{3}}x_{4}^{r_{4}}}{1 − x_{1}^{−1}x_{2}^{r_{2}}x_{3}^{r_{3}}x_{4}^{r_{4}}} ))

. x^{k_{∼}} F (x_{∼}; a, b)},

where k_{∼} = (k_{1}, k_{2}, k_{3}, k_{4}). Hence,

C.T. k_{1} x^{k_{∼}} F (4.2)

= C.T. (a ∑_{j=2}^{4} (\frac{1 + x_{1}x_{j}}{1 − x_{1}x_{j}} + \frac{1 + \frac{x_{1}}{x_{j}}}{1 − \frac{x_{1}}{x_{j}}} )

+ 2b \frac{(1 + x_{1}^{2})}{(1 − x_{1}^{2})}

+ b ∑_{r_{2},r_{3},r_{4}=±1} \frac{1 + x_{1}x_{2}^{r_{2}}x_{3}^{r_{3}}x_{4}^{r_{4}}}{1 − x_{1}x_{2}^{r_{2}}x_{3}^{r_{3}}x_{4}^{r_{4}}} ) . x^{k_{∼}} F.

This completes STEP 1.

In STEP 2 we reduce the number of different denominators appearing in STEP 1 to two (one for each root length). The reason this can be done is that each term

on the right hand side of (4.2) is of the form \frac{1 + x^{α}}{1 − x^{α}} for some α ∈ F_{4}, and so

can be converted to one of two types by using the fact that the Weyl group acts transitively on roots of equal length and by using Lemma (2.21). Hence for each α ∈ F_{4} we need to find a w ∈ W (F_{4}) such that

w(α) = \{ \begin{matrix}e_{1} − e_{3}, & α short, \\ e_{1} − e_{2} + e_{3} + e_{4}, & α long. \end{matrix}

It is clear that for α of the form e_{i} ± e_{j}, e_{1} ± e_{2} ± e_{3} ± e_{4} we may take w ∈ H, the set of signed permutations (defined in §2), and w is easy to calculate. All that remains is to find a w ∈ H such that w(2e_{1}) = e_{1} − e_{2} + e_{3} + e_{4}. Let SYM denote the reflection through the hyperplane orthogonal to e_{1} − e_{2} − e_{3} − e_{4} then

2e_{1} ⟶^{(13)} 2e_{3} ⟶^{SYM} e_{1} − e_{2} + e_{3} − e_{4} ⟶^{s_{4}} e_{1} − e_{2} + e_{3} + e_{4} .

Hence we find that (4.2) may be written as

(4.3)  0 = C.T. {k_{1}x^{k_{∼}}

+ a \frac{(x_{1} + x_{3})}{(x_{1} − x_{3})} (x^{k_{∼}} + (23)x^{k_{∼}} + (34)x^{k_{∼}}

+ s_{3}x^{k_{∼}} + (23)s_{2}x^{k_{∼}} + (34)s_{4}x^{k_{∼}})

+ b \frac{(x_{2} + x_{1}x_{3}x_{4})}{(x_{1}x_{3}x_{4} − x_{2})} (2 s_{4} SYM (13)x^{k_{∼}}

+ s_{2}x^{k_{∼}} + s_{4}s_{2}x^{k_{∼}} + s_{2}s_{3}x^{k_{∼}}

+ s_{2}s_{3}s_{4}x^{k_{∼}} + x^{k_{∼}} + s_{4}x^{k_{∼}}

+ s_{3}x^{k_{∼}} + s_{3}s_{4}x^{k_{∼}})} F (x_{∼}; a, b).

5. STEP 3: Getting rid of denominators.

The constant term expressions that arise in STEP 2 can be written as either

C.T. \frac{p_{1}(x_{∼})}{x_{1} − x_{3}} F (x_{∼}; a, b) (5.1)

or

C.T. \frac{p_{2}(x_{∼})}{x_{1}x_{3}x_{4} − x_{2}} F (x_{∼}; a, b), (5.2)

where p_{i}(x_{∼}) (i = 1, 2) are Laurent polynomials. In this section we show how each of these expressions can be written in the form

C.T.p(x_{∼}) F (x_{∼}; a, b),

for some Laurent polynomial p(x_{∼}), and how such an expression can be computed. (5.1) is easy to handle. Since F (x_{∼}; a, b) is symmetric in x_{1}, x_{3} we have

(5.3)  C.T. \frac{p_{1}(x_{∼})}{x_{1} − x_{3}} F (x_{∼}; a, b) = \frac{1}{2} C.T. (\frac{p_{1}(x_{∼}) − (13)p_{1}(x_{∼})}{x_{1} − x_{3}} ) F (x_{∼}; a, b)

and it is clear that \frac{p_{1}(x_{∼}) − (13)p_{1}(x_{∼})}{x_{1} − x_{3}} is a polynomial.

