Determinantal expressions for Macdonald polynomials

L. Lapointe, A. Lascoux, and J. Morse

Institut Gaspard Monge, C.N.R.S. Université de Marne-la-Vallée, 5 Bd Descartes, Champs sur Marne, 77454 Marne La Vallée Cedex, France

ABSTRACT: We show that the action of classical operators associated to the Macdonald polynomials on the basis of Schur functions, S_{λ} [X(t − 1)/(q − 1)], can be reduced to addition in λ−rings. This provides explicit formulas for the Macdonald polynomials expanded in this basis as well as in the ordinary Schur basis, S_{λ}[X], and the monomial basis, m_{λ}[X].

Introduction

Important developments in the theory of symmetric functions rely on the use of the Macdonald polynomial basis, {J_{λ}(X; q, t)}_{λ} [M1]. This basis specializes to several fundamental bases including the Schur, Hall-Littlewood, Zonal, and Jack. It has been conjectured that the Macdonald polynomials occur naturally in representation theory of the symmetric group [GH]. It is also known that these polynomials are eigenfunctions of a family of commuting difference operators with significance in many-body physics [RS].

We first reformulate the Macdonald operators in terms of divided differences, operators that act naturally on the Schur function basis. This enables us to show that the action of the Macdonald operators on the modified Schur basis, S_{λ}[X(t − 1)/(q − 1)], primarily amounts to addition in λ-rings (Theorem 2.1). The action provides a determinantal expression for the Macdonald polynomials expanded in this Schur basis. By involution this expression can be converted into an explicit formula for the Macdonald polynomials in the usual Schur basis, S_{λ}[X], and in the monomial basis, m_{λ}[X] (Theorems 3.1 and 3.3). Further, it is known that the Macdonald polynomials can be built recursively using ‘creation operators’. We reformulate these operators in a manner similar to our expression for the Macdonald operators enabling us to give a simple expression for their action on the modified Schur functions (Theorem 4.2).

1. Background

To begin, we clarify some notation. Sym refers to the ring of symmetric functions.

A partition will be defined as a weakly decreasing sequence of non-negative integers, λ = (λ_{1}, …, λ_{n}) with ℓ(λ), the number of parts of λ and λ′ the conjugate to λ. To a partition λ, there corresponds a Ferrers’ diagram with λ_{i} lattice squares in the i^{th} row, from the bottom to top. For each square s in the diagram of λ, we define a(s) and l(s) to be the number of squares respectively to the north and east of s. The order of λ is the sum of the parts of λ, denoted |λ|. The dominance order on λ is defined such that for |λ| = |μ|, λ ≤ μ when λ_{1} + ⋯ + λ_{i} ≤ μ_{1} + ⋯ + μ_{i} for all i.

We shall use λ-rings, needing only the formal ring of symmetric functions Sym to act on the ring of rational functions in x_{1}, …, x_{n}, q, t, with coefficients in ℝ. The ring Sym is generated by power sums Ψ_{i}, i = 1, 2, 3 . . .. The action of Ψ_{i} on a rational function ∑_{α} c_{α}u_{α}/ ∑_{β} d_{β}v_{β} is by definition

Ψ_{i} [\frac{∑_{α} c_{α}u_{α}}{∑_{β} d_{β}v_{β}} ] = \frac{∑_{α} c_{α}u_{α}^{i}}{∑_{β} d_{β}v_{β}^{i}} ,  1.1

with c_{α}, d_{β} ∈ ℝ and u_{α}, v_{β} monomials in x_{1}, …, x_{n}, q, t. Since any symmetric function is uniquely expressed in terms of the power sums, formula 1.1 extends to an action of Sym on rational functions. In particular, a symmetric function f(X) of X = {x_{1}, …, x_{n}} can be denoted f[x_{1} + ⋯ + x_{n}].

The Schur function S_{λ}, with Y, Z ∈ ℚ[x_{1}, …, x_{n}, q, t], is such that

S_{λ}[Y ± Z] = ∑_{μ} S_{λ/μ}[Y]S_{μ}[±Z] ,  1.2

where S_{μ}[−Z] = (−1)^{|μ|}S_{μ′}[Z], with μ′ the partition conjugate to μ. Schur functions will be considered as determinants of complete functions:

S_{λ}[X] = det | \begin{matrix}S_{λ_{1}}[X] & S_{λ_{1}+1}[X] & ⋯ & S_{λ_{1}+n−1}[X] \\ S_{λ_{2}−1}[X] & S_{λ_{2}}[X] & ⋯ & S_{λ_{2}+n−2}[X] \\ ⋮ & ⋮ & ... & ⋮ \\ S_{λ_{n}−n+1}[X] & S_{λ_{n}−n+2}[X] & ⋯ & S_{λ_{n}}[X] \end{matrix} | ,  1.3

as will skew Schur functions:

S_{λ/μ}[X] = det | \begin{matrix}S_{λ_{1}−μ_{1}}[X] & S_{λ_{1}+1−μ_{2}}[X] & ⋯ & S_{λ_{1}+n−1−μ_{n}}[X] \\ S_{λ_{2}−1−μ_{1}}[X] & S_{λ_{2}−μ_{2}}[X] & ⋯ & S_{λ_{2}+n−2−μ_{n}}[X] \\ ⋮ & ⋮ & ... & ⋮ \\ S_{λ_{n}−n+1−μ_{1}}[X] & S_{λ_{n}−n+2−μ_{2}}[X] & ⋯ & S_{λ_{n}−μ_{n}}[X] \end{matrix} | .  1.4

In these expressions, S_{i} = 0 for i < 0.

We denote X = x_{1} + ⋯ + x_{n} and a partial alphabet, X_{I} = x_{i_{1}} + ⋯ + x_{i_{k}}, for I = {i_{1}, …, i_{k}}. The complement of X_{I} in X will be X_{I}^{c}. For I = {1, …, k}, we have X_{k} = x_{1} + ⋯ + x_{k}. Two particular elements, X^{tq} := X(t − 1)/(q − 1) and X^{t} := X(t − 1) will be used throughout the paper.

We shall need the divided differences ∂_{μ}, indexed by elements μ of the symmetric group 𝔖(n). In particular, if μ is the simple transposition σ_{i}, we have

∂_{i} : f ⟶ (f − σ_{i}f)/(x_{i} − x_{i+1}) .  1.5

The divided differences satisfy the Coxeter relations [M2]

∂_{i}∂_{j} = ∂_{j}∂_{i}, |i − j| > 1;  ∂_{i}∂_{i+1}∂_{i} = ∂_{i+1}∂_{i}∂_{i+1} ,  1.6

which imply that ∂_{μ} can be written as the product of elementary operators ∂_{i} corresponding to any reduced decomposition of μ. It should be noted that symmetric functions in x_{i}, x_{i+1} are scalar with respect to ∂_{i} and that ∂_{i} · 1 = 0. Consequently, symmetric functions in x_{1}, …, x_{n} commute with all ∂_{μ}.

