Publ. I.R.M.A. Strasbourg, 1988, 372/S–20 Actes 20^{e} Séminaire Lotharingien, p. 131–171

A NEW CLASS OF SYMMETRIC FUNCTIONS

BY

I. G. MACDONALD

Contents.

1. Introduction

2. The symmetric functions P_{λ}(q, t)

3. Duality

4. Skew P and Q functions

5. Explicit formulas

6. The Kostka matrix

7. Another scalar product

8. Conclusion

9. Appendix

1. Introduction

I will begin by reviewing briefly some aspects of the theory of symmetric functions. This will serve to fix notation and to provide some motivation for the subject of these lectures.

Let x_{1}, …, x_{n} be independent indeterminates. The symmetric group 𝔖_{n} acts on the polynomial ring ℤ[x_{1}, …, x_{n}] by permuting the x’s, and we shall write

Λ_{n} = ℤ[x_{1}, …, x_{n}]^{𝔖_{n}}

for the subring of symmetric polynomials in x_{1}, …, x_{n}. If f ∈ Λ_{n}, we may write

f = ∑_{r≥0} f^{(r)}

where f^{(r)} is the homogeneous component of f of degree r; each f^{(r)} is itself symmetric, and so Λ_{n} is a graded ring :

Λ_{n} = ⨁_{r≥0} Λ_{n}^{r},

I.G. MACDONALD

where Λ_{n}^{r} is the additive group of symmetric polynomials of degree r in x_{1}, …, x_{n}. (By convention, 0 is homogeneous of every degree.)

If we now adjoin another indeterminate x_{n+1}, we can form Λ_{n+1} = ℤ[x_{1}, …, x_{n+1}]^{𝔖_{n+1}}, and we have a surjective homomorphism (of graded rings)

Λ_{n+1} → Λ_{n}

defined by setting x_{n+1} = 0. The mapping Λ_{n+1}^{r} → Λ_{n}^{r} is surjective for all r ≥ 0, and bijective if and only if r ≤ n.

Often it is convenient to pass to the limit. Let

Λ^{r} = lim_{⟵}_{n} Λ_{n}^{r}

for each r ≥ 0, and let

Λ = ⨁_{r≥0} Λ^{r}.

By the definition of inverse (or projective) limits, an element of Λ_{n}^{r} is a sequence (f_{n})_{n≥0} where f_{n} ∈ Λ_{n}^{r} for each n, and f_{n} is obtained from f_{n+1} by setting x_{n+1} = 0. We may therefore regard the f_{n} as the partial sums of an infinite series f of monomials of degree r in infinitely many indeterminates x_{1}, x_{2}, … For example, if f_{n} = x_{1} + ⋯ + x_{n}, then f = ∑_{i=1}^{∞} x_{i}. Thus the elements of Λ are no longer polynomials, and we call them symmetric functions. (Of course, they aren’t functions either, but they have to be called something !)

For each n there is a surjective homomorphism Λ → Λ_{n}, obtained by setting x_{n+1} = x_{n+2} = ⋯ = 0. The graded ring Λ is the ring of symmetric functions. If R is any commutative ring, we write

Λ_{R} = Λ ⊗_{ℤ} R, Λ_{n,R} = Λ_{n} ⊗_{ℤ} R

for the ring of symmetric functions (resp. symmetric polynomials in n indeterminates) with coefficients in R.

There are various ℤ-bases of the ring Λ, some of which we shall review. They all are indexed by partitions. A partitions λ is a (finite or infinite) sequence

λ = (λ_{1}, λ_{2}, λ_{3}, …)

of non-negative integers, such that λ_{1} ≥ λ_{2} ≥ ⋯ and

|λ| = ∑ λ_{i} < ∞,

A NEW CLASS OF SYMMETRIC FUNCTIONS

so that from a certain point onwards (if the sequence λ is infinite) all the λ_{i} are zero. We shall not distinguish between two such sequences which differ only by a string of zeros at the end. Thus (2, 1), (2, 1, 0), (2, 1, 0, 0, …) are all to be regarded as the same partition.

The nonzero λ_{i} are called the parts of λ, and the number of parts is the length ℓ(λ) of λ. If λ has m_{1} parts equal to 1, m_{2} parts equal to 2, and so on, we shall occasionally write λ = (1^{m_{1}}2^{m_{2}} …), although strictly speaking we should write this in reverse order.

Let P denote the set of all partitions, and P_{n} the set of all partitions of n (i.e. partitions λ such that |λ| = n). The natural (or dominance) partial ordering in P is defined as follows :

(1.1) λ ≥ μ ⟺ |λ| = |μ| and λ_{1} + ⋯ + λ_{r} ≥ μ_{1} + ⋯ + μ_{r} for all r ≥ 1.

It is a total order on P_{n} for n ≤ 5, but not for n ≥ 6.

With each partition λ we associate a diagram, consisting of the points (i, j) ∈ ℤ^{2} such that 1 ≤ j ≤ λ_{i}. We adopt the convention (as with matrices) that the first coordinate i (the row index) increases as one goes downwards, and the second coordinate j (the column index) increases from left to right. Often it is more convenient to replace the lattice points (i, j) by squares, and then the diagram of λ consists of λ_{1} boxes in the top row, λ_{2} boxes in the second row, and so on ; the whole arrangement of boxes being left-justified.

If we read the diagram of a partition λ by columns, we obtain the conjugate partition λ′. Thus λ′_{j} is the number of boxes in the j-th column of λ, and hence is equal to the number of parts of λ that are ≥ j. It is not difficult to show that

λ ≥ μ ⟺ μ′ ≥ λ′.

Bases of Λ.

1. Monomial symmetric functions. — Let λ be a partition. It defines a monomial x^{λ} = x_{1}^{λ_{1}}x_{2}^{λ_{2}} … The monomial symmetric function m_{λ} is the sum of all distinct monomials obtainable from x^{λ} by permutations of the x’s. For example, m_{(2,1)} = ∑ x_{i}^{2}x_{j}, summed over all (i, j) such that i ≠ j.

In particular, when λ = (1^{r}) we have

m_{(1^{r})} = e_{r} = ∑_{i_{1}<···<i_{r}} x_{i_{1}} … x_{i_{r}},

the r-th elementary symmetric function. Their generating function is

(1.2)  E(t) = ∑_{r≥0} e_{r}t^{r} = ∏(1 + x_{i}t),

where t is another indeterminate, and e_{0} = 1.

At the other extreme, when λ = (r) we have

m_{(r)} = p_{r} = ∑ x_{i}^{r},

the r-th power sum.

It is clear that every f ∈ Λ is uniquely expressible as a finite linear combination of the m_{λ}, so that (m_{λ})_{λ∈P} is a ℤ-basis of Λ.

2. — For any partition λ, let

e_{λ} = e_{λ_{1}}e_{λ_{2}} …

The e_{λ} form another ℤ-basis of Λ. Equivalently, we have ℤ[e_{1}, e_{2}, …] and the e_{r} are algebraically independent. Indeed, it is not difficult to show that

e_{λ′} = m_{λ} + ∑_{μ<λ} a_{λμ}m_{μ}

for suitable coefficients a_{λμ}, from which the assertion follows immediately.

3. — For each r ≥ 0 let

h_{r} = ∑_{|λ|=r} m_{λ},

the sum of all monomials of total degree r in the x’s. The generating function for the h_{r} is

(1.3)  H(t) = ∑_{r≥0} h_{r}t^{r} = ∏(1 − x_{i}t)^{−1},

as one sees by expanding each factor (1 − x_{i}t)^{−1} in the product on the right as a geometric series, and then multiplying these series together. From (1.2) and (1.3) it follows that

H(t)E(−t) = 1

so that

(1.4)  ∑_{r=0}^{n}(−1)^{r}e_{r}h_{n−r} = 0

for each n ≥ 1.

Since the e_{r} are algebraically independent, we may define a ring homomorphism ω : Λ → Λ by

ω(e_{r}) = h_{r} for all r ≥ 1. The symmetry of the relations (1.4) as between the e’s and the h’s then shows that ω(h_{r}) = e_{r}, i.e., ω^{2} = 1. Thus ω is an automorphism (of period 2) of Λ, and therefore we have Λ = ℤ[h_{1}, h_{2}, …]. Equivalently, the products

h_{λ} = h_{λ_{1}}h_{λ_{2}} … = ω(e_{λ})

form another ℤ-basis of Λ.

4. — The generating function for the power-sums p_{r} = ∑ x_{i}^{r} is

P (t) = ∑_{r≥1} p_{r}t^{r−1}

= ∑_{i} ∑_{r≥1} x_{i}^{r}t^{r−1}

= ∑_{i} \frac{x_{i}}{1 − x_{i}t}

and therefore

(1.5)  P (t) = \frac{d}{dt} log H(t) = \frac{H′(t)}{H(t)}.

Hence we have H′(t) = H(t)P (t), so that

nh_{n} = ∑_{r=1}^{n} p_{r}h_{n−r}

for all n ≥ 1. These relations enable us to express the h’s in terms of the p’s, and vice versa, and show that

h_{n} ∈ ℚ[p_{1}, …, p_{n}],
p_{n} ∈ ℤ[h_{1}, …, h_{n}]

so that

ℚ[p_{1}, …, p_{n}] = ℚ[h_{1}, …, h_{n}]

for all n ≥ 1. Letting n → ∞, we see that

Λ_{ℚ} = ℚ[h_{1}, h_{2}, …] = ℚ[p_{1}, p_{2}, …].

For each partition λ let

p_{λ} = p_{λ_{1}}p_{λ_{2}} …

(with the understanding that p_{0} = 1). The p_{λ} form a ℚ-basis of Λ_{ℚ}, but do not form a ℤ-basis of Λ.

Analogously to (1.5) we have

(1.6)  P (−t) = \frac{d}{dt} log E(t).

Since the involution ω interchanges E(t) and H(t), it follows from (1.5) and (1.6) that it interchanges P (t) and P (−t), so that

(1.7)  ω(p_{r}) = (−1)^{r−1}p_{r}

for all r ≥ 1.

Finally, we may compute h_{n} as a polynomial in the power sums, as follows : from (1.5) we have

H(t) = exp(∑_{r≥1} \frac{p_{r}t^{r}}{r} )

= ∏_{r≥1} exp \frac{p_{r}t^{r}}{r}

= ∏_{r≥1} ∑_{m_{r}≥0} \frac{1}{m_{r}!}(\frac{p_{r}t^{r}}{r} )^{m_{r}}.

Let us pick out the coefficient of p_{λ} in the product. If λ = (1^{m_{1}}2^{m_{2}} …), it is z_{λ}^{−1}, where

(1.8)  z_{λ} = ∏_{r≥1}(r^{m_{r}}.m_{r}!)

and therefore

h_{n} = ∑_{|λ|=n} z_{λ}^{−1}p_{λ}.

This numerical function z_{λ} (which will occur frequently in the sequel) has the following interpretation. Let |λ| = n, and let w ∈ 𝔖_{n} be a permutation of cycle-type λ. Then z_{λ} is the order of the centralizer of w in 𝔖_{n}.

5. Schur functions. — Let λ be a partition of length ≤ n, and form the determinant

D_{λ} = det(x_{i}^{λ_{j}+n−j})_{1≤i,j≤n}.

This vanishes whenever any two of the x’s are equal, hence is divisible by the Vandermonde determinant

D_{0} = ∏_{i<j}(x_{i} − x_{j}).

The quotient

s_{λ}(x_{1}, …, x_{n}) = D_{λ}/D_{0}

is a homogeneous symmetric polynomial of degree |λ| in x_{1}, …, x_{n}. Moreover we have

s_{λ}(x_{1}, …, x_{n}, 0) = s_{λ}(x_{1}, …, x_{n})

and hence for each partition λ a well-defined element s_{λ} ∈ Λ, homogeneous of degree |λ|. These are the Schur functions.

Let us define a scalar product ⟨ , ⟩ on Λ as follows :

(1.9)  ⟨p_{λ}, p_{μ}⟩ = δ_{λμ}z_{λ}

where δ_{λμ} = 1 if λ = μ, and δ_{λμ} = 0 otherwise. Then one can show (see e.g. [M_{1}], ch. I) that the Schur functions s_{λ} have the following properties, which characterize them uniquely :

(A) s_{λ} = m_{λ} + ∑_{λ<μ} K_{λμ}m_{μ}

for suitable coefficients K_{λμ}; (B) ⟨s_{λ}, s_{μ}⟩ = 0 if λ ≠ μ.