Before we can handle (5.2) we need to define an algorithm, FUN, whose input is a vector given in terms of the e_{i} (1 ≤ i ≤ 4) and whose output is the same vector given in terms of the α_{i} (1 ≤ i ≤ 4), defined in (2.6). FUN : ℝ^{4} ⟶ ℝ^{4} is a linear transformation whose matrix is

(5.4)  ( \begin{matrix}1 & \frac{1}{2} & \frac{1}{2} & 1 \\ 0 & \frac{1}{2} & \frac{1}{2} & 1 \\ 0 & \frac{1}{2} & \frac{1}{2} & 0 \\ 0 & \frac{1}{2} & −\frac{1}{2} & 0 \end{matrix} )

with respect to the bases {e_{i} : 1 ≤ i ≤ 4}, {α_{i} : 1 ≤ i ≤ 4}. The inverse of FUN is UNFUN with matrix

(5.5)  ( \begin{matrix}1 & −1 & 0 & 0 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 1 & −1 \\ 0 & 1 & −1 & 0 \end{matrix} )

For α = ∑_{i=1}^{4} ℓ_{i}α_{i} (ℓ_{i} ∈ ℤ) we let

(5.6)  y^{α} = ∏_{i=1}^{4} y_{i}^{ℓ_{i}} .

FUN acts on monomials by

(5.7)  FUN (x^{α}) = y^{FUN (α)},

and by linearity on Laurent polynomials that are linear combinations of the x^{α}. For example,

FUN (\frac{x_{1}x_{3}x_{4}}{x_{2}} ) = FUN (x^{e_{1}−e_{2}+e_{3}+e_{4}})
= y^{FUN (e_{1}−e_{2}+e_{3}+e_{4})}
= y^{α_{1}+α_{2}}
= y_{1}y_{2} .

Alternatively, we could describe the action of FUN as replacing x_{1} by y_{1}, x_{2} by \sqrt{y_{1}y_{2}y_{3}y_{4}} , x_{3} by \sqrt{y_{1}y_{2}y_{3}/y_{4}} and x_{4} by y_{1}y_{2}.

From Lemma (2.8) it follows that

(5.8)  F′(y_{∼}; a, b) = FUN (F (x_{∼}; a, b))

is symmetric in y_{1}, y_{2}, y_{3}. Now we can handle (5.2).

C.T. \frac{p_{2}(x_{∼})}{x_{1}x_{3}x_{4} − x_{2}} F (x_{∼}; a, b) (5.9)

=C.T. \frac{((12)p_{2}(x_{∼}))}{x_{2}x_{3}x_{4} − x_{1}} F (x_{∼}; a, b)  (by Lemma (2.21))

=C.T.(\frac{x_{1}^{−1}(12)p_{2}(x_{∼})}{x^{α_{2}−α_{1}} − 1} ) F (x_{∼}; a, b)

=C.T. \frac{q(y_{∼})}{y_{2} − y_{1}} F′(y_{∼}; a, b)  (by applying FUN)

where q(y_{∼}) is the Laurent polynomial

(5.10)  q(y_{∼}) = y_{1} FUN (x_{1}^{−1}(12)p_{2}(x_{∼})).

Hence,

C.T. \frac{p_{2}(x_{∼})}{x_{1}x_{3}x_{4} − x_{2}} F (x_{∼}; a, b) (5.11)

=\frac{1}{2} C.T.( \frac{q(y_{∼}) − (12)q(y_{∼})}{y_{2} − y_{1}} ) F′(y_{∼}; a, b)

=\frac{1}{2} C.T. UNFUN ( \frac{q(y_{∼}) − (12)q(y_{∼})}{y_{2} − y_{1}} ) F′(x_{∼}; a, b)

and UNFUN ( \frac{q(y_{∼}) − (12)q(y_{∼})}{y_{2} − y_{1}} ) is a Laurent polynomial in x_{i} (1 ≤ i ≤ 4) as

required.

We note that \frac{p_{1}(x_{∼}) − (13)p_{1}(x_{∼})}{x_{1} − x_{3}} and similarly UNFUN ( \frac{q(y_{∼}) − (12)q(y_{∼})}{y_{2} − y_{1}} )can

be easily computed by observing that if p_{1}(x_{∼}) = ∑_{a_{∼}} c_{a_{∼}} x_{∼}^{a_{∼}} = ∑_{a_{∼}} c_{a_{∼}}x_{1}^{a_{1}}x_{2}^{a_{2}}x_{3}^{a_{3}}x_{4}^{a_{4}}

then (5.12)

\frac{p_{1}(x_{∼}) − (13)p_{1}(x_{∼})}{x_{1} − x_{3}} = ∑_{a_{∼}} sgn (a_{1}−a_{3})c_{a_{∼}}x_{2}^{a_{2}}x_{4}^{a_{4}} ∑_{ℓ=0}^{|a_{1}−a_{3}|−1} x_{1}^{min(a_{1},a_{3})+ℓ}x_{3}^{max(a_{1},a_{3})−ℓ−1}.