We have that ∂_{n−1} ⋯ ∂_{1} is the Lagrange operator [L1],

∂_{n−1} ⋯ ∂_{1} : f ∈ Sym(1|n − 1) ⟶ ∑_{i} \frac{f(x_{i}, x_{i}^{c})}{R(x_{i}, x_{i}^{c})} ∈ Sym(n) ,  1.7

or more generally, ∂_{(k|n−k)} = (∂_{n−k} ⋯ ∂_{1}) ⋯ (∂_{n−2} ⋯ ∂_{k−1})(∂_{n−1} ⋯ ∂_{k}) is the Sylvester operator [L1],

∂_{(k|n−k)} : f ∈ Sym(k|n − k) ⟶ ∑_{|I|=k} \frac{f(X_{I}, X_{I}^{c})}{R(X_{I}, X_{I}^{c})} ∈ Sym(n) ,  1.8

where Sym(k|n − k) is the space of functions symmetrical in x_{1}, …, x_{k} and in x_{k+1}, …, x_{n}, and where R(X, Y) is the resultant of two alphabets X and Y ;

R(X, Y) = ∏_{x∈X,y∈Y} (x − y) .  1.9

These operators may be used to define the Euler-Poincaré characteristic χ. In fact, in his study of Riemann-Roch theorem, Hirzebruch defined a ‘χ_{y}’-characteristic, or ‘χ_{y}’-genus [H],[HBJ]. We shall use the one corresponding to a relative flag manifold, changing y into 1/t . We define the following operators on polynomials in x_{1}, …, x_{n},

χ_{(1|n−k)}^{(k)} = ∂_{n−1} ⋯ ∂_{k} R(x_{k}, X_{k}^{c}/t) ,  1.10

considered as a composition of multiplication by the resultant followed by a sequence of divided differences. Note that the image of 1 is (1 + ⋯ + t^{−n+k}) and that the superscript refers to the variable at which the symmetrization begins. We shall also use

χ_{(k|n−k)} = ∂_{(k|n−k)} R(X_{k}, X_{k}^{c}/t)  1.11

and

χ_{ω(k)} = ∂_{ω(k)} ∏_{1≤i<j≤k}(x_{i} − x_{j}/t) ,  1.12

where ω(k) is the longest permutation in 𝔖(k) and χ_{ω(k)} · 1 = k_{t}! = t^{−k(k−1)/2}(t)_{k}/(1 − t)^{k} , where (t)_{k} = (1 − t) ⋯ (1 − t^{k}). To compute with ∂_{ω(k)}, we shall use the factorization ∂_{ω(k)} = (∂_{k−1})(∂_{k−2}∂_{k−1}) ⋯ (∂_{1} ⋯ ∂_{k−1}).

When operating on elements of Sym(ℓ|k − ℓ), one has

χ_{ω(k)} f = ℓ_{t}! (k − ℓ)_{t}! χ_{(ℓ|k−ℓ)} f  ∀ f ∈ Sym(ℓ|k − ℓ) .  1.13

We shall also need the factorization

χ_{(k|n−k)}χ_{(1|k−1)}^{(1)} ⋯ χ_{(1|2)}^{(k−2)}χ_{(1|1)}^{(k−1)} = χ_{(1|n−1)}^{(1)} ⋯ χ_{(1|n−k+1)}^{(k−1)}χ_{(1|n−k)}^{(k)}  1.14

and the following lemma (proved in the appendix).

Lemma 1.1 For an alphabet C, symmetric in x_{1}, …, x_{n}, and c(k) = t^{−1} + ⋯ + t^{−n+k} , one has

χ_{(1|n−k)}^{(k)} S_{j}[C + X_{k}^{t}] = c(k) S_{j}[C + X_{k−1}^{t}] + S_{j}[C + X^{t}] .  1.15

2. Macdonald Operators

This section presents the action of the Macdonald operators on S_{λ}[X^{tq}] as a sum of determinants. The Macdonald operators [M1] are defined as

M_{k} = t^{\binom{k}{2}} ∑_{|I|=k} R(tX_{I}, X_{I}^{c}) Θ_{i_{k}} ⋯ Θ_{i_{1}}  k = 1, 2, 3, …,  2.1

where the sum is over all k-element subsets I = (i_{1}, …, i_{k}) of {1, …, n} and

Θ_{l} : f[X] ⟶ f[X + (q − 1)x_{l}] .  2.2

It is more convenient to use a reformulation for these operators in terms of divided differences,

\begin{aligned}M_{k} & = t^{\binom{k}{2}+k(n−k)}χ_{(k|n−k)}Θ_{k} ⋯ Θ_{1} \\ & = t^{\binom{k}{2}} ∂_{(k|n−k)} R(tX_{k}, X_{k}^{c}) Θ_{k} ⋯ Θ_{1} , \end{aligned}  2.3

thanks to 1.8.

We shall express the action of M_{k} on S_{λ}[X^{tq}] with formal operators, [ \begin{matrix}a_{1} & a_{2} & ⋯ & a_{n} \\ b_{1} & b_{2} & ⋯ & b_{n} \end{matrix} ], acting on the columns of an n × n matrix of Schur functions such that for all elements in any column ℓ , S_{β}[Y] → S_{β+a_{ℓ}}[Y + b_{ℓ}]. For example;

[ \begin{matrix}0 & 30 & −5 \\ b_{1} & Y & 0 \end{matrix} ] : ( \begin{matrix}S_{11}[A] & S_{4}[B] & S_{8}[C] \\ S_{0}[A] & S_{2}[B] & S_{13}[C] \\ S_{1}[A] & S_{2}[B] & S_{3}[C] \end{matrix} ) → ( \begin{matrix}S_{11}[A + b_{1}] & S_{34}[B + Y] & S_{3}[C] \\ S_{0}[A + b_{1}] & S_{32}[B + Y] & S_{8}[C] \\ S_{1}[A + b_{1}] & S_{32}[B + Y] & S_{−2}[C] \end{matrix} ) .

2.4

Theorem 2.1 The action of the Macdonald operators on S_{λ}[X^{tq}], with ℓ(λ) ≤ n, can be expressed as

M_{k}S_{λ}[X^{tq}] = ∑_{1≤i_{1}<···<i_{k}≤n} t^{(n−i_{1})+···+(n−i_{k})} [ \begin{matrix}0 & ⋯ & 0 & ⋯ & 0 & ⋯ & 0 \\ 0 & ⋯ & X^{t} & ⋯ & X^{t} & ⋯ & 0 \end{matrix} ] S_{λ}[X^{tq}] ,

2.5

where X^{t} occurs in columns i_{1}, …, i_{k}.

For example, if n = 2,

M_{1}S_{(5,2)} = t | \begin{matrix}S_{5}[X^{tq} + X^{t}] & S_{6}[X^{tq}] \\ S_{1}[X^{tq} + X^{t}] & S_{2}[X^{tq}] \end{matrix} | + | \begin{matrix}S_{5}[X^{tq}] & S_{6}[X^{tq} + X^{t}] \\ S_{1}[X^{tq}] & S_{2}[X^{tq} + X^{t}] \end{matrix} | ,  2.6

or more concisely since X^{tq} + X^{t} = qX^{tq}, using S_{i} for S_{i}[X^{tq}] ,

M_{1}S_{(5,2)} = t | \begin{matrix}q^{5}S_{5} & S_{6} \\ qS_{1} & S_{2} \end{matrix} | + | \begin{matrix}S_{5} & q^{6}S_{6} \\ S_{1} & q^{2}S_{2} \end{matrix} | .  2.7

Proof of Theorem 2.1 The Macdonald operators,

M_{k} = t^{\binom{k}{2}} ∂_{(k|n−k)} R(tX_{k}, X_{k}^{c}) Θ_{k} ⋯ Θ_{1} ,  2.8

act first with a product of Θ_{i}, defined to replace x_{i} by qx_{i}. However, this q-deformation is actually a t-deformation when acting on f[X^{tq}] since

Θ_{ℓ} : f[X^{tq}] → f [(X + (q − 1)x_{ℓ}) · \frac{t − 1}{q − 1}] = f[X^{tq} + (t − 1)x_{ℓ}] .  2.9