6. Zonal symmetric functions. — These are certain symmetric functions Z_{λ}, at present more familiar to statisticians than combinatorialists, which (when restricted to a finite number of variables x_{1}, …, x_{n}) arise naturally in connection with Fourier analysis on the homogeneous space G/K, where G = GL_{n}(ℝ) and K = O(n), the orthogonal group (so that G/K may be identified, via X ↦ XX^{t}, with the space of positive definite real symmetric n × n matrices). I shall not give a direct definition here, but will only remark that the Z_{λ} (suitably normalized) are characterized by the following two properties :

(A) Z_{λ} = m_{λ} + lower terms

where by “lower terms” is meant a linear combination of the m_{μ} such that μ < λ,

(B) ⟨Z_{λ}, Z_{μ}⟩_{2} = 0 if λ ≠ μ,

where the scalar product ⟨ , ⟩_{2} on Λ is defined by

(1.10) ⟨p_{λ}, p_{μ}⟩_{2} = δ_{λμ}.2^{ℓ(λ)}z(λ),

ℓ(λ) being the length of the partition λ.

7. Jack’s symmetric functions [S]. — These are a common generalization of the Schur functions and the zonal symmetric functions, and again I shall not give a direct construction of them at this stage. Let α ∈ ℝ, α > 0, and define a scalar product ⟨ , ⟩_{α} on Λ_{ℝ} by

(1.11)  ⟨p_{λ}, p_{μ}⟩_{α} = δ_{λμ}.α^{ℓ(λ)}z_{λ}.

Then Jack’s functions P_{λ} = P_{λ}(x; α) are characterized by the two properties

(A) P_{λ} = m_{λ} + lower terms,
(B) ⟨P_{λ}, P_{μ}⟩_{α} = 0 if λ ≠ μ.

The symmetric functions P_{λ} depend rationally on α, i.e. they lie in Λ_{F} where F is the field ℚ(α), and we may if we prefer regard the parameter α as an indeterminate rather than a real number. Clearly when α = 1 they reduce to the Schur functions s_{λ}, and when α = 2 to the zonal functions Z_{λ}. They also tend to definite limits as α → 0 and as α → ∞ (even though the scalar product (1.11) collapses), and in fact

\begin{aligned}P_{λ}(x; α) & → e_{λ′} as α → 0, \\ P_{λ}(x; α) & → m_{λ} as α → ∞. \end{aligned}

Also when α = \frac{1}{2} they occur in nature, as zonal spherical functions on the homogeneous space G/K, where now G = GL_{n}(ℍ) and K = U(n, ℍ), the quaternionic unitary group of n × n matrices.

8. Hall-Littlewood symmetric functions [M_{1}, ch. III]. — These symmetric functions arose originally in connection with the combinatorial and enumerative lattice properties of finite abelian p-groups (where p is a prime number). Let t be an indeterminate, let F = ℚ(t) and define a scalar product ⟨ , ⟩_{(t)} on Λ_{F} by

(1.13)  ⟨p_{λ}, p_{μ}⟩_{(t)} = δ_{λμ}z_{λ} ∏_{i=1}^{ℓ(λ)}(1 − t^{λ_{i}})^{−1}.

Then the Hall-Littlewood symmetric functions P_{λ}(x; t) are characterized by the two properties

(A) P_{λ} = m_{λ} + lower terms,
(B) ⟨P_{λ}, P_{μ}⟩_{(t)} = 0 if λ ≠ μ.

When t = 0, the P_{λ} reduce to the Schur functions s_{λ}, and when t = 1 to the monomial symmetric functions m_{λ}.

Remarks.

1. — Let n be a positive integer, and arrange the partitions of n in lexicographical order, so that (1^{n}) comes first and (n) comes last. For example,

(1^{4}), (2, 1^{2}), (2^{2}), (3, 1), (4),

if n = 4. This is a total ordering L_{n} of the set P_{n} of partitions of n, and correspondingly defines a totally ordered basis of Λ^{n} :

m_{(1^{n})}(= e_{n}), …, m_{λ}, …, m_{(n)}(= p_{n}).

Now suppose we are given a (positive definite) scalar product ⟨ , ⟩ on the space Λ_{ℝ}^{n}. Then by the Gram-Schmidt process we can derive a unique basis (u_{λ}) of Λ_{ℝ}^{n} with the following two properties :

(A′) u_{λ} = m_{λ} + a linear combination of the m_{μ} for partitions μ that precede λ in L_{n};

(B′) ⟨u_{λ}, u_{μ}⟩_{α} = 0 if λ ≠ μ.

If we replace L_{n} by some other total ordering of P_{n}, and apply Gram-Schmidt as before, we should expect in general to end up with a different basis (u_{λ}). What in fact happens in each of the cases (5) — (8) is that, for the appropriate scalar product, the basis obtained is independent of the total ordering chosen, provided only that it is compatible with the partial ordering (1.1). (The lexicographical order L_{n} satisfies this condition : if μ < λ then μ precedes λ in L_{n}.) In other words, the conditions (A) and (B) (in each of the cases (5) — (8)) overdetermine the corresponding family of symmetric functions.

2. — Let V be a vector space (over some field F) and let S = S(V) be the symmetric algebra of V. (If x_{1}, x_{2}, … is a basis of V then S = F [x_{1}, x_{2}, …], the polynomial algebra over F generated by the x_{i}.) Now suppose we are given a scalar product ⟨u, v⟩ on V, with values in F. This scalar product has a natural extension to S, defined by

⟨u_{1} … u_{m}, v_{1} … v_{n}⟩ = \{ \begin{matrix}0, & if m ≠ n, \\ per(⟨u_{i}, v_{j}⟩), & if m = n, \end{matrix}

where per(⟨u_{i}, v_{j}⟩) is the permanent of the matrix of scalar products ⟨u_{i}, v_{j}⟩ (1 ≤ i, j ≤ n). Here the u’s and v’s are arbitrary elements of V.

In particular, let V be the vector subspace of Λ_{F} spanned by the power sums p_{r} (r ≥ 1), so that S = F [p_{1}, p_{2}, …] = Λ_{F}. Suppose that ⟨ , ⟩ is a scalar product on V for which the p_{r} are mutually orthogonal, say

⟨p_{r}, p_{s}⟩ = δ_{rs}a_{r}.

Then the natural extension of this scalar product to S = Λ_{F} is such that

⟨p_{λ}, p_{μ}⟩ = δ_{λμ}z_{λ}a_{λ},

where a_{λ} = a_{λ_{1}}a_{λ_{2}} … for any partition λ.

For the scalar products (5) — (8) above, the a_{r} are respectively 1, 2, α, and (1 − t^{r})^{−1}.

2. The symmetric functions P_{λ}(q, t)

Let q, t be independent indeterminates and let F = ℚ(q, t) be the field of rational functions in q and t. We shall now change the scalar product yet again, and define

(2.1) ⟨p_{λ}, p_{μ}⟩_{(q,t)} = δ_{λμ}z_{λ}(q, t)

where

(2.1) z_{λ}(q, t) = z_{λ} ∏_{i=1}^{ℓ(λ)} \frac{1 − q^{λ_{i}}}{1 − t^{λ_{i}}} .

It is perhaps better to think of the parameters q and t as real variables lying in the interval (0, 1) of ℝ, so that the scalar product (2.1) is positive definite. The main result of this section is the following existence theorem.

(2.3) Theorem. — For each partition λ there is a unique symmetric function P_{λ} = P_{λ}(q, t) ∈ Λ_{F} such that

(A) P_{λ} = m_{λ} + ∑_{μ<λ} u_{λμ}m_{μ}

with coefficients u_{λμ} ∈ F ; (B) ⟨P_{λ}, P_{μ}⟩_{(q,t)} = 0 if λ ≠ μ.

As remarked in the previous section, these two conditions overdetermine the P_{λ}, and their existence therefore requires proof. Before embarking on the proof, let us consider some particular cases.

(1) When q = t, the scalar product (2.1) reduces to the ‘usual’ scalar product (1.9), and hence P_{λ}(q, q) is the Schur function s_{λ}.

(2) When q = 0, (2.1) reduces to (1.13), and hence P_{λ}(0, t) is the Hall-Littlewood function P_{λ}(t).

(3) Let q = t^{α} (α ∈ ℝ, α > 0) and let t → 1, so that q → 1 also. Then

\frac{1 − q^{m}}{1 − t^{m}} = \frac{1 − t^{αm}}{1 − t^{m}} → α,

as t → 1, for all m. Hence the scalar product (2.1) tends to (1.11) as t → 1 and hence

lim_{t→1} P_{λ}(t^{α}, t)

is the Jack symmetric function P_{λ}(α).

(4) When t = 1 (and q is arbitrary) we have P_{λ}(q, 1) = m_{λ}.

(5) When q = 1 (and t is arbitrary) we have P_{λ}(1, t) = e_{λ′}.

(6) Finally, it is clear from (2.2) that

z_{λ}(q^{−1}, t^{−1}) = (q^{−1}t)^{n}z_{λ}(q, t),

if λ is a partition of n. Hence on each Λ_{F}^{n}, the scalar products ⟨ , ⟩_{(q,t)} and ⟨ , ⟩_{q^{−1},t^{−1}} are proportional, and therefore

P_{λ}(q^{−1}, t^{−1}) = P_{λ}(q, t).

We may summarize these special cases in the following diagram, in which the point (q, t) represents the basis (P_{λ}(q, t)) of Λ_{F} (or of Λ_{ℝ}, since we are regarding q, t as real numbers).

[figure]
t
↑
(0, 1)  (m_{λ})  (1, 1)

(P_{λ}(t))  (s_{λ})  (e_{λ′})

⟶ q
(0, 0)  (1, 0)

At each point (q, q) on the diagonal of the square we have the Schur functions s_{λ}, at each point on the upper edge (t = 1) the monomial symmetric functions m_{λ}, and so on. In this scheme the Jack functions P_{λ}(α), for varying α, correspond to the points in the infinitesimal neighbourhood of the point (1, 1), and more precisely P_{λ}(α) corresponds to the direction through (1, 1) with slope 1/α. Notice that the bottom edge (t = 0) of the square remains unmarked ; I do not know if the P_{λ}(q, 0) have any reasonable interpretation (except that they can be derived from the Hall-Littlewood functions P_{λ}(0, t) by duality (§ 3)).

Let x = (x_{1}, x_{2}, …) and y = (y_{1}, y_{2}, …) be two sequences of independent indeterminates over F = ℚ(q, t), and define

(2.4) Π = Π(x, y; q, t) = ∏_{i,j} \frac{(tx_{i}y_{j}; q)_{∞}}{(x_{i}y_{j}; q)_{∞}} ,

where as usual

(a; q)_{∞} = ∏_{r=0}^{∞}(1 − aq^{r})

for any a for which the product on the right makes sense.

Then we have

(2.5)  Π(x, y; q, t) = ∑_{λ} z_{λ}(q, t)^{−1}p_{λ}(x)p_{λ}(y).

Proof. — We compute exp(log Π) ; first of all,

log Π = ∑_{i,j} ∑_{r=0}^{∞}(log(1 − x_{i}y_{j}q^{r})^{−1} − log(1 − tx_{i}y_{j}q^{r})^{−1})

= ∑_{i,j} ∑_{r≥0} ∑_{n≥1} \frac{1}{n}(x_{i}y_{j}q^{r})^{n}(1 − t^{n})

= ∑_{n≥1} \frac{1}{n} \frac{1 − t^{n}}{1 − q^{n}} p_{n}(x)p_{n}(y)

and therefore

Π = ∏_{n≥1} exp(\frac{1}{n} \frac{1 − t^{n}}{1 − q^{n}} p_{n}(x)p_{n}(y))

= ∏_{n≥1} ∑_{m_{n}=0}^{∞} \frac{1}{m_{n}!}(\frac{1}{n} \frac{1 − t^{n}}{1 − q^{n}} p_{n}(x)p_{n}(y))^{m_{n}}

in which the coefficient of p_{λ}(x)p_{λ}(y), where λ = (1^{m_{1}}2^{m_{2}} …) is seen to be z_{λ}(q, t)^{−1}.