6. STEP 4: Obtaining equations in the an(i).

In this section we describe the final step needed in converting the constant term equation, (3.7), into an equation involving the an(i). An examination of STEPS 1-4 will then yield an algorithm whose input is a 4-tuple k_{∼} = (k_{1}, k_{2}, k_{3}, k_{4}) ∈ ℕ^{4} and whose output is an equation involving the an(i), or an error message.

By STEP 3 we can write the constant term equation (3.7), in terms of constant terms of certain Laurent polynomials times F (x_{∼}; a, b). In STEP 4 we would like to find such expressions in terms of the an(i). To do this we use funch, defined in (2.17). Suppose we are given such an expression, say p(x_{∼}) F (x_{∼}; a, b), where

(6.1)  p(x_{∼}) = ∑_{α∈L′} a_{α}x^{α}  (for some finite set L′).

Then,

C.T. p(x_{∼})F (x_{∼}; a, b) (6.2)
= C.T. ∑_{α∈L′} a_{a}x^{funch(α)} F (x_{∼}; a, b)  (by Lemma (2.13), (2.17) and Lemma (2.20))

= C.T. ∑_{α∈S} a′_{α}x^{α} F (x_{∼}; a, b)  (where S ⊆ funch (L′) ⊆ \overline{C} de- fined in (2.12))

= ∑_{i=1}^{37} a′_{v(i)} an(i) (where an(i) is defined in (3.6).

Note that here we are assuming S ⊆ {v(i)}_{i=1}^{37} which may not necessarily be the case. However, for all values of the input k_{∼}, that we use, this condition is satisfied. We leave it to the reader to write a subroutine that will do the reduction described in (6.2). This subroutine should check whether S ⊆ {v(i)}_{i=1}^{37}. If this condition is not satisfied the output of the subroutine should be some error message.

7. STEP 5: Generating the equations and completing the proof. In the previous section we noted that STEPS 1-4 yield an algorithm whose input

is a 4-tuple k_{∼} = (k_{1}, k_{2}, k_{3}, k_{4}) ∈ ℕ^{4} and whose output is an equation involving the an(i). We have written a FORTRAN program that incorporates this algorithm. We leave it to the reader to use equation (4.3) and STEPS 3 and 4 to write such a program. We have found that the set of inputs k_{∼} = v(i) (2 ≤ i ≤ 37), given in Appendix B, yield a system of 36 independent equations in the an(i), as required. In fact a certain sequence of such inputs will yield a certain sequence of equations that can be solved easily using an algebra package like MAPLE. Our sequences have the following form:

INPUT  OUTPUT
k_{∼}_{1} = v(2)  an(2) = \frac{−a}{(5a + 6b + 1)} an(1)
k_{∼}_{2} = v(3)  an(3) = \frac{−1}{(3a + 5b + 1)} \{b an(1) + 3a an(2)\}
k_{∼}_{3} = v(4)  an(4) = \frac{−1}{(3a + 4b + 1)} \{2(a + b)an(2) + a an(3)\}
k_{∼}_{4} = v(5)  an(5) = \frac{−1}{(5a + 6b + 2)} \{(a + 2b)an(2) + 2a an(3)
+4(a + b) an(4)}
k_{∼}_{5} = v(7)  an(7) = \frac{−1}{(5a + 6b + 2)} \{a an(1) + 2a an(2)
+6b an(3) + 4a an(4)}
⋮  ⋮
⋮  ⋮
⋮  ⋮

k_{∼}_{36} = v(37)  an(37) = \frac{−1}{(6a + 8b + 6)} \{a an(4) + 2a an(6)
+ . . . . . . + a an(36)}

We note that each input v(i_{0}) produces an equation whose left hand side is an(i_{0}). Each equation in the output is a linear equation in the an(i) (1 ≤ i ≤ 37) and the an(i) that appear on the right hand side occur as left hand sides of equations that appear earlier in the output sequence. In other words, the system of equations is triangular in shape. Since this paper was first written we have found that this triangularity extends to all root systems. In fact, if the set of inputs has the following form:

\{ γ : γ ≺ γ_{0}\} ∩ L ∩ \overline{C},

then the corresponding system of equations is triangular with respect to any order that preserves the root order. Here ≺ is the usual root order, γ_{0} ∈ L ∩ \overline{C}, L is the root lattice and C is the fundamental chamber.