Now for Y = X^{tq} − X, if we define a formal operator Θ_{ℓ}^{t} that sends x_{i} → x_{i} and x_{ℓ} → tx_{ℓ} while leaving Y invariant; i.e. Y + X → Y + X + (t − 1)x_{ℓ}, we obtain

R(tX_{k}, X_{k}^{c}) Θ_{k} ⋯ Θ_{1} S_{λ}[Y + X] = Θ_{k}^{t} ⋯ Θ_{1}^{t} R(X_{k}, X_{k}^{c}) S_{λ}[Y + X] ,  2.10

where Θ_{i} acts on both Y and X on the left hand side of this expression. Further, we have the following determinantal expression for R(X_{k}, X_{k}^{c}) S_{λ}[Y + X];

\begin{matrix}R(X_{k}, X_{k}^{c}) & S_{λ}[Y + X] = \\ |S_{α_{i}−k+n}[Y + X_{k}] ⋯ S_{α_{i}−1+n}[Y + X_{k}]S_{α_{i}+k}[Y + X_{k}^{c}] ⋯ S_{α_{i}+n−1}[Y + X_{k}^{c}]|_{i=1···n} , \end{matrix}

2.11

where α_{i} → λ_{i} − i + 1 (to obtain this expression see the appendix). Since Θ_{k}^{t} ⋯ Θ_{1}^{t} acts exclusively on the first k columns of this determinant, we have that

\begin{matrix}t^{−\binom{k}{2}}M_{k}S_{λ}[Y + X] = \\ ∂_{(k|n−k)}|S_{α_{i}−k+n}[Y + tX_{k}] ⋯ S_{α_{i}−1+n}[Y + tX_{k}]S_{α_{i}+k}[Y + X_{k}^{c}] ⋯ S_{α_{i}+n−1}[Y + X_{k}^{c}]|_{i=1..n} . \end{matrix}

2.12

As it remains to show that this is equivalent to expression 2.5, we proceed by giving the action of ∂_{(k|n−k)} on products of arbitrary minors of order k on the first k columns and of order n − k on the last n − k columns of this determinant. The action of ∂_{(k|n−k)} on the whole determinant will then follow by linearity.

Lemma 2.2 For α = (α_{1}, …, α_{k}), β = (β_{1}, …, β_{n−k}), and A, B invariant under 𝔖(n), we have

\begin{matrix}∂_{(k|n−k)}( & S_{α}[A + tX_{k}]S_{β}[B + X_{k}^{c}]) = t^{k(n−k)} ∑_{I}(−1/t)^{|I|} \\ × ([ \begin{matrix}−n + k + i_{k} & ⋯ & −n + k + i_{1} \\ tX & ⋯ & tX \end{matrix} ] S_{α}[A]) ([ \begin{matrix}−i′_{1} & ⋯ & −i′_{n−k} \\ X & ⋯ & X \end{matrix} ] S_{β}[B]) , \end{matrix}

2.13

where I = (i_{1}, …, i_{k}) and (i′_{1}, …, i′_{n−k}) are pairs of conjugate partitions.

Proof of lemma Define the elements A′ = A + tX and B′ = B + X. The product S_{α}[A + tX_{k}]S_{β}[B + X_{k}^{c}] can be expanded, thanks to 1.2, into products of Schur functions in −X_{k}^{c} and X_{k} (with coefficients in A′ and B′),

\begin{aligned}S_{α}[A + tX_{k}] S_{β}[B + X_{k}^{c}] & = S_{α}[A′ − tX_{k}^{c}] S_{β}[B′ − X_{k}] \\ & = ∑_{I,J} S_{α/I}[A′]t^{|I|}S_{I}[−X_{k}^{c}]S_{β/J}[B′](−1)^{|J|}S_{J′}[X_{k}] , \end{aligned}  2.14

where I and J are partitions and J′ is conjugate to J. Since ℓ(I), ℓ(J′) ≤ k and ℓ(I′), ℓ(J) ≤ n − k, where I′ is conjugate to I, we have I, J′ ⊆ (n − k, …, n − k). Thus the following property describing two adjoint bases of Sym(k, n − k) as a free module over Sym(n) ([L2], Corollary 2.2),

∂_{(k|n−k)}(S_{I}[−X_{k}^{c}] S_{J}[X_{k}]) = \{ \begin{matrix}1 & if I = (n − k − J_{k}, …, n − k − J_{1}) \\ 0 & else \end{matrix} ,  2.15

gives

∂_{(k|n−k)}(S_{α}[A + tX_{k}] S_{β}[B + X_{k}^{c}]) = t^{k(n−k)} ∑_{J}(−1/t)^{|J|}S_{α/(n−k−J′_{k},...,n−k−J′_{1})}[A′]S_{β/J}[B′].

2.16

The lemma is proved by letting J → I′ in the last expression and using the definition 1.4 of skew Schur functions.

Now applying Lemma 2.2 directly to expression 2.12, we have

M_{k}S_{λ}[Y + X]

= ∑_{I}(−1/t)^{|I|}t^{k(n−k)+k(k−1)/2} [ \begin{matrix}−n + k + i_{k} & ⋯ & −n + k + i_{1} & −i′_{1} & ⋯ & −i′_{n−k} \\ tX & ⋯ & tX & X & ⋯ & X \end{matrix} ]

× |S_{α_{i}−k+n}[Y] ⋯ S_{α_{i}−1+n}[Y]S_{α_{i}+k}[Y] ⋯ S_{α_{i}+n−1}[Y]|_{i=1..n}

= ∑_{I}(−1/t)^{|I|}t^{k(n−k)+k(k−1)/2} [ \begin{matrix}i_{k} & ⋯ & i_{1} + k − 1 & −i′_{1} + k & ⋯ & −i′_{n−k} + n − 1 \\ tX & ⋯ & tX & X & ⋯ & X \end{matrix} ]

× |S_{α_{i}}[Y] ⋯ S_{α_{i}}[Y]S_{α_{i}}[Y] ⋯ S_{α_{i}}[Y]|_{i=1..n} .

2.17

For ρ = (0, …, n − 1) and λ^{ω} the reverse reading of a partition λ, we have from [M1] that ρ + λ^{ω} ∪ −λ′ = σρ for some permutation σ. The top row of the formal operator in 2.17, ρ + (i_{k}, …, i_{1}, −i′_{1}, …, −i′_{n−k}), may thus be rearranged to ρ. The rearrangement sends column ℓ + 1, containing tX and corresponding to index i_{k−ℓ} + ℓ, to column i_{k−ℓ} + ℓ + 1. This requires i_{k−ℓ} commutations with X-columns inducing a (−1)^{i_{k−ℓ}}. Proceeding on all tX-columns yields

M_{k} S_{λ}[Y + X] = ∑_{I} t^{k(n−k)+k(k−1)/2−|I|} [ \begin{matrix}0 & ⋯ & 0 & ⋯ & 0 & ⋯ & 0 \\ X & ⋯ & tX & ⋯ & tX & ⋯ & X \end{matrix} ] S_{λ}[Y] , 2.18

where the tX are in columns i_{k} + 1, …, i_{1} + k. Finally, the substitution Y = X^{tq} − X with the shift i_{k} + 1 → i_{k}, …, i_{1} + k → i_{1} gives Theorem 2.1 .