(2.6) For each integer n ≥ 0 let (u_{λ}), (v_{λ}) be F -bases of Λ_{F}^{n}, indexed by the partitions λ of n. Then the following statements are equivalent :

(a) ⟨u_{λ}, v_{μ}⟩_{(q,t)} = δ_{λμ} for all λ, μ (i.e., (u_{λ}), (v_{λ}) are dual bases of Λ_{F}^{n} for the scalar product (2.1)) ;

(b) ∑_{λ} u_{λ}(x)v_{λ}(y) = Π(x, y; q, t).

Proof. — Let p_{λ}^{∗} = z_{λ}(q, t)^{−1}p_{λ}, so that

⟨p_{λ}^{∗}, p_{μ}⟩_{(q,t)} = δ_{λμ}.

Suppose that

u_{λ} = ∑_{ρ} a_{λρ}p_{ρ}^{∗}, v_{λ} = ∑_{σ} a_{μσ}p_{σ}.

Then we have

⟨u_{λ}, v_{μ}⟩_{(q,t)} = ∑_{ρ} a_{λρ}b_{μρ},

so that (a) is equivalent to

(a′)  ∑_{ρ} a_{λρ}b_{μρ} = δ_{λμ}.

On the other hand, by (2.5), (b) is equivalent to

∑_{λ} u_{λ}(x)v_{λ}(y) = ∑_{ρ} p_{ρ}^{∗}(x)p_{ρ}(y)

and hence to

(b′)  ∑_{λ} a_{λρ}b_{λσ} = δ_{ρσ}.

Since (a′) and (b′) are equivalent ((a′) says that AB^{t} = 1, where A is the matrix (a_{λμ}) and B the matrix (b_{λμ}), and (b′) says that A^{t}B = 1), it follows that (a) and (b) are equivalent.

After these preliminaries we can embark on the proof of the existence theorem (2.3). The idea of the proof is as follows : we shall work initially with a finite set of variables x = (x_{1}, …, x_{n}) and construct an F -linear map (or operator)

D = D_{q,t} : Λ_{n,F} → Λ_{n,F}

having the following properties :

(2.7.1)  Dm_{λ} = ∑_{μ≤λ} c_{λμ}m_{μ}

for each partition λ of length ≤ n;

(2.7.2)  ⟨Df, g⟩_{(q,t)} = ⟨f, Dg⟩_{(q,t)}

for all f, g ∈ Λ_{F} ;

(2.7.3)  λ ≠ μ ⇒ c_{λλ} ≠ c_{μμ}.

These three properties say respectively that the matrix of D relative to the basis (m_{λ}) is triangular (2.7.1) ; that D is self-adjoint (2.7.2) ; and that the eigenvalues of D are distinct (2.7.3).

The P_{λ} are then just the eigenfunctions (or eigenvectors) of the operator D. Namely we have (2.8) For each partition λ (of length ≤ n) there is a unique symmetric polynomial P_{λ} ∈ Λ_{n,F} satisfying the two conditions

(A) P_{λ} = ∑_{μ≤λ} u_{λμ}m_{μ}

where u_{λμ} ∈ F and u_{λλ} = 1 ; (C) DP_{λ} = c_{λλ}P_{λ}.

Proof. — From (A) and (2.7.1) we have

DP_{λ} = ∑_{μ≤λ} u_{λμ}Dm_{μ}

= ∑_{ν≤μ≤λ} u_{λμ}c_{μν}m_{ν}

and

c_{λλ}P_{λ} = ∑_{ν≤λ} c_{λλ}u_{λν}m_{ν},

so that (A) and (C) are satisfied if and only if

c_{λλ}u_{λν} = ∑_{ν≤μ≤λ} u_{λμ}c_{μν},

that is to say, if and only if

(c_{λλ} − c_{νν})u_{λν} = ∑_{ν<μ≤λ} u_{λμ}c_{μν}

whenever ν < λ. Since c_{λλ} ≠ c_{νν} by (2.7.3), this relation determines u_{λμ} uniquely in terms of the u_{λμ} such that ν < μ ≤ λ. Hence the coefficients u_{λμ} in (A) are uniquely determined, given that u_{λλ} = 1.

With the P_{λ} as defined in (2.8) we have, by the self-adjointness of D,

\begin{aligned}c_{λλ}⟨P_{λ}, P_{μ}⟩_{q,t} & = ⟨DP_{λ}, P_{μ}⟩_{q,t} \\ & = ⟨P_{λ}, DP_{μ}⟩_{q,t} = c_{μμ}⟨P_{λ}, P_{μ}⟩_{q,t}. \end{aligned}

But c_{λλ} ≠ c_{μμ} if λ ≠ μ, hence ⟨P_{λ}, P_{μ}⟩_{q,t} = 0 if λ ≠ μ.

This establishes (2.3) when the number of variables is finite (i.e. in Λ_{n,F} rather Λ_{F}). As explained in § 1, we may then compute the coefficients u_{λμ} in (A) by Gram-Schmidt ; they will involve only the scalar products ⟨m_{μ}, m_{ν}⟩_{q,t}, which are independent of n. Hence the P_{λ} are well-defined as elements of Λ_{F}.

The proof of (2.3) therefore reduces to the construction of an operator D satisfying (2.7.1) — (2.7.3). Let

Δ = ∏_{1≤i<j≤n}(x_{i} − x_{j})

= ∑_{w∈𝔖_{n}} ϵ(w)x^{wδ} (2.9)

be the Vandermonde determinant in x_{1}, …, x_{n}, where ϵ(w) is the sign of the permutation w, and δ = (n−1, n−2, …, 1, 0). Next, for any polynomial f(x_{1}, …, x_{n}), symmetric or not, we define

(T_{q,x_{i}}f)(x_{1}, …, x_{n}) = f(x_{1}, …, qx_{i}, …, x_{n}),
(T_{t,x_{i}}f)(x_{1}, …, x_{n}) = f(x_{1}, …, tx_{i}, …, x_{n})

for 1 ≤ i ≤ n. Then D is defined as follows :

D = Δ^{−1} ∑_{i=1}^{n}(T_{t,x_{i}}Δ)T_{q,x_{i}}

= ∑_{i=1}^{n}(∏_{j≠i} \frac{tx_{i} − x_{j}}{x_{i} − x_{j}} )T_{q,x_{i}}. (2.10)

A more useful expression for D is, from (2.9) above,

(2.10′)  D = Δ^{−1} ∑_{w∈𝔖_{n}} ϵ(w) ∑_{i=1}^{n} t^{(wδ)_{i}}x^{wδ}T_{q,x_{i}}.

We must now verify that D satisfies (2.7.1) — (2.7.3). Let λ be a partition of length ≤ n, and let 𝔖_{n}^{λ} be the subgroup of 𝔖_{n} that fixes λ, so that

m_{λ} = |𝔖_{n}^{λ}|^{−1} ∑_{w_{1}∈𝔖_{n}} x^{w_{1}λ}.

From (2.10′) we have

|𝔖_{n}^{λ}| Dm_{λ} = Δ^{−1} ∑_{w,w_{1}} ϵ(w) ∑_{i=1}^{n} t^{(wδ)_{i}}q^{(w_{1}λ)_{i}}x^{wδ+w_{1}λ}.

In this sum w and w_{1} run independently through 𝔖_{n}. Put w_{1} = ww_{2}, and we obtain :

|𝔖_{n}^{λ}| Dm_{λ} = Δ^{−1} ∑_{w,w_{2}} ϵ(w)(∑_{i=1}^{n} q^{(w_{2}λ)_{i}}t^{δ_{i}})x^{w(w_{2}λ+δ)}

= ∑_{w_{2}∈𝔖_{n}}(∑_{i=1}^{n} q^{(w_{2}λ)_{i}}t^{n−i})s_{w_{2}λ}.

Hence if we define

u(μ) = ∑_{i=1}^{n} q^{μ_{i}}t^{n−i}

for any μ = (μ_{1}, …, μ_{n}) ∈ ℕ^{n}, we have

Dm_{λ} = ∑_{μ} u(μ)s_{μ}

summed over all distinct derangements μ of λ = (λ_{1}, …, λ_{n}). In this sum μ is not a partition (unless μ = λ), but s_{μ} is defined for all μ ∈ ℕ^{n} and is either zero or equal to ±s_{ν} for some partition ν < λ. Hence

Dm_{λ} = u(λ)s_{λ} + ⋯

where the terms not written are a linear combination of the Schur functions s_{ν} such that ν < λ, and therefore finally

Dm_{λ} = ∑_{μ≤λ} c_{λμ}m_{μ}

with

(2.11) c_{λλ} = u(λ) = ∑_{i=1}^{n} q^{λ_{i}}t^{n−i}.

This establishes (2.7.1) and also (2.7.3), since the eigenvalues c_{λλ} of D given by (2.11) are visibly all distinct.

It remains to show that D is self-adjoint (2.7.2). The proof is in several stages :

(2.12) D is self-adjoint when q = t.

Proof. — We have, when q = t,

D = Δ^{−1} ∑_{i=1}^{n}(T_{t,x_{i}}Δ)T_{t,x_{i}}

so that

Ds_{λ} = Δ^{−1} ∑_{i=1}^{n} T_{t,x_{i}}(Δs_{λ}).

Since

Δs_{λ} = ∑_{w∈𝔖_{n}} ϵ(w)x^{w(λ+δ)}

it follows that

Ds_{λ} = (∑_{i=1}^{n} t^{λ_{i}+n−i})s_{λ}

and hence that ⟨Ds_{λ}, s_{μ}⟩ = 0 if λ ≠ μ. So

⟨Ds_{λ}, s_{μ}⟩ = ⟨s_{λ}, Ds_{μ}⟩

for all λ, μ, and therefore D is self-adjont.

(2.13) D = D_{q,t} is self-adjoint if and only if

D_{x}Π = D_{y}Π,

where Π = Π(x, y; q, t) (2.4) and D_{x} (resp. D_{y}) means D operating on symmetric functions in the x (resp. y) variables.

Proof. — Let (u_{λ}), (v_{λ}) be dual bases of Λ_{n,F} as in (2.6), and let

(2.14)  a_{λμ} = ⟨Du_{λ}, u_{μ}⟩_{q,t}.

Clearly, D is self-adjoint if and only if a_{λμ} = a_{μλ} for all λ, μ.

From (2.14) we have

Du_{λ} = ∑_{μ} a_{λμ}v_{μ}

and therefore, since Π = ∑ u_{λ}(x)v_{λ}(y),

D_{x}Π = ∑_{λ,μ} a_{λμ}v_{μ}(x)v_{λ}(y).

Likewise

D_{y}Π = ∑_{λ,μ} a_{λμ}v_{μ}(y)v_{λ}(x)

and therefore D_{x}Π = D_{y}Π if and only if a_{λμ} = a_{μλ} for all λ, μ.

From the definition (2.4) of Π we have

Π^{−1}T_{q,x_{i}}Π = ∏_{j=1}^{n} \frac{1 − x_{i}y_{j}}{1 − tx_{i}y_{j}}

which is independent of q. Hence Π^{−1}D_{x}Π is independent of q. Hence we may assume that q = t. But then by (2.13) and (2.14) we have Π^{−1}D_{x}Π = Π^{−1}D_{y}Π, and so by (2.14) again D_{q,t} is self-adjoint for all q, t. This completes the proof of (2.3).

We shall see in the subsequent sections that the formal properties of the symmetric functions P_{λ}(x; q, t) generalize to a remarkable extent familiar properties of Schur functions. What is lacking (at any rate at present) is any sort of usable “closed formula” for P_{λ}. This has the effect that the proofs we shall give are usually indirect and quite complicated in detail, compared to the usual proofs of the corresponding properties of Schur functions.

3. Duality

It is well known (see, e.g., [M_{1}], ch. I, (3.8)) that the involution ω : Λ → Λ of § 1 permutes the Schur functions, namely that

ω(s_{λ}) = s_{λ′}.