Below we give our complete sequence of inputs:

k_{∼}_{1} = v(2), k_{∼}_{2} = v(3), k_{∼}_{3} = v(4), k_{∼}_{4} = v(5), k_{∼}_{5} = v(7),
k_{∼}_{6} = v(8), k_{∼}_{7} = v(6), k_{∼}_{8} = v(11), k_{∼}_{9} = v(14), k_{∼}_{10} = v(9),
k_{∼}_{11} = v(10), k_{∼}_{12} = v(12), k_{∼}_{13} = v(13), k_{∼}_{14} = v(15), k_{∼}_{15} = v(16),
k_{∼}_{16} = v(17), k_{∼}_{17} = v(18), k_{∼}_{18} = v(19), k_{∼}_{19} = v(20), k_{∼}_{20} = v(21),
k_{∼}_{21} = v(22), k_{∼}_{22} = v(23), k_{∼}_{23} = v(24), k_{∼}_{24} = v(25), k_{∼}_{25} = v(29),
k_{∼}_{26} = v(26), k_{∼}_{27} = v(27), k_{∼}_{28} = v(28), k_{∼}_{29} = v(30), k_{∼}_{30} = v(31),
k_{∼}_{31} = v(32), k_{∼}_{32} = v(33), k_{∼}_{33} = v(34), k_{∼}_{34} = v(35), k_{∼}_{35} = v(36),

k_{∼}_{36} = v(37).

We can now complete the proof. From (3.4) and (3.5) we have

f′(a + 1, b) = 192 an(1) − 768 an(2) + . . . . . . + 192 an(37) (7.1)

(a complete list of the coefficients is given in Appendix B)

= 18432 \frac{(3a + 2)(3a + 1)(2a + 1)(6a + 6b + 5)(4a + 4b + 3)}{(3a + 4b + 3)(3a + 5b + 3)(5a + 6b + 5)(2a + 3b + 1)(3a + 5b + 2)}

·\frac{(2a + 6b + 1)(4a + 2b + 3)(4a + 4b + 1)(6a + 6b + 1)(2a + 4b + 1)}{(5a + 6b + 3)(5a + 6b + 4)(3a + 4b + 2)(5a + 6b + 1)(5a + 6b + 2)}

·\frac{(4a + 2b + 1)(2a + 2b + 1)^{2} an(1)}{(3a + 4b + 1)(3a + 5b + 1)(2a + 3b + 2)} (via MAPLE)

= \frac{f(a + 1, b)}{f(a, b)} f′(a, b),

which is (3.2) as required.

8. Other results.

In this section we give other results that have to do with adding roots to the F_{4} case of the Macdonald-Morris root system conjecture. Recently we [5] have found a new proof of the G_{2} case of the Macdonald-Morris root system conjecture that is solely in terms of integrals. Our proof was motivated by some conjectures of Askey [2], that have to do with adding roots to the G_{2} case, and is analogous to Aomoto’s [1] proof of Selberg’s integral. We have been able to extend our integral-type proof, mentioned above, to the F_{4} case. This proof involves converting equations involving integrals into equations analogous to (4.3) given in STEP 3 and then the proof is completed by proceeding as in STEPS 4 and 5. The proof given in this paper is more straightforward and direct.

We consider sets of the form S = T ∪ −T where T is a subset of the short roots of F_{4} (i.e: D_{4}). We call two such subsets S_{1} and S_{2} equivalent if there is a w ∈ W (F_{4}) such that S_{2} = w(S_{1}). This defines an equivalence relation on such sets. By utilizing a FORTRAN program we have found all the equivalence classes:

\frac{|S|}{2}  # of equivalence classes  representative of each equivalence class
1  1  ±{β_{1}}
2  2  ±{β_{1}, β_{2}}, ±{β_{1}, β_{4}}
3  4  ±{β_{1}, β_{2}, β_{3}}, ±{β_{1}, β_{2}, β_{4}}
±{β_{1}, β_{4}, β_{5}}, ±{β_{2}, β_{4}, β_{5}}
4 6 ±{β_{1}, β_{2}, β_{3}, β_{4}}, ±{β_{1}, β_{2}, β_{4}, β_{5}}
±{β_{2}, β_{3}, β_{4}, β_{5}}, ±{β_{1}, β_{2}, β_{5}, β_{6}}
±{β_{1}, β_{3}, β_{5}, β_{6}}, ±{β_{4}, β_{8}, β_{9}, β_{10}}
5 7 ±{β_{1}, β_{2}, β_{3}, β_{4}, β_{5}}, ±{β_{1}, β_{2}, β_{3}, β_{5}, β_{6}}
±{β_{1}, β_{2}, β_{4}, β_{5}, β_{6}}, ±{β_{1}, β_{3}, β_{4}, β_{5}, β_{6}}
±{β_{3}, β_{4}, β_{5}, β_{6}, β_{8}}, ±{β_{1}, β_{2}, β_{3}, β_{7}, β_{8}, }
±{β_{1}, β_{4}, β_{8}, β_{9}, β_{10}}
6 9 ±{β_{1}, β_{2}, β_{3}, β_{4}, β_{5}, β_{6}}, ±{β_{1}, β_{2}, β_{3}, β_{5}, β_{6}, β_{7}}

±{β_{1}, β_{2}, β_{4}, β_{5}, β_{6}, β_{8}}, ±{β_{1}, β_{3}, β_{4}, β_{5}, β_{6}, β_{8}} ±{β_{1}, β_{2}, β_{3}, β_{4}, β_{7}, β_{8}}, ±{β_{2}, β_{3}, β_{4}, β_{5}, β_{7}, β_{8}} ±{β_{2}, β_{3}, β_{6}, β_{7}, β_{8}, β_{9}}, ±{β_{1}, β_{2}, β_{4}, β_{8}, β_{9}, β_{10}}

±{β_{1}, β_{4}, β_{5}, β_{8}, β_{9}, β_{10}}

Here the β_{k} are given in Appendix A. The results for 6 < \frac{|S|}{2} < 12 follow easily from the results in the table by taking complements. We denote

(8.1)  [x^{α}] = (1 − x^{α})(1 − x^{−α} ).