Remark: The image of S_{λ}[X^{tq}] under the Macdonald operator M_{k} in the basis of products of complete functions S^{μ}[X^{tq}] = S_{μ_{1}}[X^{tq}]S_{μ_{2}}[X^{tq}] … can be derived from 2.5. However, there exists a scalar product for which the S^{μ}[X^{tq}] basis is adjoint to the m_{μ}[X] basis, the S_{λ}[X^{tq}] basis is adjoint to the S_{λ′}[X] basis, and M_{k} is self-adjoint. This allows that formula 2.5 be connected to the action of the Macdonald operators on monomial functions given in [M1],VI 3.6.

3. Macdonald Polynomials

This section contains determinantal expressions for the Macdonald polynomials in the bases, S_{μ}[X^{tq}] , S_{μ}[X] , and m_{μ}[X] , obtained from the action of the operator M_{1} on S_{μ}[X^{tq}].

The Macdonald polynomials [M1] can be defined to be the eigenfunctions of the Macdonald operators. In particular we have , for any partition λ,

M_{1} J_{λ}(X; q, t) = [|λ|] J_{λ}(X; q, t) ,  3.1

where the symbol

[|α|] := q^{α_{1}}t^{n−1} + q^{α_{2}}t^{n−2} + ⋯ + q^{α_{n}} ,  3.2

will be used for any composition α ∈ ℕ^{n}. Since the eigenvalues are distinct for each λ, any polynomial that satisfies 3.1 must be proportional to J_{λ}(X; q, t). It is with this in mind that we proceed to construct such polynomials using the action of M_{1} on S_{λ}[X^{tq}].

First, we have that the ordered expansion of S_{μ} in terms of complete functions is given, for S^{α} = S_{α_{1}} ⋯ S_{α_{n}}, by

S_{μ}[X] = ∑_{σ∈𝔖(n)}(−1)^{ℓ(σ)} S^{σ(μ+ρ)−ρ}[X] := ∑_{α} ϵ(μ, α) S^{α}[X] ,  3.3

summed over all vectors α ∈ ℕ^{n}, where ϵ(μ, α) ∈ {0, +1, −1}. The set of all α with a corresponding nonzero ϵ(μ, α) is included in the set {σ̄μ | σ̄ ∈ 𝔖(ℓ(μ))} where we define σ̄_{i}μ = −(…, μ_{i+1} − 1, μ_{i} + 1, …). For example, if n = 3,

\begin{aligned}& & & −(0, 4, 1) & \underrightarrow{σ̄_{2}} & (0, 0, 5) \\ & & σ̄_{1} ; ↗ & & & & σ̄_{1} ; ↘ \\ μ = (3, 1, 1) : & (3, 1, 1) & & & & & & −(−1, 1, 5) \\ & & ↘ ; σ̄_{2} & & & & ↗ ; σ̄_{2} \\ & & & −(3, 0, 2) & \underrightarrow{σ̄_{1}} & (−1, 4, 2) \end{aligned} . 3.4

For μ = (3, 1, 1), ϵ(μ, α) α = {(3, 1, 1), −(0, 4, 1), (0, 0, 5), −(3, 0, 2)}, since α ∈ ℕ^{n} cannot have negative components.

Now, recall that Theorem 2.1 gives explicitly

M_{1} S_{μ}[X^{tq}] = ∑_{1≤i≤n} t^{n−i} [ \begin{matrix}0 & ⋯ & 0 & ⋯ & 0 \\ 0 & ⋯ & X^{t} & ⋯ & 0 \end{matrix} ] S_{μ}[X^{tq}] .  3.5

The expansion of the right hand side, taking into account that S_{k}[X^{tq} + X^{t}] = q^{k}S_{k}[X^{tq}], is

M_{1} S_{μ}[X^{tq}] = ∑_{σ∈𝔖(n)}(−1)^{ℓ(σ)} [|σ(μ + ρ) − ρ|] S^{σ(μ+ρ)−ρ}[X^{tq}] = ∑_{α} ϵ(μ, α) [|α|] S^{α}[X^{tq}] . 3.6

The action of the Macdonald operator M_{1} is triangular on the basis, S_{λ}[X^{tq}], giving that

M_{1} S_{μ}[X^{tq}] = [|μ|] S^{μ}[X^{tq}] + ∑_{α} ϵ(μ, α) S^{ν}[X^{tq}] ,  3.7

where the sum is over all α a permutation of partitions ν > μ. Formula 3.7 demonstrates that the eigenspaces of M_{1} are 1-dimensional and reveals the eigenvalues.

For any pair of partitions μ, ν ∈ ℕ^{n} of the same weight, let us write

[λ]_{μν}(q, t) = ∑_{α} ϵ(μ, α) ([|λ|] − [|α|]) ,  3.8

where α runs over all distinct permutations of ν ∈ ℕ^{n}. Note that, in the following, [λ]_{μν} shall stand for [λ]_{μν}(q, t).

Theorem 3.1 Expanded in terms of S_{λ}[X^{tq}], the Macdonald polynomials are

J_{λ}(X; q, t) = \frac{c_{λ′}(t, q)}{v_{λ}(q, t)} det | \begin{matrix}S_{λ}[X^{tq}] & … & S_{μ}[X^{tq}] & … \\ ⋮ & & ⋮ \\ [λ]_{λν} & … & [λ]_{μν} & … \\ ⋮ & & ⋮ \end{matrix} |_{{ν>λ \atop μ≥λ}} .  3.9

Columns are indexed by all partitions μ ≥ λ. Aside from the first row, the entries are the polynomials in q, t, [λ]_{μν}, ν > λ. The normalization factor consists of

v_{λ}(q, t) = ∏_{μ>λ}([|λ|] − [|μ|]) = ∏_{μ>λ}[λ]_{μμ} ,  3.10

and

c_{λ}(q, t) = ∏_{s∈λ} (1 − q^{a(s)}t^{l(s)+1}) .  3.11

For example, up to a scalar,

J_{2,2,1}(X; q, t) ≐ | \begin{matrix}S_{2,2,1}[X^{tq}] & S_{3,1,1}[X^{tq}] & S_{3,2}[X^{tq}] & S_{4,1}[X^{tq}] & S_{5}[X^{tq}] \\ −[1, 3, 1] & [3, 1, 1] & 0 & 0 & 0 \\ −[2, 0, 3] & −[3, 0, 2] & [3, 2, 0] & 0 & 0 \\ [1, 0, 4] & −[0, 4, 1] & −[1, 4, 0] & [4, 1, 0] & 0 \\ 0 & [0, 0, 5] & 0 & −[0, 5, 0] & [5, 0, 0] \end{matrix} | \begin{matrix}(3,1,1) \\ (3,2,0) \\ (4,1,0) \\ (5,0,0) \end{matrix} 3.12

where an entry ±[α] stands for ±([|2, 2, 1|] − [|α|]). Each column of such determinant is obtained from the ordered expansion of a Schur function. For example, the second column is given by the expansion of S_{3,1,1} computed in 3.4. Notice we can limit ourselves to vectors of length ℓ(λ) since each entry is the difference of two vectors of which the last n − ℓ(λ) components are equal.

Proof of Theorem 3.1 It suffices to check that ([|λ|] − M_{1})J′_{λ}(X; q, t) = 0, where J′_{λ}(X; q, t) is the determinantal expression 3.9, given that the eigenvalues are distinct for each λ. The operator acts only on the first row of the determinant allowing us to use formula 3.6 for the action of M_{1} on S_{μ}[X^{tq}]. This gives

([|λ|] − M_{1})J′_{λ}(X; q, t) = det | \begin{matrix}∑_{β≥λ}[λ]_{λβ} S^{β} & … & ∑_{β≥μ}[λ]_{μβ} S^{β} & … \\ ⋮ & & ⋮ \\ [λ]_{λν} & … & [λ]_{μν} & … \\ ⋮ & & ⋮ \end{matrix} | .  3.13

The determinant can be shown to vanish by examining the coefficient of all S^{β}’s. The coefficient of S^{λ} is proportional to [λ]_{λλ} = 0. For β > λ, the coefficient of S^{β} is equal to the determinant

det | \begin{matrix}[λ]_{λβ} & … & [λ]_{μβ} & … \\ ⋮ & & ⋮ \\ [λ]_{λβ} & … & [λ]_{μβ} & … \\ ⋮ & & ⋮ \end{matrix} | \begin{matrix}row 1 \\ row β \end{matrix} ,  3.14

which has two identical rows, and thus also vanishes.