The duality theorem to be stated below generalizes this fact. Let

(3.1)  b_{λ} = b_{λ}(q, t) = ⟨P_{λ}, P_{μ}⟩_{q,t}^{−1} ∈ F.

(We shall later (§ 5) obtain an explicit formula for b_{λ}(q, t) in terms of λ, q and t.) Now define

(3.2) Q_{λ} = b_{λ}P_{λ} so that

⟨P_{λ}, P_{μ}⟩_{q,t} = δ_{λμ},

i.e., (P_{λ}), (Q_{λ}) are dual bases of Λ_{F} for the scalar product ⟨ , ⟩_{q,t}. We also define an automorphism

ω_{q,t} : Λ_{F} → Λ_{F}

by

(3.3) ω_{q,t}(p_{r}) = (−1)^{r−1} \frac{1 − q^{r}}{1 − t^{r}} p_{r}.

Clearly ω_{q,t}^{−1} = ω_{t,q}, and ω_{t,t} = ω. Also we have

(3.4)  ⟨ω_{q,t}f, g⟩_{t,q} = ⟨ωf, g⟩

for all f, g ∈ Λ_{F}, where the scalar product on the right is that defined by (1.9).

It is enough to check (3.4) when f = p_{λ} and g = p_{μ}, and then it is immediate from the definitions.

We can now state the duality theorem :

(3.5) Theorem. — For all partitions λ we have

ω_{q,t}P_{λ}(q, t) = Q_{λ′}(t, q)

or equivalently

ω_{q,t}Q_{λ}(q, t) = P_{λ′}(t, q).

(The equivalence of these two statements follows from the fact that ω_{q,t}^{−1} = ω_{t,q}.)

For the proof of (3.5) we require the following lemma, whose proof we leave as an exercise.

(3.6) Let λ be a partition, thought of as an infinite sequence, and let

f_{λ}(q, t) = (1 − t) ∑_{i=1}^{∞} q^{λ_{i}}t^{i−1}.

Then f_{λ}(q, t) = f_{λ′}(t, q).

Let D = D_{q,t} be the operator defined in § 2, acting on symmetric polynomials in n variables x_{1}, …, x_{n}. We need to modify it slightly : we define

(3.7)  E = E_{q,t} = t^{−n}(1 + (t − 1)D_{q,t}).

From (2.11) we have, for any partition λ of length ≤ n,

EP_{λ}(q, t) = t^{−n}(1 + (t − 1) ∑_{i=1}^{n} q^{λ_{i}}t^{n−i})P_{λ}(q, t)

= f_{λ}(q, t^{−1})P_{λ}(q, t). (3.8)

Next we have

(3.9)  ω_{q,t}E_{q,t}ω_{q,t}^{−1} = E_{t^{−1},q^{−1}}.

The only proof I have of (3.9) at present is rather messy, and I shall not give it here. Assuming (3.9), the proof of (3.5) proceeds as follows : we have

E_{t^{−1},q^{−1}}ω_{q,t}P_{λ}(q, t) = ω_{q,t}E_{q,t}P_{λ}(q, t)
= f_{λ}(q, t^{−1})ω_{q,t}P_{λ}(q, t)
= f_{λ′}(t^{−1}, q)ω_{q,t}P_{λ}(q, t)

by (3.9), (3.8) and (3.6). Hence ω_{q,t}P_{λ}(q, t) is an eigenfunction of E_{t^{−1},q^{−1}} with eigenvalue f_{λ′}(t^{−1}, q). Hence it must be a scalar multiple of P_{λ′}(t^{−1}, q^{−1}) = P_{λ′}(t, q). We want to show that it is actually Q_{λ′}(t, q), so we must show that

⟨ω_{q,t}P_{λ}(q, t), P_{λ′}(t, q)⟩_{t,q} = 1.

By (3.4) this is equivalent to showing that

(3.10)  ⟨ωP_{λ}(q, t), P_{λ′}(t, q)⟩ = 1

for the “usual” scalar product (1.9).

I.G. MACDONALD

To prove (3.10), we shall express P_{λ} and P_{λ′} as linear combinations of Schur functions : say

\begin{aligned}P_{λ}(q, t) & = s_{λ} + ∑_{μ<λ} a_{λμ}s_{μ}, \\ P_{λ′}(t, q) & = s_{λ′} + ∑_{ν′<λ′} b_{λν}s_{ν′}. \end{aligned}

Since ωs_{μ} = s_{μ′} it follows that

ωP_{λ}(q, t) = s_{λ′} + ∑_{μ′>λ′} a_{λμ}s_{μ′}

and therefore P_{λ′}(t, q) and ωP_{λ}(q, t) have only s_{λ′} in common, so that

⟨ωP_{λ}(q, t), P_{λ′}(t, q)⟩ = ⟨s_{λ′}, s_{λ′}⟩ = 1

as required. This completes the proof of (3.5).

Since (P_{λ}), (Q_{λ}) are dual bases of Λ_{F}, we have from (2.6)

(3.11)  ∑_{λ} P_{λ}(x; q, t)Q_{λ}(y; q, t) = Π(x, y; q, t).

Apply ω_{q,t} to the y-variables ; from (2.5) it is easily computed that

ω_{q,t}Π(x, y; q, t) = ∏_{i,j}(1 + x_{i}y_{j}).

Hence it follows from (3.5) that

(3.12)  ∑_{λ} P_{λ}(x; q, t)P_{λ′}(y; t, q) = ∏_{i,j}(1 + x_{i}y_{j}).

When q = t, (3.11) and (3.12) reduce to the familiar identities

∑_{λ} s_{λ}(x)s_{λ}(y) = ∏_{i,j}(1 − x_{i}y_{j})^{−1}

and

∑_{λ} s_{λ}(x)s_{λ′}(y) = ∏_{i,j}(1 + x_{i}y_{j})

respectively.

Finally, we can now identify the symmetric functions P_{λ}(q, t) when either q = 1 or t = 1. Suppose first that t = 1. Then from (2.10) we have

D_{q,1} = ∑_{i=1}^{n} T_{q,x_{i}}

so that (for any partition λ of length ≤ n)

D_{q,1}m_{λ} = (∑_{i=1}^{n} q^{λ_{i}})m_{λ}.

Hence the m_{λ} are the eigenfunctions of the operator D_{q,1}, and therefore

(3.13)  P_{λ}(q, 1) = m_{λ}

for all partitions λ.

Next, it follows from (3.5) that

⟨ω_{q,t}P_{λ}(q, t), P_{μ′}(t, q)⟩_{t,q} = δ_{λμ}

and hence by (3.4) that

⟨P_{λ}(q, t), ωP_{μ′}(t, q)⟩ = δ_{λμ}.

By (3.13) this gives

⟨m_{λ}, ωP_{μ′}(1, q)⟩ = δ_{λμ}

when t = 1. But the basis dual to (m_{λ}) for the usual scalar product is (h_{λ}) ([M_{1}], ch. I). Hence ωP_{μ′}(1, q) = h_{μ}, i.e., P_{μ′}(1, q) = ωh_{μ} = e_{μ}. Hence (replacing q, μ′ by t, λ)

(3.14)  P_{λ}(1, t) = e_{λ′}.

4. Skew P and Q functions

Let μ and ν be two partitions. Then the product P_{μ}P_{ν} is a linear combination of the P_{λ}, say

(4.1) P_{μ}P_{ν} = ∑_{λ} f_{μν}^{λ}P_{μ}

where

f_{μν}^{λ} = f_{μν}^{λ}(q, t) = ⟨Q_{λ}, P_{μ}P_{ν}⟩_{q,t} ∈ F.

In particular

(1) f_{μν}^{λ}(t, t) is the coefficient c_{μν}^{λ} of s_{λ} in s_{μ}s_{ν}, which may be calculated by the Littlewood-Richardson rule.

(2) f_{μν}^{λ}(0, t) is the Hall polynomial f_{μν}^{λ}(t) ([M_{1}], ch. II). It may be written as a sum

f_{μν}^{λ}(t) = ∑_{T} f_{T} (t)

of monic polynomials, where T runs through the set of LR-tableaux of shape λ′ − μ′ and weight ν′ (loc. cit.)

(3) f_{μν}^{λ}(q, 1) is the coefficient of m_{λ} in m_{μ}m_{ν}, and so is independent of q.

(4) f_{μν}^{λ}(1, t) = 1 if λ = μ + ν, and is zero otherwise. For P_{μ}(1, t) = e_{μ′}, and e_{μ′}e_{ν′} = e_{μ′∪ν′} = e_{(μ+ν)′}.

(5) f_{μν}^{λ}(q^{−1}, t^{−1}) = f_{μν}^{λ}(q, t), since P_{λ}(q^{−1}, t^{−1}) = P_{λ}(q, t) (§ 2).

(6) By duality (3.5) we have

f_{μν}^{λ}(q, t) = f_{μ′ν′}^{λ′}(t, q)b_{μ′}(t, q)b_{ν′}(t, q)/b_{λ′}(t, q).

In view of (1) and (2) above, it is natural to ask whether it is possible to attach to each LR-tableau T a non zero rational function f_{T} (q, t) so that

f_{μν}^{λ}(q, t) = ∑_{T} f_{T} (q, t),

where T runs through the set of LR-tableaux of shape λ−μ and weight ν. (If so, then by duality ((6) above) there will be likewise a decomposition f_{μν}^{λ} over the LR-tableaux of shape λ′ − μ′ and weight ν′.) Again, is it true that f_{μν}^{λ}(q, t) ≠ 0 if and only if c_{μν}^{λ} ≠ 0 ? The answers to these questions are not known, at any rate to the author.

Clearly we shall have f_{μν}^{λ} = 0 unless |λ| = |μ| + |ν|. In fact, more is true :

(4.2) f_{μν}^{λ} = 0 unless λ ⊃ μ and λ ⊃ ν (i.e., the diagram of λ contains those of μ and ν).

STANLEY [S] gives a proof of (4.2) in the context of Jack’s symmetric functions. His proof can be transposed to the present context without difficulty.

We now define skew Q-functions as follows. If λ, μ are partitions, define

(4.3) Q_{λ/μ} = ∑_{ν} f_{μν}^{λ}Q_{ν}

so that

⟨Q_{λ/μ}, P_{ν}⟩_{q,t} = ⟨Q_{λ}, P_{μ}P_{ν}⟩_{q,t}.

From (4.2) if follows that Q_{λ/μ} = 0 unless λ ⊃ μ.

Let x = (x_{1}, x_{2}, …) and y = (y_{1}, y_{2}, …) be two sequences of independent indeterminates. If f is any symmetric function, f(x, y) shall mean f(x_{1}, y_{1}, x_{2}, y_{2}, …), and likewise we define f(x, y, z, …) for three or more sequences x, y, z, … Then we have

(4.4)  Q_{λ/μ}(x, y) = ∑_{ν} Q_{λ/ν}(x)Q_{ν/μ}(y)

summed over partitions ν such that λ ⊃ ν ⊃ μ.

Proof. — We have

∑_{λ} Q_{λ/μ}(x)P_{λ}(y) = ∑_{λ,ν} f_{μν}^{λ}Q_{ν}(x)P_{λ}(y)

= ∑_{ν} Q_{ν}(x)P_{μ}(y)P_{ν}(y)

= P_{μ}(y)Π(x, y)

by (3.11). Hence

∑_{λ,μ} Q_{λ/μ}(x)P_{λ}(y)Q_{μ}(z) = Π(x, y)Π(z, y)

= ∑_{λ} P_{λ}(y)Q_{λ}(x, z)

by (3.11) again. By comparing the coefficients of P_{λ}(y) on either side, we obtain

(4.5)  Q_{λ}(x, z) = ∑_{μ} Q_{λ/μ}(x)Q_{μ}(z).

If we now replace x by x, y in (4.5), we have

∑_{μ} Q_{λ/μ}(x, y)Q_{μ}(z) = Q_{λ}(x, y, z)

= ∑_{ν} Q_{λ/ν}(x)Q_{ν}(y, z)

= ∑_{μ,ν} Q_{λ/ν}(x)Q_{ν/μ}(y)Q_{μ}(z)

by two applications of (4.5). If we now equate the coefficients of Q_{μ}(s) at either end of this string of equalities, we shall obtain (4.4).