We have calculated

(8.2)  C.T. ∏_{α∈T} [x^{α}] F (x_{∼}; a, b)

for all possible subsets T of the short roots of F_{4} such that T ∩−T = ϕ. By Lemma (2.21) it is enough to consider only those T where S = T ∪−T is a representative of an equivalence class. The same FORTRAN program, that calculated f′(a + 1, b) in terms of the an(i) (see (3.4), (3.5)), was used to calculate (8.2) in terms of the an(i) and hence as a product of a rational function of a and b, and f(a, b). To our surprise many of these rational functions factored completely into linear functions. In fact, for each 1 ≤ k ≤ 11 there exists at least one T with |T | = k such that the expression (8.2) factors completely into linear factors. Most of these linear factors seem to glue onto f(a, b) to become factorials. Askey [2] observed a similar phenomenon in G_{2}. We do not see how to generalize these results to all root systems. At the very

A PROOF OF THE MACDONALD-MORRIS ROOT SYSTEM CONJECTURE FOR F_{4} 15

least our results seem to indicate that an easier non-computer proof of F_{4} may be possible. The results that only involve linear factors are given below.

For 1 ≤ i_{j} ≤ 12 we define

(8.3)  [i_{1}, i_{2}, …, i_{k}] = ∏_{j=1}^{k}[β_{i_{j}}].

C.T.[1]F = 2\frac{(6a + 6b + 1)}{(5a + 6b + 1)}f(a, b), (8.4)
C.T.[1, 2]F = 4\frac{(4a + 4b + 1)(6a + 6b + 1)}{(3a + 5b + 1)(5a + 6b + 1)}f(a, b), (8.5)
C.T.[1, 4]F = 2\frac{(4a + 4b + 1)(6a + 6b + 1)(5a + 10b + 2)}{(3a + 4b + 1)(3a + 5b + 1)(5a + 6b + 1)}f(a, b), (8.6)
C.T.[1, 2, 3]F = 8\frac{(4a + 2b + 1)(4a + 4b + 1)(6a + 6b + 1)}{(3a + 4b + 1)(3a + 5b + 1)(5a + 6b + 1)}f(a, b), (8.7)
C.T.[1, 4, 5]F = 12\frac{(4a + 4b + 1)(4a + 2b + 1)(2a + 6b + 1)(6a + 6b + 1)}{(5a + 6b + 2)(3a + 4b + 1)(3a + 5b + 1)(5a + 6b + 1)}f(a, b), (8.8)
C.T.[1, 2, 4, 5]F (8.9)

= 8\frac{(2a + 6b + 1)(4a + 2b + 1)(4a + 4b + 1)(7a + 8b + 3)}{(2a + 3b + 1)(5a + 6b + 2)(3a + 4b + 1)(3a + 5b + 1)}

· \frac{(6a + 6b + 1)}{(5a + 6b + 1)}f(a, b),

C.T.[4, 8, 9, 10]F (8.10)

= 32\frac{(3a + 1)(6a + 6b + 1)(4a + 4b + 1)(4a + 2b + 1)}{(5a + 6b + 2)(3a + 4b + 1)(3a + 5b + 1)(5a + 6b + 1)}f(a, b),

C.T.[1, 4, 8, 9, 10]F (8.11)

= 48\frac{(3a + 1)(2a + 2b + 1)(4a + 2b + 1)(6a + 6b + 1)}{(5a + 6b + 3)(2a + 3b + 1)(5a + 6b + 2)(3a + 4b + 1)}

· \frac{(4a + 4b + 1)(7a + 12b + 4)}{(3a + 5b + 1)(5a + 6b + 1)}f(a, b),

C.T.[1, 2, 4, 5, 6, 8]F (8.12)

= 288\frac{(3a + 1)(6a + 6b + 1)(2a + 6b + 1)(2a + 2b + 1)}{(5a + 6b + 3)(3a + 5b + 2)(2a + 3b + 1)(5a + 6b + 2)}

· \frac{(4a + 4b + 1)(2a + 4b + 1)(4a + 2b + 1)}{(3a + 4b + 1)(3a + 5b + 1)(5a + 6b + 1)}f(a, b),

C.T.[1, 2, 7, 9, 10, 11, 12]F (8.13)

= 384\frac{(3a + 1)(2a + 2b + 1)(2a + 4b + 1)(2a + 6b + 1)(5a + 5b + 3)}{(3a + 4b + 2)(5a + 6b + 3)(3a + 5b + 2)(2a + 3b + 1)(5a + 6b + 2)}