To verify the normalization, we recall [M1]

J_{λ}(X; q, t) = c_{λ′}(t, q) S_{λ}[X^{tq}] + ∑_{μ>λ} c_{λμ}S_{μ}[X^{tq}]  3.15

and observe that because, from 3.7, [λ]_{μν} = 0 for μ > ν, the sub-determinant giving the coefficient of S_{λ}[X^{tq}] in J′_{λ}(X; q, t) is triangular and equals v_{λ}(q, t).

The formula for J_{λ}(X; q, t) in terms of S_{λ}[X^{tq}] can be converted into a similar expression in terms of the S_{μ}[X] basis by applying to expression 3.9, the involution ω_{q,t} satisfying properties [M1]:

ω_{q,t}S_{λ}[X^{tq}] = S_{λ′}[X]  and  ω_{q,t}J_{λ}(X; q, t) ≐ J_{λ′}(X; t, q), up to a scalar .

Renormalizing using

J_{λ}(X; q, t) = c_{λ}(q, t) S_{λ}[X] + ∑_{μ<λ} c_{λμ} S_{μ}[X] ,  3.16

we have

Corollary 3.2

J_{λ}(X; q, t) = \frac{c_{λ}(q, t)}{u_{λ}(t, q)} det | \begin{matrix}S_{λ}[X] & … & S_{μ}[X] & … \\ ⋮ & & ⋮ \\ [λ′]_{λ′ν′}(t, q) & … & [λ′]_{μ′ν′}(t, q) & … \\ ⋮ & & ⋮ \end{matrix} |_{{ν<λ \atop μ≤λ}} ,  3.17

where u_{λ}(q, t) = ∏_{μ<λ}([|λ′|] − [|μ′|]).

The first row is the list of Schur fonctions S_{μ}, μ ≤ λ, the other entries are the polynomials [λ′]_{μ′ν′}(t, q) for ν < λ.

A determinantal formula for the Macdonald polynomials in terms of the monomial basis may now be obtained from formula 3.17 . The interpretation of the entries of the matrix in 3.17 (aside from the top row) as scalar products will allow that we pass to other bases of symmetric functions.

We use the space of polynomials in x_{1}, …, x_{n} as a free module over the ring of symmetric polynomials. Under the scalar product, ⟨f, g⟩ = ∂_{ω}(fg) , this space has the following two adjoint bases

{x̄^{I} = (−x_{1})^{I_{1}} ⋯ (−x_{m})^{I_{m}}}_{0^{m}⊆I⊆ρ} and {E_{I} = e_{I_{1}}[0]e_{I_{2}}[X_{1}] ⋯ e_{I_{m}}[X_{m−1}]}_{0^{m}⊆I⊆ρ} ,

3.18

for ρ = (0, …, m − 1) and 0^{m} = (0, …, 0) ∈ ℕ^{m}. That is,

⟨x̄^{I}, E_{ρ−J}⟩ = ∂_{ω}(x̄^{I}E_{ρ−J}) = δ_{IJ},  I, J ⊆ ρ .  3.19

This result can be used to determine the coefficients in the following matrix;

S_{λ′}[X_{n}] = det | \begin{matrix}e_{λ_{1}}[X_{n}] & e_{λ_{1}+1}[X_{n+1}] & ⋯ & e_{λ_{1}+n−1}[X_{2n−1}] \\ e_{λ_{2}−1}[X_{n}] & e_{λ_{2}}[X_{n+1}] & ⋯ & e_{λ_{2}+n−2}[X_{2n−1}] \\ ⋮ & ⋮ & ... & ⋮ \\ e_{λ_{n}−n+1}[X_{n}] & e_{λ_{n}−n+2}[X_{n+1}] & ⋯ & e_{λ_{n}}[X_{2n−1}] \end{matrix} | ,  3.20

which is obtained by applying the ω involution, ωS_{k}[X] = e_{k}[X], ωS_{λ}[X] = S_{λ′}[X], to 1.3 and increasing the alphabets in each column with the relation, e_{k}[X_{l} + x_{l+1} + ⋯ + x_{2n−1}] = e_{k}[X_{l}] + ∑_{j=1}^{2n−l−1} e_{k−j}[X_{l}] e_{j}[x_{l+1} + ⋯ + x_{2n−1}]. Note that this relation does not change the value of the determinant since it corresponds to multiplication by a unitriangular matrix. Now using 3.19 with m = 2n, the coefficients in the expansion of this matrix are

S_{λ′}[X_{n}] = ∑_{α′∈ρ′}⟨x̄^{ρ−α′}, S_{λ′}[X_{n}]⟩ E_{α′} ,  3.21

where α′ = (0, …, 0, α_{1}, …, α_{n}) and ρ′ = (0, …, 0, n, …, 2n − 1) for α′, ρ′ ∈ ℕ^{2n} . Since the ordered expansion of 3.20, and thus 3.21, must coincide with the ordered expansion 3.3 of S_{λ}[X] in terms of complete functions, we have

ϵ(λ, α) = ⟨x̄^{ρ−α′}, S_{λ′}[X_{n}]⟩ .  3.22

This implies that 3.17 can be rewritten, using

⟨f⟩_{λν} = ∑_{α}⟨x̄^{ρ−α′}, f⟩ ([|λ|]_{t,q} − [|α|]_{t,q}) ,  3.23

where the sum is over all distinct permutations α of ν ∈ ℕ^{n}, as

J_{λ}(X_{n}; q, t) = \frac{c_{λ}(q, t)}{u_{λ}(t, q)} det | \begin{matrix}S_{λ}[X_{n}] & … & S_{μ}[X_{n}] & … \\ ⋮ & & ⋮ \\ ⟨S_{λ}[X_{n}]⟩_{λ′ν′} & … & ⟨S_{μ}[X_{n}]⟩_{λ′ν′} & … \\ ⋮ & & ⋮ \end{matrix} |_{{μ≤λ \atop ν<λ}} .  3.24

With ϵ(μ, α) now defined as a scalar product, we can write Macdonald polynomials in any linear basis of the space generated by {S_{μ}}_{μ≤λ}. In particular, we have the following theorem:

Theorem 3.3 The Macdonald polynomials in terms of monomial symmetric functions are

J_{λ}(X_{n}; q, t) = \frac{c_{λ}(q, t)}{u_{λ}(t, q)} det | \begin{matrix}m_{λ}[X_{n}] & … & m_{μ}[X_{n}] & … \\ ⋮ & & ⋮ \\ ⟨m_{λ}[X_{n}]⟩_{λ′ν′} & … & ⟨m_{μ}[X_{n}]⟩_{λ′ν′} & … \\ ⋮ & & ⋮ \end{matrix} |_{{μ≤λ \atop ν<λ}} .  3.25

For example, J_{2,2}(X; q, t) is given by

J_{2,2}(X_{4}; q, t) = \frac{c_{2,2}(q, t)}{u_{2,2}(t, q)} det | \begin{matrix}m_{2,2}[X_{4}] & m_{2,1,1}[X_{4}] & m_{1,1,1,1}[X_{4}] \\ −[3, 1] − [1, 3] & [3, 1] & 0 \\ [4, 0] + [0, 4] & −3[4, 0] − [0, 4] & [4, 0] \end{matrix} | ,  3.26

with [α] = ([|2, 2|]_{t,q} − [|α|]_{t,q}). In this expression, the term −3[4, 0] appears, for instance, because ⟨x̄^{(0,1,2,3,0,5)}, m_{2,1,1}[X_{4}]⟩ = −3. Again, we work with vectors of length ℓ(λ), since adding zeros to α does not change the result. We skip the problem of computing efficiently all the scalar products in the matrix.