The identity (4.4) clearly generalizes to n sets of variables x^{(1)}, …, x^{(n)} : if λ, μ are partitions, then

(4.6)  Q_{λ/μ}(x^{(1)}, …, x^{(n)}) = ∑_{(ν)} ∏_{i=1}^{n} Q_{ν^{i}/ν^{i−1}}(x^{(i)})

summed over all sequences (ν) = (ν^{0}, ν^{1}, …, ν^{n}) of partitions such that μ = ν^{0} ⊂ ν^{1} ⊂ ⋯ ⊂ ν^{n} = λ.

Let us apply (4.6) in the case where each set of variables x^{(i)} consists of a single element x_{i}. For a single x, we have

(4.7)  Q_{λ/μ}(x) = φ_{λ/μ} x^{|λ−μ|}

say, where φ_{λ/μ} = φ_{λ/μ}(q, t) ∈ F, and in fact

(4.8) φ_{λ/μ} = 0 unless λ ⊃ μ and λ − μ is a horizontal strip, i.e., unless the partitions λ, μ are interlaced :

λ_{1} ≥ μ_{1} ≥ λ_{2} ≥ μ_{2} ≥ ⋯

Again we refer to [S] for a proof of this. As in the case of (4.2), STANLEY’s proof (for Jack’s functions) can be transposed without difficulty (and indeed (4.2) is a a consequence of (4.8)).

From (4.6) and (4.7) we obtain

(4.9)  Q_{λ/μ}(x_{1}, …, x_{n}) = ∑_{(ν)} ∏_{i=1}^{n} φ_{ν^{i}/ν^{i−1}} x_{i}^{|ν^{i}−ν^{i−1}|}

with (ν) as in (4.6) and each skew diagram ν^{i} − ν^{i−1} a horizontal strip. The sequence (ν) of partitions determines a (column strict) tableau T of shape λ − μ, in which the symbol i occurs in each square of ν^{i} − ν^{i−1}, for 1 ≤ i ≤ n. If we now define

(4.10) φ_{T} = ∏_{i≥1} φ_{ν^{i}/ν^{i−1}}

then we shall have

(4.11) Q_{λ/μ} = ∑_{T} φ_{T} x^{T}

summed over all tableaux T of shape λ − μ, where x^{T} is the monomial determined by the tableau T, i.e., x^{T} = x^{α}, where α is the weight of T.

Later (§ 5) we shall derive an explicit formula for φ_{λ/μ} and hence also for φ_{T}, and then (4.11) will provide an explicit (if complicated) expression for Q_{λ/μ} as a sum of monomials.

Finally, one can define skew P -functions P_{λ/μ} by interchanging the roles of the P ’s and the Q’s throughout. The relation between the two is

(4.12)  P_{λ/μ} = b_{λ}^{−1}b_{μ}Q_{λ/μ}

with b_{λ} as defined by (3.1).

5. Explicit formulas

We have introduced various scalars b_{λ}, φ_{λ/μ}, φ_{T} in the preceding sections, but so far we have no way of computing them explicitly. The key to this is a specialization theorem, which will be stated in a moment.

Let u be a new indeterminate, and define a homomorphism (or specialization)

ϵ_{u,t} : Λ_{F} → F [u]

by

(5.1) ϵ_{u,t}(p_{r}) = \frac{1 − u^{r}}{1 − t^{r}}

for each r ≥ 1.

To motivate this definition, suppose that u = t^{n}, n a positive integer. Then

ϵ_{t^{n},t}(p_{r}) = \frac{1 − t^{nr}}{1 − t^{r}}
= 1 + t^{r} + ⋯ + t^{(n−1)r}
= p_{r}(1, t, …, t^{n−1})

so that

ϵ_{t^{n},t}(f) = f(1, t, …, t^{n−1})

for any symmetric function f : i.e., the effect of ϵ_{t^{n},t} is to evaluate at (x_{1}, …, x_{n}, x_{n+1}, …) = (1, t, …, t^{n−1}, 0, 0, …).

There is a nice formula for ϵ_{u,t}(P_{λ}(q, t)). In order to state it in a convenient form, let us introduce the following notation. For each square s = (i, j) in the diagram of a partition λ, let

(5.2)  \begin{cases}a(s) = λ_{i} − j, & a′(s) = j − 1, \\ l(s) = λ′_{j} − i, & l′(s) = i − 1, \end{cases}

so that l′(s), l(s), a(s) and a′(s) are respectively the numbers of squares in the diagram of λ to the north, south, east and west of the square s. The numbers a(s) and a′(s) may be called respectively the arm-length and the arm-colength of s, and l(s), l′(s) the leg-length and leg-colength. The hook-length at s is a(s) + l(s) + 1.

(5.3) Theorem. — We have

ϵ_{u,t}(P_{λ}(q, t)) = ∏_{s∈λ} \frac{q^{a′(s)}u − t^{l′(s)}}{q^{a(s)}t^{l(s)+1} − 1}.

Proof. — Since by (5.1) ϵ_{u,t}(p_{r}) is a polynomial of degree r in u with coefficients in F, it follows that ϵ_{u,t}(P_{λ}) is a polynomial of degree ≤ |λ|, say

ϵ_{u,t}(P_{λ}(q, t)) = Φ_{λ}(u; q, t) ∈ F [u].

The idea of the proof is first to locate the zeros of this polynomial, which will give the numerator of the expression in (5.3). For this we require two relations. The first comes from the formula (4.5) (with Q’s replaced by P ’s), which gives

(5.4)  P_{λ}(x_{1}, …, x_{n}) = ∑_{μ} P_{λ/μ}(x_{1})P_{μ}(x_{2}, …, x_{n}).

Set (x_{1}, …, x_{n}) = (1, t, …, t^{n−1}) and let

(5.5)  ψ_{λ/μ} = P_{λ/μ}(1; q, t).

Then (5.4) becomes

Φ_{λ}(t^{n}; q, t) = ∑_{μ} ψ_{λ/μ}(q, t)t^{|μ|}Φ_{μ}(t^{n−1}; q, t)

for all n ≥ 1, and hence we have

(A)  Φ_{λ}(u; q, t) = ∑_{μ} ψ_{λ/μ}(q, t)t^{|μ|}Φ_{μ}(ut^{−1}; q, t).

Next we have

(5.6)  ϵ_{u,t}ω_{t,q}(f) = (−q)^{−n}ϵ_{u,q^{−1}}(f)

if f ∈ Λ_{F}^{n}. (Since both ϵ and ω are ring homomorphisms, it is enough to check (5.6) when f = p_{n}.) Hence by duality (3.5) we have

ϵ_{u,t}P_{λ}(q, t) = ϵ_{u,t}ω_{t,q}Q_{λ′}(t, q)
= (−q)^{−|λ|}ϵ_{u,q^{−1}}Q_{λ}(t, q)
= (−q)^{−|λ|}b_{λ′}(t, q)ϵ_{u,q^{−1}}P_{λ′}(t^{−1}, q^{−1})

(since P_{λ′}(t, q) = P_{λ′}(t^{−1}, q^{−1})) and therefore

(B)  Φ_{λ}(u; q, t) = (−q)^{−|λ|}b_{λ′}(t, q)Φ_{λ′}(u; t^{−1}, q^{−1}).

We observe next that

(5.7) P_{λ}(x_{1}, …, x_{n}) = 0 if n < ℓ(λ).

For P_{λ} is a linear combination of the m_{μ} such that μ ≤ λ, and

μ ≤ λ ⇒ μ′ ≥ λ′ ⇒ ℓ(μ) = μ′_{1} ≥ λ′_{1} = ℓ(λ) > n

⇒ ℓ(μ) > n ⇒ m_{μ}(x_{1}, …, x_{n}) = 0.

It follows from (5.7) that

Φ_{λ}(u; q, t) = 0 for u = 1, t, …, t^{ℓ(λ)−1}.

Hence by (B) the polynomial Φ_{λ′}(u; t^{−1}, q^{−1}) vanishes also for these values of u. By replacing (λ, q, t) by (λ′, t^{−1}, q^{−1}), we see that

Φ_{λ}(u; q, t) = 0 for u = 1, q^{−1}, …, q^{1−λ_{1}}

and therefore

(C)  Φ_{λ}(u; q, t) is divisible in F [u] by ∏_{j=1}^{λ_{1}}(q^{j−1}u − 1).

Now we consider the relation (A) above. By (C), each term on the right of (A) is divisible by

∏_{j=1}^{λ_{2}}(q^{j−1}u − t)

(because the sum in (A) is over partitions μ that interlace λ). Hence Φ_{λ}(u; q, t) is divisible by this product. We now repeat the argument : each term on the right-hand side of (A) is divisible by

∏_{j=1}^{λ_{3}}(q^{j−1}u − t^{2})

(since μ_{2} ≥ λ_{3}) and therefore Φ_{λ}(u; q, t) is also divisible by this product. Thus finally Φ_{λ}(u; q, t) is divisible in F [u] by

∏_{i=1}^{ℓ(λ)} ∏_{j=1}^{λ_{i}}(q^{j−1}u − t^{i−1}) = ∏_{s∈λ}(q^{a′(s)}u − t^{l′(s)}).

(Observe that all these linear factors are distinct.) But we know that Φ_{λ} has degree at most |λ| in u. Hence we have

(D)  Φ_{λ}(u; q, t) = v_{λ}(q, t) ∏_{s∈λ}(q^{a′(s)}u − t^{l′(s)}),

and it remains to identify the scalar factor v_{λ}(q, t). For this purpose we require

(5.8) Let λ be a partition of length n, and let λ = (λ_{1} − 1, …, λ_{n} − 1). Then

P_{λ}(x_{1}, …, x_{n}) = x_{1} … x_{n}P_{μ}(x_{1}, …, x_{n}).

(Compute D_{q,t}(x_{1} … x_{n}P_{μ}) and show that x_{1} … x_{n}P_{μ} is an eigenfunction of D_{q,t} with eigenvalue ∑ q^{λ_{i}}t^{n−i}.)

From (5.8) it follows that

Φ_{λ}(t^{n}; q, t) = t^{n(n−1)/2}Φ_{μ}(t^{n}; q, t)

if ℓ(λ) = n, and hence from (D) that

\begin{aligned}\frac{v_{λ}(q, t)}{v_{μ}(q, t)} & = t^{n(n−1)/2} ∏_{s∈λ−μ}(q^{a′(s)}t^{n} − t^{l′(s)})^{−1} \\ & = ∏_{i=1}^{n}(q^{λ_{i}−1}t^{n−i+1} − 1)^{−1}. \end{aligned}

Now the factors in this product are precisely (q^{a(s)}t^{l(s)+1} − 1)^{−1} for s in the first column of λ. Hence by induction on the number of columns of λ we conclude that

v_{λ}(q, t) = ∏_{s∈λ}(q^{a(s)}t^{l(s)+1} − 1)^{−1}

and the proof of (5.3) is complete.

From (B) we have

b_{λ′}(t, q) = (−q)^{|λ|} \frac{Φ_{λ}(u; q, t)}{Φ_{λ′}(u; t^{−1}, q^{−1})}

which together with (5.3) leads to the formula

b_{λ}(q, t) = ∏_{s∈λ} \frac{1 − q^{a(s)}t^{l(s)+1}}{1 − q^{a(s)+1}t^{l(s)}} .

So we know now the value of ⟨P_{λ}, P_{μ}⟩_{q,t} = b_{λ}(q, t)^{−1}.

Knowing the b_{λ}, it turns out that it is now quite straightforward to calculate the scalars

φ_{λ/μ} = Q_{λ/μ}(1)

defined in § 4. I will omit the details and merely state the result. For each square s and each partition λ, define

(5.10)  b_{λ}(s) = b_{λ}(s; q, t) = \frac{1 − q^{a(s)}t^{l(s)+1}}{1 − q^{a(s)+1}t^{l(s)}}

if s ∈ λ, and b_{λ}(s) = 1 if s ∉ λ.

Next, if S is any set of squares (contained in the diagram of λ or not), let

(5.11)  b_{λ}(S) = ∏_{s∈S} b_{λ}(s).