· \frac{(4a + 4b + 1)(6a + 6b + 1)(4a + 2b + 1)}{(3a + 4b + 1)(3a + 5b + 1)(5a + 6b + 1)}f(a, b),

C.T.[1, 2, 3, 5, 6, 7, 11, 12]F (8.14)

= 1152\frac{(3a + 2)(3a + 1)(4a + 4b + 3)(2a + 2b + 1)(4a + 6b + 3)}{(5a + 6b + 4)(3a + 4b + 2)(5a + 6b + 3)(3a + 5b + 2)(2a + 3b + 1)}

· \frac{(4a + 4b + 1)(4a + 2b + 1)(6a + 6b + 1)(4a + 2b + 3)}{(5a + 6b + 2)(3a + 4b + 1)(3a + 5b + 1)(5a + 6b + 1)}f(a, b),

C.T.[1, 6, 7, 8, 9, 10, 11, 12]F (8.15)

= 384\frac{(3a + 1)(4a + 2b + 1)(2a + 4b + 1)(4a + 4b + 3)(2a + 2b + 1)}{(5a + 6b + 4)(3a + 4b + 2)(5a + 6b + 3)(3a + 5b + 2)(2a + 3b + 1)}

· \frac{(6a + 6b + 1)(2a + 6b + 1)(4a + 4b + 1)(13a + 10b + 7)}{(5a + 6b + 2)(3a + 4b + 1)(3a + 5b + 1)(5a + 6b + 1)}f(a, b),

C.T.[2, 4, 7, 8, 9, 10, 11, 12]F (8.16)

= 768\frac{(3a + 1)(6a + 6b + 1)(2a + 6b + 1)(4a + 2b + 1)(4a + 4b + 3)}{(5a + 6b + 4)(3a + 4b + 2)(5a + 6b + 3)(3a + 5b + 2)(2a + 3b + 1)}

· \frac{(4a + 4b + 1)(2a + 4b + 1)(2a + 2b + 1)(7a + 4b + 4)}{(5a + 6b + 2)(3a + 4b + 1)(3a + 5b + 1)(5a + 6b + 1)}f(a, b),

C.T.[1, 3, 6, 7, 8, 9, 10, 11, 12]F (8.17)

= 2304\frac{(3a + 1)(2a + 4b + 1)(6a + 6b + 5)(4a + 4b + 3)(2a + 6b + 1)}{(5a + 6b + 4)(3a + 5b + 3)(3a + 4b + 2)(5a + 6b + 3)(3a + 5b + 2)}

· \frac{(4a + 4b + 1)(6a + 6b + 1)(4a + 2b + 1)(2a + 2b + 1)^{2}}{(2a + 3b + 1)(5a + 6b + 2)(3a + 4b + 1)(3a + 5b + 1)(5a + 6b + 1)}

· f(a, b),

C.T.[3, 5, 6, 7, 8, 9, 10, 11, 12]F (8.18)

= 768\frac{(3a + 1)(4a + 2b + 1)(2a + 6b + 1)(6a + 6b + 1)(6a + 6b + 5)}{(5a + 6b + 4)(3a + 5b + 3)(3a + 4b + 2)(5a + 6b + 3)(3a + 5b + 2)}

· \frac{(4a + 4b + 1)(4a + 4b + 3)(2a + 4b + 1)(2a + 2b + 1)(7a + 4b + 4)}{(2a + 3b + 1)(5a + 6b + 2)(3a + 4b + 1)(3a + 5b + 1)(5a + 6b + 1)}

· f(a, b),

C.T.[4, 5, 6, 7, 8, 9, 10, 11, 12]F (8.19)

= 2304\frac{(3a + 2)(3a + 1)(6a + 5 + 6b)(4a + 4b + 3)(6a + 6b + 1)}{(5a + 6b + 5)(5a + 6b + 4)(3a + 4b + 2)(5a + 6b + 3)(3a + 5b + 2)}

· \frac{(4a + 4b + 1)(2a + 6b + 1)(2a + 2b + 1)(4a + 2b + 3)(4a + 2b + 1)}{(2a + 3b + 1)(5a + 6b + 2)(3a + 4b + 1)(3a + 5b + 1)(5a + 6b + 1)}

· f(a, b),

C.T.[2, 3, 5, 6, 7, 8, 9, 10, 11, 12]F (8.20)

= 2304\frac{(3a + 1)(2a + 4b + 1)(6a + 6b + 5)(4a + 2b + 1)(2a + 6b + 1)}{(2a + 3b + 2)(5a + 6b + 4)(3a + 5b + 3)(3a + 4b + 2)(5a + 6b + 3)}

· \frac{(6a + 6b + 1)(4a + 4b + 1)(4a + 2b + 3)(4a + 4b + 3)(2a + 2b + 1)^{2}}{(3a + 5b + 2)(2a + 3b + 1)(5a + 6b + 2)(3a + 4b + 1)(3a + 5b + 1)}