4. Creation Operators

A Macdonald polynomial associated to any partition can be constructed by repeated application of creation operators, B_{k}^{(n)}. Specifically,

B_{k}^{(n)} J_{λ}(X; q, t) = J_{λ+1^{k}}(X; q, t) .  4.1

The creation operators were defined originally to be [LV],[KN]

B_{k}^{(n)} = ∑_{|I|=k} ∑_{l=0}^{k}(−t)^{l}x^{I} \frac{R(X_{I}, X_{I}^{c}/t)}{R(X_{I}, X_{I}^{c})} M_{l}^{(I)} ,  4.2

where I is a k subset of {1, …, n} and M^{(I)} acts only on x_{i} for all i ∈ I. We will see that the action of these operators on the modified Schur function basis may be expressed in a manner similar to that of the Macdonald operators on this basis, though the former increases degrees. We begin by giving a new expression for the creation operators.

Proposition 4.1 The creation operators acting on the space of symmetric functions are

B_{k}^{(n)} = \frac{1}{k_{t}!} χ_{(k|n−k)} χ_{ω(k)} x_{1} ⋯ x_{k} (1 − tΩ_{k}) ⋯ (1 − t^{k}Ω_{k})  4.3

where Ω_{k} = σ_{1} ⋯ σ_{k−1}Θ_{k}.

Proof The binomial expansion of the right hand side of expression 4.3 becomes

rhs = \frac{1}{k_{t}!} ∑_{l=0}^{k} \frac{(−1)^{l}t^{\binom{l+1}{2}}(t)_{k}}{(t)_{k−l}(t)_{l}} χ_{(k|n−k)} x_{1} ⋯ x_{k} χ_{ω(k)} Θ_{l} ⋯ Θ_{1} ,  4.4

since Ω_{k}^{l} = Θ_{l} ⋯ Θ_{1} on symmetric functions. Further, the image of such a function under Ω_{k}^{l} belongs to Sym(l|k − l) allowing that we use property 1.12 of χ_{ω(k)} to obtain

rhs = \frac{1}{k_{t}!} ∑_{l=0}^{k} \frac{(−1)^{l}t^{\binom{l+1}{2}+l(k−l)}(t)_{k}}{(t)_{k−l}(t)_{l}} l_{t}!(k − l)_{t}!χ_{(k|n−k)}x_{1} ⋯ x_{k} χ_{(l|k−l)} Θ_{l} ⋯ Θ_{1} . 4.5

The definition of the Macdonald operators and χ_{(k|n−k)} then imply that

rhs = ∑_{l=0}^{k}(−t)^{l}∂_{(k|n−k)} R(X_{k}, X_{k}^{c}/t) x_{1} ⋯ x_{k} M_{l}^{(k)} .  4.6

Finally, the action of ∂_{(k|n−k)} as described in 1.8 confirms that this expression, and therefore 4.3, are in fact equivalent to the original definition for the creation operators.

Theorem 4.2 The action of B_{k}^{(n)} on S_{λ}[X^{tq}], for ℓ(λ) ≤ k, can be expressed as

B_{k}^{(n)} S_{λ}[X^{tq}] = ∏_{i=1}^{k} ([ \begin{matrix}⋯ & 0 & 1 & 0 & ⋯ \\ ⋯ & 0 & 0 & 0 & ⋯ \end{matrix} ] − t^{k−i} [ \begin{matrix}⋯ & 0 & 1 & 0 & ⋯ \\ ⋯ & 0 & X^{t} & 0 & ⋯ \end{matrix} ]) S_{λ}[X^{tq}], 4.7

where the i^{th} column is the only non-zero column of the operators. The relation, S_{j}[X^{tq} + X^{t}] = S_{j}[qX^{tq}] = q^{j}S_{j}[X^{tq}] , yields immediately

Corollary 4.3 Let f(i, j) = λ_{i} + (j − i + 1). For any ℓ(λ) ≤ k we have

B_{k}^{(n)} S_{λ}[X^{tq}] = |(1 − q^{f(i,j)}t^{k−j})S_{f(i,j)}[X^{tq}]|_{i,j≤k} .  4.8

For example, when k = 2, we have

\begin{aligned}B_{2}^{(n)} S_{λ_{1},λ_{2}}[X^{tq}] & = ([ \begin{matrix}1 & 0 \\ 0 & 0 \end{matrix} ] − t [ \begin{matrix}1 & 0 \\ X^{t} & 0 \end{matrix} ]) ([ \begin{matrix}0 & 1 \\ 0 & 0 \end{matrix} ] − t^{0} [ \begin{matrix}0 & 1 \\ 0 & X^{t} \end{matrix} ]) S_{λ_{1},λ_{2}}[X^{tq}] \\ & = | \begin{matrix}(1 − q^{λ_{1}+1}t)S_{λ_{1}+1}[X^{tq}] & (1 − q^{λ_{1}+2})S_{λ_{1}+2}[X^{tq}] \\ (1 − q^{λ_{2}}t)S_{λ_{2}}[X^{tq}] & (1 − q^{λ_{2}+1})S_{λ_{2}+1}[X^{tq}] \end{matrix} | . \end{aligned}

Proof of Theorem 4.2 In what follows, as we have done in the case of the Macdonald operators, we shall use Θ^{t} instead of Θ in the operator B_{k}. This will allow us to use the action of M_{k} on S_{λ}[Y + X] (with Y an alphabet invariant under Θ^{t}) obtained in the previous section. Letting Y = X^{tq} − X in the final result will prove the theorem.

First, notice from 4.3 that B_{k}^{(n)} = χ_{(k|n−k)}B_{k}^{(k)}. This given, we will first describe the action of B_{k}^{(k)} on S_{λ}[Y + X].

Claim 4.4 For ℓ(λ) ≤ k and α_{i} → λ_{i} − i + 1, we have

\begin{matrix}B_{k}^{(k)} S_{λ}[Y + X] = & ∏_{i=1}^{k} ([ \begin{matrix}⋯ & 0 & 1 & 0 & ⋯ \\ ⋯ & 0 & 0 & 0 & ⋯ \end{matrix} ] − t^{k−i} [ \begin{matrix}⋯ & 0 & 1 & 0 & ⋯ \\ ⋯ & 0 & X_{k}^{t} & 0 & ⋯ \end{matrix} ]) S_{λ}[Y + X] \\ = & |G_{α_{i}+1}(t^{k−1}, X_{k})G_{α_{i}+2}(t^{k−2}, X_{k}) ⋯ G_{α_{i}+k}(t^{0}, X_{k})|_{i=1..k} . \end{matrix}

4.9

where G_{h}(t^{r}, X_{k}) = S_{h}[Y + X] − t^{r}S_{h}[Y + X + X_{k}^{t}].