Now let λ, μ be partitions such that λ_{1} ≥ μ_{1} ≥ λ_{2} ≥ μ_{2} ≥ ⋯ or equivalently such that λ ⊃ μ and λ − μ is a horizontal strip. Let C_{λ/μ} denote the union of the columns that contain squares of λ − μ. Then the formula for φ_{λ/μ} is

(5.12)  φ_{λ/μ} = b_{λ}(C_{λ/μ})/b_{μ}(C_{λ/μ})

and for a tableau T

(5.13)  φ_{T} = ∏_{i≥0} b_{λ^{i}}(C_{i})/b_{λ^{i}}(C_{i+1}),

where λ^{0} ⊂ λ^{1} ⊂ ⋯ is the sequence of partitions defined by the tableau, and C_{i} is the union of the columns that contain a symbol i (so that C_{0} is empty).

In the case of Jack’s symmetric functions, the results of this section are all due to Richard STANLEY [S], and the transposition of his proofs into the present context is quite straightforward. Also when q = 0 the formulas (5.9), (5.12) and (5.13) reduce respectively to [M_{1}], ch. III, (2.12), (5.8) and (5.9).

Finally we may remark that the identities of Schur and Littlewood

\begin{aligned}∑_{λ} s_{λ} & = ∏(1 − x_{i})^{−1} ∏_{i<j}(1 − x_{i}x_{j})^{−1}, \\ ∑_{μ} s_{μ} & = ∏_{i≤j}(1 − x_{i}x_{j})^{−1}, \\ ∑_{ν} s_{ν} & = ∏_{i<j}(1 − x_{i}x_{j})^{−1} \end{aligned}

in which λ runs through all partitions, μ through all even partitions (i.e. with all parts even) and ν through all partitions such that ν′ is even, generalize to identities for the P_{λ}(q, t) as follows. For any partition λ, let

b_{λ}^{el} = ∏_{{s∈λ \atop l(s) even}} b_{λ}(s), b_{λ}^{oa} = ∏_{{s∈λ \atop a(s) odd}} b_{λ}(s),

(so the superscripts el and oa stand for “even legs” and “odd arms,” respectively. Then there are product expressions for the four series

(a) ∑_{λ} b_{λ}^{el}(q, t)P_{λ}(q, t), (b) ∑_{λ} b_{λ}^{oa}(q, t)P_{λ}(q, t),

(c) ∑_{μ} b_{μ}^{oa}(q, t)P_{μ}(q, t), (d) ∑_{ν} b_{ν}^{el}(q, t)P_{ν}(q, t),

where as before λ runs through all partitions, μ through all even partitions and ν through partitions such that ν′ is even. For example, the sum (a) is equal to the product

∏_{i} \frac{(tx_{i}; q)_{∞}}{(x_{i}; q)_{∞}} ∏_{i<j} \frac{(tx_{i}x_{j}; q)_{∞}}{(x_{i}x_{j}; q)_{∞}}

and (d) is equal to

∏_{i<j} \frac{(tx_{i}x_{j}; q)_{∞}}{(x_{i}x_{j}; q)_{∞}} .

The products for (b) and (c) are a little more complicated : they may be derived from (a) and (b) by duality (3.5). In the case of Jack’s symmetric functions, (a) is due to K. KADELL.

6. The Kostka matrix

Let

(6.1)  h_{λ}(q, t) = ∏_{s∈λ}(1 − q^{a(s)}t^{l(s)+1}),
(6.2)  h′_{λ}(q, t) = ∏_{s∈λ}(1 − q^{a(s)+1}t^{l(s)}),
= h_{λ′}(t, q),

so that by (5.9) we have

b_{λ}(q, t) = h_{λ}(q, t)/h′_{λ}(q, t).

Now define

(6.3)  J_{λ}(q, t) = h_{λ}(q, t)P_{λ}(q, t)
= h′_{λ}(q, t)Q_{λ}(q, t).

It seems likely that when the J_{λ}(q, t) are expressed in terms of the monomial symmetric functions, the coefficients are polynomials, i.e. elements of ℤ[q, t]. I shall make a more precise conjecture later.

When q = t, we have

J_{λ}(t, t) = H_{λ}(t)s_{λ},

where

H_{λ}(t) = ∏_{s∈λ}(1 − t^{h(s)})

is the hook-length polynomial (h(s) = a(s) + l(s) + 1).

When q = 0,

(6.4)  J_{λ}(0, t) = Q_{λ}(0, t)

because h′_{λ}(0, t) = 1.

Duality (3.5) now takes the form

(6.5)  ω_{q,t}J_{λ}(q, t) = J_{λ′}(t, q)

and the specialization theorem (5.3) takes the form

ϵ_{u,t}J_{λ}(q, t) = ∏_{s∈λ}(t^{l′(s)} − q^{a′(s)}u).

From the fact (§ 2) that P_{λ}(q, t) = P_{λ}(q^{−1}, t^{−1}) we deduce that

(6.6) J_{λ}(q^{−1}, t^{−1}) = (−1)^{|λ|}q^{−n(λ′)}t^{−n(λ)−|λ|}J_{λ}(q, t), where

n(λ) = ∑_{i≥1}(i − 1)λ_{i} = ∑_{i≥1} \binom{λ′_{i}}{2}.

Recall ([M_{1}], ch. I, § 6) that the Kostka numbers K_{λμ} are defined by

(6.7)  s_{λ} = ∑_{μ} K_{λμ}m_{μ}.

We have K_{λμ} = 0 unless μ ≤ λ, and K_{λλ} = 1. They generalize to the Kostka-Foulkes polynomials K_{λμ}(t) ([M_{1}], ch. III) defined as follows :

(6.8)  s_{λ}(x) = ∑_{μ} K_{λμ}(t)P_{μ}(x; t),

where the P_{μ}(x; t) = P_{μ}(0; x, t) are the Hall-Littlewood functions. Let S_{λ}(x; t) denote the Schur functions associated with the product

∏(1 − tx_{i})/(1 − x_{i}) ;

they form a basis of Λ_{ℚ(t)} dual to the basis (s_{λ}(x)), relative to the scalar product (1.13), and therefore (6.8) is equivalent to

(6.8′)  Q_{μ}(x; t) = ∑_{λ} K_{λμ}(t)S_{λ}(x; t).

Since P_{μ}(x; 1) = m_{μ} it follows from (6.7) and (6.8) that K_{λμ}(1) = K_{λμ}. Now K_{λμ} is the number of tableaux of shape λ and weight μ. FOULKES conjectured, and LASCOUX and SCH^{}ÜTZENBERGER proved, that K_{λμ}(t) is a polynomial in t with positive integral coefficients, and more precisely that

K_{λμ}(t) = ∑_{T} t^{c(T)}

summed over all tablelaux T of shape λ and weight μ, where c(T) (the charge of T) is a well-defined ℕ-valued function of the tableau T.

In the present context we now define K_{λμ}(q, t) ∈ F by

(6.9)  J_{μ}(x; q, t) = ∑_{λ} K_{λμ}(q, t)S_{λ}(x; t).

Computations of the K_{λμ}(q, t) suggest the

Conjecture. — K_{λμ}(q, t) is a polynomial in q and t with positive integral coefficients.

Here are some partial results which tend to confirm this conjecture.

(1) We have K_{λμ}(0, t) = K_{λμ}(t) (by (6.4) and (6.8′)). In particular, K_{λμ}(0, 0) = δ_{λμ} and K_{λμ}(0, 1) = K_{λμ}.

(2) From (6.5) and (6.6) we deduce that

\begin{aligned}K_{λμ}(q, t) & = K_{λ′μ′}(t, q), \\ K_{λμ}(q, t) & = q^{n(μ′)}t^{n(μ)}K_{λ′μ}(q^{−1}, t^{−1}). \end{aligned}

(3) Another special case is (λ, μ partitions of n)

K_{λμ}(1, 1) = \frac{n!}{h(λ)},

where h(λ) is the product of the hook lengths of λ. In other words, K_{λμ}(1, 1) is the number of standard tableaux of shape λ (and in particular does not depend on μ). This prompts the following question, which refines the conjecture above : can one find ℕ-valued functions a_{μ}(T), b_{μ}(T), defined for standard tableaux T, such that

K_{λμ}(q, t) = ∑_{T} q^{a_{μ}(T)}t^{b_{μ}(T)}

summed over the standard tableaux T of shape λ? Of course this question, as I have stated it, is not well-posed. The point is to find some “natural” bijection between the monomials q^{a}t^{b} that occur in K_{λμ}(q, t) (assuming the truth of the conjecture) and the standard tableaux of shape λ.

(4) When q = t we have K_{λμ}(t, t) ∈ ℤ[t], by a result of STANLEY.

(5) K_{λμ}(1, t) is a polynomial in t with positive integer coefficients. (Recall that P_{λ}(1, t) = e_{λ′}; this makes it feasible to compute K_{λμ}(1, t).) By duality ((2) above) K_{λμ}(q, 1) is a polynomial in q with positive integral coefficients.

(6) Let λ = (r, 1^{s}). Then for each partition μ of r + s, K_{λμ}(q, t) is the coefficient of u^{s} in the product

∏(t^{i−1} + q^{j−1}u)

over all (i, j) ∈ μ with the exception of (1,1).

The symmetric functions S_{λ}(x, t) are linear combinations of the m_{μ}(x) with coefficients in ℤ[t], from the table on p. 128 of [M_{1}]. Hence the conjecture would imply that

J_{λ}(q, t) = ∑_{μ≤λ} v_{λμ}(q, t)m_{μ}

with coefficients v_{λμ} ∈ ℤ[q, t], divisible by (1−t)^{l(μ)}. Another consequence of the conjecture is that

⟨J_{λ}, J_{μ}J_{ν}⟩_{q,t} ∈ ℤ[q, t]

for any three partitions λ, μ, ν.

Let K_{n}(q, t) denote the matrix (K_{λμ}(q, t))_{λ,μ∈P_{n}}. For n = 1, …, 6 the matrices K_{n} (or rather their transposes K′_{n}) are shown in the appendix.

7. Another scalar product

We have seen that the symmetric functions P_{λ}(q, t) are pairwise orthogonal with respect to the scalar product ⟨ , ⟩_{q,t}. It will appear that they are also pairwise orthogonal with respect to another scalar product ⟨ , ⟩′_{q,t}, which I shall now define.

We shall work throughout with a finite number of indeterminates x = (x_{1}, …, x_{n}) (i.e., in Λ_{n,F} rather Λ_{F}). Moreover we shall assume (although it is not strictly necessary) that t = q^{k} where k ∈ ℕ.

Let

L_{n} = F [x_{1}^{±1}, …, x_{n}^{±1}]

be the F -algebra of Laurent polynomials in x_{1}, …, x_{n}, i.e. of polynomials in the x_{i} and x_{i}^{−1}. If f ∈ L_{n} let \overline{f} = f(x_{1}^{−1}, …, x_{n}^{−1}) and let [f]_{1} denote the constant term in f. Moreover let

(7.1)  Δ = Δ(x; q, t) = ∏_{{i,j=1 \atop i≠j}}^{n} \frac{(x_{i}x_{j}^{−1}; q)_{∞}}{(tx_{i}x_{j}^{−1}; q)_{∞}}

= ∏_{i≠j} ∏_{r=0}^{k−1}(1 − q^{r}x_{i}x_{j}^{−1})

so that Δ ∈ L_{n}.

For f, g ∈ Λ_{n,F} we now define

(7.2)  ⟨f, g⟩′_{q,t} = \frac{1}{n!}[f\overline{g}Δ]_{1}.

This is a symmetric, positive definite scalar product on Λ_{n,F}. It is not difficult (in fact, it is a good deal easier than it was in § 2) to show that

(7.3) The operator D_{q,t} (2.10) is self-adjoint for this scalar product :

⟨Df, g⟩′_{q,t} = ⟨f, Dg⟩′_{q,t}

for all f, g ∈ Λ_{n,F}.

From (7.3) it follows, just as in § 2, that

⟨P_{λ}, P_{μ}⟩′_{q,t} = 0

if λ ≠ μ.

Remark. — When q = t (i.e., when k = 1) the two scalar products are the same. Otherwise they are different.