· \frac{f(a, b)}{(5a + 6b + 1)},

C.T.[3, 4, 5, 6, 7, 8, 9, 10, 11, 12]F (8.21)

= 4608\frac{(3a + 2)(3a + 1)(6a + 6b + 1)(4a + 4b + 1)(2a + 6b + 1)}{(5a + 6b + 5)(5a + 6b + 4)(3a + 5b + 3)(3a + 4b + 2)(5a + 6b + 3)}

· \frac{(4a + 2b + 1)(2a + 4b + 1)(6a + 6b + 5)(4a + 2b + 3)(2a + 2b + 1)}{(3a + 5b + 2)(2a + 3b + 1)(5a + 6b + 2)(3a + 4b + 1)(3a + 5b + 1)}

· \frac{(4a + 4b + 3)}{(5a + 6b + 1)}f(a, b),

C.T.[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11]F (8.22)

= 3072\frac{(3a + 2)(3a + 1)(6a + 6b + 5)(2a + 4b + 1)(6a + 6b + 1)}{(3a + 5b + 1)(5a + 6b + 1)(3a + 4b + 2)(5a + 6b + 3)(3a + 5b + 2)}

· \frac{(4a + 4b + 1)(4a + 2b + 1)(2a + 6b + 1)(4a + 4b + 3)(4a + 2b + 3)}{(2a + 3b + 1)(5a + 6b + 2)(3a + 4b + 1)(5a + 6b + 5)(5a + 6b + 4)}

· \frac{(2a + 2b + 1)^{2}(7a + 9b + 7)}{(3a + 5b + 3)(2a + 3b + 2)(3a + 4b + 3)}f(a, b).

It is a little unsettling that not all of the above results can be written as factorials. Since this paper was first written we have found nicer results. In fact, if we restrict to subsets of positive roots there is a chain of subsets in which each corresponding constant term formula can be written as a product of factorials. This chain seems to be related to the root order but we have been unable to generalize to other root systems.

Acknowledgements.

I would like to thank the following: Richard Askey for encouraging me to tackle F_{4}, Kevin Kadell for pointing out the approach of working with Laurent polynomials rather than with integrals and Dennis Stanton for suggesting many improvements and for helping me with Figure 1. It should be pointed out that Kevin Kadell has obtained independently some of the results of this paper. In particular he has calculated the values of an(2) and an(3) given in the table in §7.

Note added in proof. Robert Gustafson (A generalization of Selberg’s beta integral, preprint) has proved q-Macdonald-Morris for the affine root systems of types S(C_{ℓ}) and S(C_{ℓ})^{∨}.

A PROOF OF THE MACDONALD-MORRIS ROOT SYSTEM CONJECTURE FOR F_{4} 19

APPENDIX A

We write the roots of Φ^{(2)}(F_{4}) as ℤ-linear combinations of α_{i} (1 ≤ i ≤ 4) defined in (2.6).

\begin{matrix}β_{1} = α_{1} = e_{1} − e_{2}, & β_{5} = α_{1} + α_{4} = e_{1} − e_{3}, & β_{8} = α_{1} + α_{2} + α_{4} = e_{1} + e_{4}, \\ β_{2} = α_{2} = e_{3} + e_{4}, & β_{6} = α_{2} + α_{4} = e_{2} + e_{4}, & β_{9} = α_{1} + α_{3} + α_{4} = e_{1} − e_{4}, \\ β_{3} = α_{3} = e_{3} − e_{4}, & β_{7} = α_{3} + α_{4} = e_{2} − e_{4}, & β_{10} = α_{2} + α_{3} + α_{4} = e_{2} + e_{3}, \\ β_{4} = α_{4} = e_{2} − e_{3}, & β_{11} = α_{1} + α_{2} + α_{3} + α_{4} = e_{1} + e_{3}, & β_{12} = α_{1} + α_{2} + α_{3} + 2α_{4} = e_{1} + e_{2}, \end{matrix}

α_{1} + α_{2} = e_{1} − e_{2}+e_{3} + e_{4},  α_{1} − α_{2} = e_{1} − e_{2} − e_{3} − e_{4},
α_{1} + α_{3} = e_{1} − e_{2}+e_{3} − e_{4},  α_{1} − α_{3} = e_{1} − e_{2} − e_{3} + e_{4},
α_{2} + α_{3} = 2e_{3},  α_{2} − α_{3} = 2e_{4},
α_{1} + α_{2}+2α_{4} = e_{1} + e_{2} − e_{3} + e_{4}, 2α_{1} + α_{2} + α_{3} + 2α_{4} = 2e_{1},
α_{1} + α_{3}+2α_{4} = e_{1} + e_{2} − e_{3} − e_{4},  α_{1} + 2α_{2} + α_{3} + 2α_{4} = e_{1} + e_{2} + e_{3} + e_{4},
α_{2} + α_{3}+2α_{4} = 2e_{2},  α_{1} + α_{2} + 2α_{3} + 2α_{4} = e_{1} + e_{2} + e_{3} − e_{4}.