Proof of Claim With Y′ = Y + X_{k}^{c}, expression 4.2, where n = k, yields

B_{k}^{(k)} S_{λ}[Y + X] = x_{1} ⋯ x_{k} ∑_{l=0}^{k}(−t)^{l}M_{l}^{(k)}S_{λ}[Y′ + X_{k}],  4.10

which, using the Macdonald operator action on S_{λ}[Y′ + X_{k}], gives

\begin{aligned}B_{k}^{(k)} S_{λ}[Y + X] & = x_{1} ⋯ x_{k} ∑_{l=0}^{k}(−t)^{l} ∑_{1≤i_{1}<···<i_{l}≤k} t^{(n−i_{1})+···+(n−i_{l})} \\ & × [ \begin{matrix}0 & ⋯ & 0 & ⋯ & 0 & ⋯ & 0 \\ 0 & ⋯ & X_{k}^{t} & ⋯ & X_{k}^{t} & ⋯ & 0 \end{matrix} ] S_{λ}[Y′ + X_{k}] . \end{aligned}  4.11

In the last expression, for a fixed l, X^{t} occurs in positions i_{1}, ..., i_{l}. Comparing such terms, we obtain

\begin{matrix}B_{k}^{(k)} S_{λ}[Y + X] \\ = x_{1} ⋯ x_{k} ∏_{i=1}^{k} ([ \begin{matrix}0 & ⋯ & 0 \\ 0 & ⋯ & 0 \end{matrix} ] − t^{k+1−i} [ \begin{matrix}0 & ⋯ & 0 & 0 & 0 & ⋯ & 0 \\ 0 & ⋯ & 0 & X_{k}^{t} & 0 & ⋯ & 0 \end{matrix} ]) S_{λ}[Y + X] , \end{matrix}

4.12

or more compactly

B_{k}^{(k)} S_{λ}[Y + X] = x_{1} ⋯ x_{k} |G_{α_{i}}(t^{k}, X_{k})G_{α_{i}+1}(t^{k−1}, X_{k}) ⋯ G_{α_{i}+k−1}(t^{1}, X_{k})|_{i=1..k} . 4.13

The proof of the claim is completed using the following lemma(proved in the appendix):

Lemma 4.5 For F_{j}(t^{r}, X_{k}) = S_{j}[C + X_{k}] − t^{r}S_{j}[C + tX_{k}], we have

\begin{matrix}|F_{j+1}(t^{k−1}, X_{k})F_{j+2}(t^{k−2}, X_{k}) … F_{j+k}(1, X_{k})| \\ = x_{1} ⋯ x_{k}|F_{j}(t^{k}, X_{k})F_{j+1}(t^{k−1}, X_{k}) … F_{j+k−1}(t, X_{k})| , \end{matrix}  4.14

where the expressions between brackets must be understood as k × k determinants. In effect, letting C → Y′, we have F_{j}(t^{k}, X_{k}) → G_{j}(t^{k}, X_{k}), and thus using 4.14 in 4.13 proves the claim.

Expression 4.9 gives that the action of B_{k}^{(n)} = χ_{(k|n−k)} B_{k}^{(k)} on S_{λ}[Y + X] is
B_{k}^{(n)} S_{λ}[Y + X] = χ_{(k|n−k)}|G_{α_{i}+1}(t^{k−1}, X_{k}) … G_{α_{i}+k}(1, X_{k})|_{i=1..k} .  4.15

The proof of Theorem 4.2 is now equivalent to showing that χ_{(k|n−k)} acts on this determinant by extending the alphabets from X_{k} to X. This will be achieved with the following lemma (proved in the appendix);

Lemma 4.6 For any l ≥ k, we have

\begin{matrix}|G_{j+1}(t^{k−1}, X_{l})G_{j+2}(t^{k−2}, X_{l}) … G_{j+k}(1, X_{l})| \\ = χ_{(1|l−1)}^{(1)} ⋯ χ_{(1|l−k+1)}^{(k−1)}χ_{(1|l−k)}^{(k)} |G_{j+1}(t^{k−1}, X_{k})G_{j+2}(t^{k−2}, X_{k−1}) … G_{j+k}(1, X_{1})| \end{matrix} ,

4.16

where, again, the expression between brackets are k × k determinants.

Application of this lemma to expression 4.15, with l = k, gives

\begin{aligned}B_{k}^{(n)} & S_{λ}[Y + X] = \\ & χ_{(k|n−k)} χ_{(1|k−1)}^{(1)} ⋯ χ_{(1|1)}^{(k−1)}|G_{α_{i}+1}(t^{k−1}, X_{k})G_{α_{i}+2}(t^{k−2}, X_{k−1}) … G_{α_{i}+k}(1, X_{1})|_{i=1,...,k} \end{aligned} .

4.17

Property 1.14 allows a refactorization of χ that transforms 4.17 into

\begin{aligned}B_{k}^{(n)} & S_{λ}[Y + X] = \\ & χ_{(1|n−1)}^{(1)} ⋯ χ_{(1|n−k)}^{(k)} |G_{α_{i}+1}(t^{k−1}, X_{k})G_{α_{i}+2}(t^{k−2}, X_{k−1}) … G_{α_{i}+k}(1, X_{1})|_{i=1,...,k} \end{aligned} .

4.18

Lemma 4.6 may be applied again, with l = n, giving

B_{k}^{(n)} S_{λ}[Y + X] = |G_{α_{i}+1}(t^{k−1}, X)G_{α_{i}+2}(t^{k−2}, X) … G_{α_{i}+k}(1, X)|_{i=1,...,k} ,  4.19

which proves Theorem 4.2.

Appendix

Proof of Lemma 1.1 Using formula 1.2, we have

S_{j}[X^{tq} + X_{k}^{t}] = S_{j}[D + x_{k}^{t}] = ∑_{l=0}^{j} S_{j−l}[D] S_{l}[x_{k}^{t}] ,  A.1

where D = X^{tq} + X_{k−1}^{t}. Further properties of Schur functions yield, as x_{k}^{t} = tx_{k} − x_{k},

\begin{aligned}S_{j}[X^{tq} + X_{k}^{t}] & = S_{j}[D] + ∑_{l=1}^{j}(1 − 1/t)S_{j−l}[D] (tx_{k})^{l} \\ & = S_{j}[D] + ∑_{l=0}^{j−1} t(x_{k} − x_{k}/t)S_{j−l−1}[D] (tx_{k})^{l} . \end{aligned}  A.2

The alphabet D is invariant under permutations of x_{k}, …, x_{n}, and thus the first term is invariant up to a constant under χ_{(1|n−k)}^{(k)}. Thus, using 1.10, we have

χ_{(1|n−k)}^{(k)}S_{j}[X^{tq}+X_{k}^{t}] = (1+t^{−1}+⋯+t^{−n+k})S_{j}[D]+χ_{(1|n−k)}^{(k)} ∑_{l=0}^{j−1} t(x_{k}−x_{k}/t)S_{j−l−1}[D] (tx_{k})^{l} ,

A.3 and are left only to show

χ_{(1|n−k)}^{(k)} ∑_{l=0}^{j−1} t(x_{k} − x_{k}/t)S_{j−l−1}[D] (tx_{k})^{l} = −S_{j}[D] + S_{j}[X^{tq} + X^{t}] .  A.4

The definition of χ_{(1|n−k)}^{(k)} and (x_{k} − x_{k}/t)R(x_{k}, X_{k}^{c}/t) = R(x_{k}, X_{k−1}^{c}/t) convert the left hand side of this expression into

lhs = ∂_{n−1} ⋯ ∂_{k} R(x_{k}, X_{k−1}^{c}/t) ∑_{l=0}^{j−1} tS_{j−l−1}[D](tx_{k})^{l} .  A.5