It remains to compute ⟨P_{λ}, P_{λ}⟩′_{q,t}, and the answer is as follows :

(7.4)  \begin{aligned}⟨P_{λ}, P_{λ}⟩′_{q,t} & = ∏_{1≤i<j≤n} ∏_{r=1}^{k−1} \frac{1 − q^{λ_{i}−λ_{j}+r} t^{j−i}}{1 − q^{λ_{i}−λ_{j}−r} t^{j−i}} \\ & = c_{n} ∏_{s∈λ} \frac{1 − q^{a′(s)} t^{n−l′(s)}}{1 − q^{a′(s)+1} t^{n−l′(s)−1}} , \end{aligned}

where

(7.5) c_{n} = ⟨1, 1⟩′_{q,t} = \frac{1}{n!}[Δ]_{1} = ∏_{i=2}^{n} \left[{ik − 1 \atop k − 1}\right],

a product of q-binomial coefficients. (Notice that when λ = 0, (7.4) reduces to a special case of the q-Dyson conjecture.)

8. Conclusion

In the definition (7.1) of Δ, and in the first of the two products (7.4) the structure of the root system of type A_{n−1} is clearly visible. In fact this aspect of the theory generalizes to other root systems, and I shall conclude these lectures with a brief and simplified account of this generalization. For full details and proofs, see [M_{3}].

So let R be a reduced root system, R^{+} a system of positive roots in R; let Q be the root lattice of R, and Q^{+} the positive cone in Q, spanned by R^{+} ; let P be the weight lattice, and P^{++} the cone of dominant weights ; and let W be the Weyl group of R. Define a partial order on P by

λ ≥ μ ⟺ λ − μ ∈ Q^{+}.

Now let q and t be indeterminates, and F as before the field ℚ(q, t). Let A = F [P] be the group algebra of the lattice (or free abelian group) P over F. To each λ ∈ P there corresponds an element e^{λ} of A, such that e^{λ}.e^{μ} = e^{λ+μ}, and e^{0} = 1 is the identity element of A. The Weyl group W acts on P, hence on A : w(e^{λ}) = e^{wλ} (w ∈ W, λ ∈ P). Let A^{W} be the subalgebra of W -invariants in A.

We can easily describe two F -bases of A^{W}. One consists of the orbitsums

m_{λ} = ∑_{μ∈W λ} e^{μ} (λ ∈ P^{++})

and is the counterpart of the monomial symmetric functions denoted by the same symbols. The other basis consists of the Weyl characters : let

ρ = \frac{1}{2} ∑_{α∈R^{+}} α

and let

δ = ∏_{α∈R^{+}}(e^{α/2} − e^{−α/2}) = ∑_{w∈W} ϵ(w)e^{wρ}

(where ϵ(w) = det(w) = ±1). For each λ ∈ P^{++} we define the Weyl character

χ_{λ} = δ^{−1} ∑_{w∈W} ϵ(w)e^{w(λ+ρ)}

which is the counterpart of the Schur function s_{λ}. We have

χ_{λ} = m_{λ} + ∑_{{μ<λ \atop μ∈P^{++}}} K_{λμ}m_{μ}

with coefficients K_{λμ} ∈ ℕ.

If f ∈ A, say f = ∑_{λ∈P} f_{λ}e^{λ}, let

\overline{f} = ∑ f_{λ}e^{−λ}

and let [f]_{1} denote the constant term f_{0} of f.

For simplicity we shall assume as in § 7 that t = q^{k}, k ∈ ℕ, and define, in analogy with (7.1),

\begin{aligned}Δ = Δ(q, t) & = ∏_{α∈R} \frac{(e^{a}; q)_{∞}}{(te^{α}; q)_{∞}} \\ & = ∏_{α∈R}(e^{α}; q)_{k} \end{aligned}

so that Δ ∈ A^{W}. Now define a scalar product on A^{W} by

⟨f, g⟩ = |W |^{−1} [f\overline{g}Δ]_{1}

for f, g ∈ A^{W}. Then we have Theorem. — There exists a unique basis (P_{λ})_{λ∈P^{++}} of A^{W} such that

(A) P_{λ} = m_{λ} + ∑_{{μ<λ \atop μ∈P^{++}}} u_{λμ}m_{μ}

with coefficients u_{λμ} ∈ F ; (B) ⟨P_{λ}, P_{μ}⟩ = 0 if λ ≠ μ.

When R is of type A_{n−1}, these P_{λ} are essentially the symmetric functions P_{λ}(q, t), restricted to n variables x_{1}, …, x_{n}. For arbitrary R, when k = 0 (i.e., t = 1) we have P_{λ} = m_{λ}, and when k = 1 (i.e., q = t) we have P_{λ} = χ_{λ}.

I will conclude with two conjectures which generalize (7.4) and the specialization theorem (5.3) respectively. For each root α ∈ R let α^{∨} be the corresponding co-root, and let

σ = \frac{1}{2} ∑_{α∈R^{+}} α^{∨}.

Conjecture 1. — For all λ ∈ P^{++},

⟨P_{λ}, P_{μ}⟩ = ∏_{α∈R^{+}} ∏_{i=1}^{k−1} \frac{1 − q^{α^{∨}(λ+kρ)+i}}{1 − q^{α^{∨}(λ+kρ)−i}} .

This is non trivial even when λ = 0 (so that P_{λ} = 1) ; in that case it reduces to the constant term conjectures of [M_{2}].

Conjecture 2. — Let P_{λ}(kσ) denote the image of P_{λ} under the mapping e^{μ} ↦ q^{kσ(μ)} = t^{σ(μ)} (μ ∈ P). Then for all λ ∈ P^{++},

P_{λ}(kσ) = q^{−kσ(λ)} ∏_{α∈R^{+}} \frac{(q^{α^{∨}(λ+kρ)}; q)_{k}}{(q^{α^{∨}(kρ)}; q)_{k}} .

REFERENCES

[M_{1}] MACDONALD (I.G.). — Symmetric functions and Hall polynomials. — Oxford

University Press, 1979. [M_{2}] MACDONALD (I.G.). — Some conjectures for root systems, SIAM J. Math. Anal.,

t. 13, 1982, p. 988–1007. [M_{3}] MACDONALD (I.G.). — Orthogonal polynomials associated with root systems,

preprint, 1988.

[S] STANLEY (Richard). — Some combinatorial properties of Jack symmetric functions, preprint, 1988.

I. G. Macdonald,

School of Mathematical Sciences, Queen Mary College, University of London, Mile End Road, London E1 4NS, United Kingdom.

A NEW CLASS OF SYMMETRIC FUNCTIONS

9. Appendix The matrices K(q, t)′, n ≤ 6

1 1 1  \begin{aligned}& 2 & 1^{2} \\ 2 & 1 & q \\ 1^{2} & t & 1 \end{aligned}  \begin{aligned}& 3 & 21 & 1^{3} \\ 3 & 1 & q + q^{2} & q^{3} \\ 21 & t & 1 + qt & q \\ 1^{3} & t^{3} & t + t^{2} & 1 \end{aligned}
4  31  2^{2}  21^{2}  1^{4}
4 1 q + q^{2} + q^{3} q^{2} + q^{4} q^{3} + q^{4} + q^{5} q^{6}
31 t 1 + qt + q^{2}t q + q^{2}t q + q^{2} + q^{3}t q^{3}
2^{2} t^{2} t + qt + qt^{2} 1 + q^{2}t^{2} q + qt + q^{2}t q^{2}
21^{2} t^{3} t + t^{2} + qt^{3} t + qt^{2} 1 + qt + qt^{2} q
1^{4} t^{6} t^{3} + t^{4} + t^{5} t^{2} + t^{4} t + t^{2} + t^{3} 1
5  41  32
5 1 q + q^{2} + q^{3} + q^{4} q^{2} + q^{3} + q^{4} + q^{5} + q^{6}
41 t 1 + qt + q^{2}t + q^{3}t q + q^{2} + q^{2}t + q^{3}t + q^{4}t
32 t^{2} t + qt + qt^{2} + q^{2}t^{2} 1 + qt + q^{2}t + q^{2}t^{2} + q^{3}t^{2}
31^{2} t^{3} t + t^{2} + qt^{3} + q^{2}t^{3} t + qt + qt^{2} + q^{2}t^{2} + q^{2}t^{3}
2^{2}1 t^{4} t^{2} + t^{3} + qt^{3} + qt^{4} t + t^{2} + qt^{2} + qt^{3} + q^{2}t^{4}
21^{3} t^{6} t^{3} + t^{4} + t^{5} + qt^{6} t^{2} + t^{3} + t^{4} + qt^{4} + qt^{5}
1^{5} t^{10} t^{6} + t^{7} + t^{8} + t^{9}  t^{4} + t^{5} + t^{6} + t^{7} + t^{8}
31^{2}  2^{2}1  21^{3}  1^{5}
5  q^{3} + q^{4} + 2q^{5} + q^{6} + q^{7}  q^{4} + q^{5} + q^{6} + q^{7} + q^{8} q^{6} + q^{7} + q^{8} + q^{9} q^{10}
41 q + q^{2} + q^{3} + q^{3}t + q^{4}t + q^{5}t q^{2} + q^{3} + q^{4} + q^{4}t + q^{5}t q^{3} + q^{4} + q^{5} + q^{6}t q^{6}
32 q + qt + 2q^{2}t + q^{3}t + q^{3}t^{2} q + q^{2} + q^{2}t + q^{3}t + q^{4}t^{2} q^{2} + q^{3} + q^{3}t + q^{4}t q^{4}
31^{2} 1 + qt + q^{2}t + qt^{2} + q^{2}t^{2} + q^{3}t^{3} q + qt + q^{2}t + q^{2}t^{2} + q^{3}t^{2} q + q^{2} + q^{3}t + q^{3}t^{2} q^{3}
2^{2}1 t + qt + 2qt^{2} + qt^{3} + q^{2}t^{3} 1 + qt + qt^{2} + q^{2}t^{2} + q^{2}t^{3} q + qt + q^{2}t + q^{2}t^{2} q^{2}
21^{3} t + t^{2} + t^{3} + qt^{3} + qt^{4} + qt^{5} t + t^{2} + qt^{2} + qt^{3} + qt^{4} 1 + qt + qt^{2} + qt^{3} q
1^{5}  t^{3} + t^{4} + 2t^{5} + t^{6} + t^{7}  t^{2} + t^{3} + t^{4} + t^{5} + t^{6}  t + t^{2} + t^{3} + t^{4} 1