APPENDIX B

The vectors v(i) (1 ≤ i ≤ 37) that appear in (3.5) are listed below.

v(1) = (0, 0, 0, 0)  v(14) = (2, 2, 2, 2)  v(27) = (5, 3, 3, 1)
v(2) = (1, 1, 0, 0)  v(15) = (4, 2, 2, 0)  v(28) = (5, 4, 1, 0)
v(3) = (1, 1, 1, 1)  v(16) = (4, 4, 0, 0)  v(29) = (4, 4, 2, 2)
v(4) = (2, 1, 1, 0)  v(17) = (3, 3, 2, 2)  v(30) = (5, 4, 2, 1)
v(5) = (2, 2, 1, 1)  v(18) = (4, 2, 2, 2)  v(31) = (6, 2, 2, 2)
v(6) = (3, 2, 1, 0)  v(19) = (4, 3, 2, 1)  v(32) = (6, 3, 2, 1)
v(7) = (2, 2, 0, 0)  v(20) = (4, 4, 1, 1)  v(33) = (6, 3, 3, 0)
v(8) = (3, 1, 1, 1)  v(21) = (5, 2, 2, 1)  v(34) = (5, 5, 0, 0)
v(9) = (3, 2, 2, 1)  v(22) = (5, 3, 1, 1)  v(35) = (5, 5, 1, 1)
v(10) = (3, 3, 1, 1)  v(23) = (5, 3, 2, 0)  v(36) = (6, 4, 1, 1)
v(11) = (3, 3, 0, 0)  v(24) = (3, 3, 3, 3)  v(37) = (6, 4, 2, 0)
v(12) = (4, 2, 1, 1)  v(25) = (4, 3, 3, 2)
v(13) = (4, 3, 1, 0)  v(26) = (5, 3, 2, 2)

The complete version of (3.5) is given below.

∑_{α∈S} a′_{α}x^{α} = 192x^{v(1)} − 768x^{v(2)} + 576x^{v(3)} + 960 x^{v(4)} − 1152 x^{v(5)}

+ 2688 x^{v(6)} − 576 x^{v(7)} − 1152 x^{v(8)} − 576 x^{v(10)} − 576 x^{v(11)}
− 1152 x^{v(12)} + 1152 x^{v(13)} + 576 x^{v(14)} − 192 x^{v(15)} − 192 x^{v(16)}

− 1152 x^{v(17)} + 2304 x^{v(18)} − 1152 x^{v(21)} − 1152 x^{v(22)} + 1152 x^{v(23)}

+ 576 x^{v(24)} − 1728 x^{v(25)} + 2304 x^{v(26)} − 1152 x^{v(27)} + 384 x^{v(28)}
+ 576 x^{v(29)} − 1152 x^{v(30)} − 576 x^{v(31)} + 1152 x^{v(32)} − 192 x^{v(33)}
− 192 x^{v(34)} + 576 x^{v(35)} − 576 x^{v(36)} + 192 x^{v(37)}.

References

1. K. Aomoto, Jacobi polynomials associated with Selberg’s integral, SIAM J. Math. Anal, 18 (1987), 545-549.

2. R. Askey, Integration and Computers, to appear in the proceedings of a computer algebra conference edited by D. and G. Chudnowsky, and R. Jenks.

3. N. Bourbaki, Groupes et algèbres de Lie, Chap 4-6, Hermann, Paris, 1968.

4. R.W. Carter, Simple Groups of Lie Type, Wiley, London and New York, 1972.

5. F.G. Garvan, A beta integral associated with the root system G_{2}, SIAM J. Math. Anal, (to appear).

6. I.J. Good, Short proof a conjecture of Dyson, J. Math. Phys. 11 (1970), 1884.

7. L. Habsieger, La q-conjecture de Macdonald-Morris pur G_{2}, C.R. Acad. Sci., 303 (1986), 211-213.

8. J.H. Humphreys, Introduction to Lie Algebras and Representation Theory, Springer-Verlag, New York, 1972.

9. K.W.J. Kadell, A proof of the q-Macdonald-Morris conjecture for BC_{n}, preprint.

10. I.G. Macdonald, Some conjectures for root systems and finite reflection groups, SIAM J. of Math. Anal., 13 (1982), 988-1007.

11. W.G. Morris, Constant Term Identities for Finite and Affine Root Systems, Ph.D. thesis, Univ. of Wisconsin-Madison, 1982.

12. A. Selberg, Bermerkninger om et multiplet integral, Norske Mat. Tidsskr, 26 (1944), 71-78.

13. D. Zeilberger, A proof of the G_{2} case of Macdonald’s root system-Dyson conjecture, SIAM J. Math Anal., 18 (1987), 880-883.

14. D. Zeilberger, A unified approach to Macdonald’s root-system conjectures, SIAM J. Math. Anal., (to appear).