6  51  42
6 1  q + q^{2} + q^{3} + q^{4} + q^{5}  q^{2} + q^{3} + 2q^{4} + q^{5} + 2q^{6} + q^{7} + q^{8}
51 t 1 + qt + q^{2}t + q^{3}t + q^{4}t  q + q^{2} + q^{3} + q^{2}t + q^{3}t + 2q^{4}t + q^{5}t + q^{6}t
42 t^{2} t + qt + qt^{2} + q^{2}t^{2} + q^{3}t^{2}  1 + qt + 2q^{2}t + q^{3}t + q^{2}t^{2} + q^{3}t^{2} + 2q^{4}t^{2}
41^{2} t^{3} t + t^{2} + qt^{3} + q^{2}t^{3} + q^{3}t^{3} t + qt + q^{2}t + qt^{2} + q^{2}t^{2} + q^{3}t^{2} + q^{2}t^{3} + q^{3}t^{3} + q^{4}t^{3}
3^{2} t^{3} t^{2} + qt^{2} + q^{2}t^{2} + qt^{3} + q^{2}t^{3} t + qt + q^{2}t + qt^{2} + q^{2}t^{2} + q^{3}t^{2} + q^{2}t^{3} + q^{3}t^{3} + q^{4}t^{3}
321 t^{4} t^{2} + t^{3} + qt^{3} + qt^{4} + q^{2}t^{4}  t + t^{2} + 2qt^{2} + qt^{3} + 2q^{2}t^{3} + q^{2}t^{4} + q^{3}t^{4}
31^{3} t^{6} t^{3} + t^{4} + t^{5} + qt^{6} + q^{2}t^{6} t^{2} + t^{3} + qt^{3} + t^{4} + qt^{4} + q^{2}t^{4} + qt^{5} + q^{2}t^{5} + q^{2}t^{6}
2^{3} t^{6} t^{4} + qt^{4} + t^{5} + qt^{5} + qt^{6} t^{2} + t^{3} + qt^{3} + t^{4} + qt^{4} + q^{2}t^{4} + qt^{5} + q^{2}t^{5} + q^{2}t^{6}
2^{2}1^{2} t^{7} t^{4} + t^{5} + t^{6} + qt^{6} + qt^{7}  2t^{3} + t^{4} + qt^{4} + t^{5} + 2qt^{5} + qt^{6} + q^{2}t^{7}
21^{4} t^{10} t^{6} + t^{7} + t^{8} + t^{9} + qt^{10}  t^{4} + t^{5} + 2t^{6} + t^{7} + qt^{7} + t^{8} + qt^{8} + qt^{9}
1^{6} t^{15} t^{10} + t^{11} + t^{12} + t^{13} + t^{14}  t^{7} + t^{8} + 2t^{9} + t^{10} + 2t^{11} + t^{12} + t^{13}
41^{2}  3^{2}
6  q^{3} + q^{4} + 2q^{5} + 2q^{6} + 2q^{7} + q^{8} + q^{9}  q^{3} + q^{5} + q^{6} + q^{7} + q^{9}
51  q + q^{2} + q^{3} + q^{4} + q^{3}t + q^{4}t + 2q^{5}t + q^{6}t + q^{7}t  q^{2} + q^{4} + q^{3}t + q^{5}t + q^{6}t
42  q + qt + 2q^{2}t + 2q^{3}t + q^{4}t + q^{3}t^{2} + q^{4}t^{2} + q^{5}t^{2}  q + q^{2}t + q^{3}t + q^{3}t^{2} + q^{5}t^{2}
41^{2} 1 + qt + q^{2}t + q^{3}t + qt^{2} + q^{2}t^{2} + q^{3}t^{2} + q^{3}t^{3} + q^{4}t^{3} + q^{5}t^{3} qt + q^{2}t + q^{2}t^{2} + q^{4}t^{2} + q^{3}t^{3}
3^{2}  qt + q^{2}t + q^{3}t + qt^{2} + 2q^{2}t^{2} + 2q^{3}t^{2} + q^{4}t^{2} + q^{3}t^{3}  1 + q^{2}t^{2} + q^{3}t^{2} + q^{4}t^{2} + q^{3}t^{3}
321  t + qt + 2qt^{2} + q^{2}t^{2} + qt^{3} + 2q^{2}t^{3} + q^{3}t^{3} + q^{3}t^{4}  t + qt^{2} + q^{2}t^{2} + q^{2}t^{3} + q^{3}t^{4}
31^{3} t + t^{2} + t^{3} + qt^{3} + q^{2}t^{3} + qt^{4} + q^{2}t^{4} + qt^{5} + q^{2}t^{5} + q^{3}t^{6} qt^{2} + t^{3} + qt^{4} + q^{2}t^{4} + q^{2}t^{5}
2^{3}  qt^{2} + t^{3} + 2qt^{3} + q^{2}t^{3} + 2qt^{4} + q^{2}t^{4} + qt^{5} + q^{2}t^{5}  qt^{2} + t^{3} + qt^{3} + qt^{4} + q^{3}t^{6}
2^{2}1^{2}  t^{2} + t^{3} + qt^{3} + t^{4} + 2qt^{4} + 2qt^{5} + qt^{6} + q^{2}t^{6}  t^{2} + t^{4} + qt^{4} + qt^{5} + q^{2}t^{6}
21^{4}  t^{3} + t^{4} + 2t^{5} + t^{6} + qt^{6} + t^{7} + qt^{7} + qt^{8} + qt^{9}  t^{4} + t^{5} + qt^{6} + t^{7} + qt^{8}
1^{6}  t^{6} + t^{7} + 2t^{8} + 2t^{9} + 2t^{10} + t^{11} + t^{12}  t^{6} + t^{8} + t^{9} + t^{10} + t^{12}

321

6  q^{4} + 2q^{5} + 2q^{6} + 3q^{7} + 3q^{8} + 2q^{9} + 2q^{10} + q^{11}
51  q^{2} + 2q^{3} + 2q^{4} + 2q^{5} + q^{6} + q^{4}t + 2q^{5}t + 2q^{6}t + 2q^{7}t + q^{8}t
42  q + 2q^{2} + q^{3} + q^{2}t + 3q^{3}t + 3q^{4}t + q^{5}t + q^{4}t^{2} + 2q^{5}t^{2} + q^{6}t^{2}
41^{2} q + q^{2} + qt + 2q^{2}t + 2q^{3}t + q^{4}t + q^{2}t^{2} + 2q^{3}t^{2} + 2q^{4}t^{2} + q^{5}t^{2} + q^{4}t^{3} + q^{5}t^{3}
3^{2} q + q^{2} + qt + 2q^{2}t + 2q^{3}t + q^{4}t + q^{2}t^{2} + 2q^{3}t^{2} + 2q^{4}t^{2} + q^{5}t^{2} + q^{4}t^{3} + q^{5}t^{3}
321  1 + 3qt + q^{2}t + qt^{2} + 4q^{2}t^{2} + q^{3}t^{2} + q^{2}t^{3} + 3q^{3}t^{3} + q^{4}t^{4}
31^{3} t + qt + t^{2} + 2qt^{2} + q^{2}t^{2} + 2qt^{3} + 2q^{2}t^{3} + qt^{4} + 2q^{2}t^{4} + q^{3}t^{4} + q^{2}t^{5} + q^{3}t^{5}
2^{3} t + qt + t^{2} + 2qt^{2} + q^{2}t^{2} + 2qt^{3} + 2q^{2}t^{3} + qt^{4} + 2q^{2}t^{4} + q^{3}t^{4} + q^{2}t^{5} + q^{3}t^{5}
2^{2}1^{2}  t + 2t^{2} + qt^{2} + t^{3} + 3qt^{3} + 3qt^{4} + q^{2}t^{4} + qt^{5} + 2q^{2}t^{5} + q^{2}t^{6}
21^{4}  t^{2} + 2t^{3} + 2t^{4} + qt^{4} + 2t^{5} + 2qt^{5} + t^{6} + 2qt^{6} + 2qt^{7} + qt^{8}
1^{6}  t^{4} + 2t^{5} + 2t^{6} + 3t^{7} + 3t^{8} + 2t^{9} + 2t^{10} + t^{11}

\begin{aligned}& 31^{3} & 2^{3} \\ 6 & q^{6} + q^{7} + 2q^{8} + 2q^{9} + 2q^{10} + q^{11} + q^{12} & q^{6} + q^{8} + q^{9} + q^{10} + q^{12} \\ 51 & q^{3} + q^{4} + 2q^{5} + q^{6} + q^{7} + q^{6}t + q^{7}t + q^{8}t + q^{9}t & q^{4} + q^{5} + q^{7} + q^{6}t + q^{8}t \\ 42 & q^{2} + q^{3} + q^{4} + q^{3}t + 2q^{4}t + 2q^{5}t + q^{6}t + q^{6}t^{2} & q^{2} + q^{4} + q^{4}t + q^{5}t + q^{6}t^{2} \\ 41^{2} & q + q^{2} + q^{3} + q^{3}t + q^{4}t + q^{5}t + q^{3}t^{2} + q^{4}t^{2} + q^{5}t^{2} + q^{6}t^{3} & q^{3} + q^{2}t + q^{4}t + q^{4}t^{2} + q^{5}t^{2} \\ 3^{2} & q^{3} + q^{2}t + 2q^{3}t + 2q^{4}t + q^{5}t + q^{3}t^{2} + q^{4}t^{2} + q^{5}t^{2} & q^{3} + q^{2}t + q^{3}t + q^{4}t + q^{6}t^{3} \\ 321 & q + qt + 2q^{2}t + q^{3}t + q^{2}t^{2} + 2q^{3}t^{2} + q^{3}t^{3} + q^{4}t^{3} & q + q^{2}t + q^{2}t^{2} + q^{3}t^{2} + q^{4}t^{3} \\ 31^{3} & 1 + qt + q^{2}t + qt^{2} + q^{2}t^{2} + qt^{3} + q^{2}t^{3} + q^{3}t^{3} + q^{3}t^{4} + q^{3}t^{5} & qt + qt^{2} + q^{2}t^{2} + q^{3}t^{3} + q^{2}t^{4} \\ 2^{3} & qt + q^{2}t + qt^{2} + 2q^{2}t^{2} + qt^{3} + 2q^{2}t^{3} + q^{3}t^{3} + q^{2}t^{4} & 1 + q^{2}t^{2} + q^{2}t^{3} + q^{3}t^{3} + q^{2}t^{4} \\ 2^{2}1^{2} & t + qt + 2qt^{2} + 2qt^{3} + q^{2}t^{3} + qt^{4} + q^{2}t^{4} + q^{2}t^{5} & t + qt^{2} + qt^{3} + q^{2}t^{3} + q^{2}t^{5} \\ 21^{4} & t + t^{2} + t^{3} + qt^{3} + t^{4} + qt^{4} + 2qt^{5} + qt^{6} + qt^{7} & t^{2} + qt^{3} + t^{4} + qt^{5} + qt^{6} \\ 1^{6} & t^{3} + t^{4} + 2t^{5} + 2t^{6} + 2t^{7} + t^{8} + t^{9} & t^{3} + t^{5} + t^{6} + t^{7} + t^{9} \end{aligned} \begin{aligned}& 2^{2}1^{2} & 21^{4} & 1^{6} \\ 6 & q^{7} + q^{8} + 2q^{9} + q^{10} + 2q^{11} + q^{12} + q^{13} & q^{10} + q^{11} + q^{12} + q^{13} + q^{14} & q^{15} \\ 51 & q^{4} + q^{5} + 2q^{6} + q^{7} + q^{8} + q^{7}t + q^{8}t + q^{9}t & q^{6} + q^{7} + q^{8} + q^{9} + q^{10}t & q^{10} \\ 42 & 2q^{3} + q^{4} + q^{5} + q^{4}t + 2q^{5}t + q^{6}t + q^{7}t^{2} & q^{4} + q^{5} + q^{6} + q^{6}t + q^{7}t & q^{7} \\ 41^{2} & q^{2} + q^{3} + q^{4} + q^{3}t + q^{4}t + q^{5}t + q^{4}t^{2} + q^{5}t^{2} + q^{6}t^{2} & q^{3} + q^{4} + q^{5} + q^{6}t + q^{6}t^{2} & q^{6} \\ 3^{2} & q^{2} + q^{3} + q^{4} + q^{3}t + q^{4}t + q^{5}t + q^{4}t^{2} + q^{5}t^{2} + q^{6}t^{2} & q^{4} + q^{5} + q^{4}t + q^{5}t + q^{6}t & q^{6} \\ 321 & q + q^{2} + 2q^{2}t + q^{3}t + 2q^{3}t^{2} + q^{4}t^{2} + q^{4}t^{3} & q^{2} + q^{3} + q^{3}t + q^{4}t + q^{4}t^{2} & q^{4} \\ 31^{3} & q + qt + q^{2}t + qt^{2} + q^{2}t^{2} + q^{3}t^{2} + q^{2}t^{3} + q^{3}t^{3} + q^{3}t^{4} & q + q^{2} + q^{3}t + q^{3}t^{2} + q^{3}t^{3} & q^{3} \\ 2^{3} & q + qt + q^{2}t + qt^{2} + q^{2}t^{2} + q^{3}t^{2} + q^{2}t^{3} + q^{3}t^{3} + q^{3}t^{4} & q^{2} + q^{2}t + q^{3}t + q^{2}t^{2} + q^{3}t^{2} & q^{3} \\ 2^{2}1^{2} & 1 + qt + 2qt^{2} + q^{2}t^{2} + qt^{3} + q^{2}t^{3} + 2q^{2}t^{4} & q + qt + q^{2}t + q^{2}t^{2} + q^{2}t^{3} & q^{2} \\ 21^{4} & t + t^{2} + qt^{2} + t^{3} + qt^{3} + 2qt^{4} + qt^{5} + qt^{6} & 1 + qt + qt^{2} + qt^{3} + qt^{4} & q \\ 1^{6} & t^{2} + t^{3} + 2t^{4} + t^{5} + 2t^{6} + t^{7} + t^{8} & t + t^{2} + t^{3} + t^{4} + t^{5} & 1 \end{aligned}