The identity S_{n}[x − X/t] = R(x, X/t) may be used to eliminate the resultant, and we obtain

lhs = ∂_{n−1} ⋯ ∂_{k} ∑_{l=0}^{j−1} tS_{j−l−1}[D](tx_{k})^{l}S_{n−k}[x_{k} − X_{k−1}^{c}/t] .  A.6

A further Schur function property allows that the factor x_{k}^{l} be used to increase the index of S_{n−k};

lhs = ∂_{n−1} ⋯ ∂_{k} ∑_{l=0}^{j−1} t^{l+1}S_{j−l−1}[D]S_{n−k+l}[x_{k} − X_{k−1}^{c}/t] .  A.7

Now we can let the divided differences act (see [L1] for similar computations). Using the property ∂_{k}S_{l}[x_{k} − X_{k−1}^{c}/t] = S_{l−1}[x_{k} + x_{k+1} − X_{k−1}^{c}/t], we arrive at the expression,

\begin{aligned}lhs = ∑_{l=0}^{j−1} t^{l+1}S_{j−l−1}[D]S_{l+1}[x_{k} + ⋯ + x_{n} − X_{k−1}^{c}/t] & = ∑_{l=0}^{j−1} S_{j−l−1}[D]S_{l+1}[(X_{k−1}^{c})^{t}] \\ & = −S_{j}[D] + S_{j}[D + (X_{k−1}^{c})^{t}] . \end{aligned} A.8

Clearly D + (X_{k−1}^{c})^{t} = X^{tq} + X^{t}, which implies that expression A.4 holds.

Proof of formula 2.11 We want to show that

\begin{matrix}S_{λ}[Y + X]R(X_{k}, X_{k}^{c}) \\ = |S_{α_{i}−k+n}[Y + X_{k}] ⋯ S_{α_{i}−1+n}[Y + X_{k}]S_{α_{i}+k}[Y + X_{k}^{c}] ⋯ S_{α_{i}+n−1}[Y + X_{k}^{c}]|_{i=1..n} , \end{matrix}

A.9

where α_{i} → λ_{i} − i + 1. Let f_{i}, i = 1, …, n, denote arbitrary one variable polynomials in the space of polynomials in x_{1}, …, x_{n} taken as a free module over the ring of symmetric polynomials. From the usual Newton’s interpolation in one variable ([L1], Lemma Ni5), given such f_{i}, we have

det |∂_{j−1} … ∂_{1}f_{i}(x_{1})|_{1≤i,j≤n} = \frac{1}{Δ(X)} det |f_{i}(x_{j})|_{1≤i,j≤n},  A.10

where Δ(X) = ∏_{i<j}(x_{j} − x_{i}) is the Vandermonde determinant. If in this expression we let f_{i}(x) = S_{α_{i}+n−1}[Y + x], using the identity ∂_{k}S_{j}[Y + X_{k}] = S_{j−1}[Y + X_{k+1}], we obtain

\frac{1}{Δ(X)} det |S_{α_{i}+n−1}[Y + x_{j}]|_{1≤i,j≤n} = det |∂_{j−1} … ∂_{1}S_{α_{i}+n−1}[Y + x_{1}]|_{1≤i,j≤n}
= det |S_{α_{i}+n−j}[Y + X_{j}]|_{1≤i,j≤n}.

A.11

The alphabets can be increased in each column of the determinant on the right hand side of this expression by using the relation

S_{α_{i}+n−j}[Y + X_{j}] = S_{α_{i}+n−j}[Y + X] + ∑_{ℓ=1}^{n−j} S_{α_{i}+n−j−ℓ}[Y + X]S_{ℓ}[−X_{j}^{c}] ,

giving that

det |S_{α_{i}+n−j}[Y + X_{j}]|_{1≤i,j≤n} = det |S_{α_{i}+n−j}[Y + X]|_{1≤i,j≤n} .  A.12

The right hand side may thus be substituted into formula A.11 implying that

S_{λ}[Y + X]Δ(X) = |S_{α_{i}+n−1}[Y + x_{n}]S_{α_{i}+n−1}[Y + x_{n−1}] ⋯ S_{α_{i}+n−1}[Y + x_{1}]|_{i=1..n} . A.13

This result applied to arbitrary minors of order n − k in the first n − k columns and of order k in the last k columns of the right hand side of A.13, gives by linearity,

S_{λ}[Y + X]Δ(X)

= |S_{α_{i}+n−1}[Y + x_{n}] ⋯ S_{α_{i}+n−1}[Y + x_{k+1}]S_{α_{i}+n−1}[Y + x_{k}] ⋯ S_{α_{i}+n−1}[Y + x_{1}]|_{i=1..n}

= |S_{α_{i}+k}[Y + X_{k}^{c}] ⋯ S_{α_{i}+n−1}[Y + X_{k}^{c}]S_{α_{i}−k+n}[Y + X_{k}] ⋯ S_{α_{i}+n−1}[Y + X_{k}]|
× Δ(X_{k}^{c})Δ(X_{k}).
A.14

Finally, with Δ(X)/(Δ(X_{k})Δ(X_{k}^{c})) = R(X_{k}, X_{k}^{c})(−1)^{k(n−k)}, one sees that A.14 is equivalent to A.9, which proves the assertion.

Proof of Lemma 4.5 As a result of ∂_{i}S_{j}[A + b_{i}] = S_{j−1}[A + b_{i} + b_{i+1}], we have

∂_{i} F_{j}(t^{r}, A_{i}) = F_{j−1}(t^{r+1}, A_{i+1}) ,  A.15

allowing that the left hand side of 4.14 be rewritten as

lhs = |∂_{k−1}..∂_{1}F_{j+k}(1, X_{1}), ∂_{k−1}..∂_{2}F_{j+k}(1, X_{2}), …, ∂_{k−1}F_{j+k}(1, X_{k−1}), F_{j+k}(1, X_{k})| . A.16 We can factor out ∂_{k−1} ⋯ ∂_{1}, since all the columns, except the first one are symmetrical in x_{1}, …, x_{k}. Similarly we can factor out successively (∂_{k−1} ⋯ ∂_{2}), …, ∂_{k−1} to get

lhs = (∂_{k−1}..∂_{1})(∂_{k−1}..∂_{2}) ⋯ (∂_{k−1})|F_{j+k}(1, X_{1}), …, F_{j+k}(1, X_{k−1}), F_{j+k}(1, X_{k})| . A.17

With F_{j}(t^{r}, X_{k}) = x_{k}F_{j−1}(t^{r+1}, X_{k}) + F_{j}(t^{r}, X_{k−1}), we can substitute x_{k}F_{j+k−1}(t, X_{k}) + F_{j+k}(1, X_{k−1}) in the last column of the matrix. This produces a sum of two determinants, of which the second vanishes thanks to column k − 1. We thus obtain

lhs = (∂_{k−1}..∂_{1})(∂_{k−1}..∂_{2}) ⋯ (∂_{k−1})|F_{j+k}(t^{0}, X_{1}), …, F_{j+k}(t^{0}, X_{k−1}), x_{k}F_{j+k−1}(t, X_{k})| ,

A.18

and by iteration;

lhs = (∂_{k−1}..∂_{1}) ⋯ (∂_{k−1})|x_{1}F_{j+k−1}(t, X_{1}), …, x_{k−1}F_{j+k−1}(t, X_{k−1}), x_{k}F_{j+k−1}(t, X_{k})| .

A.19

The symmetric factor x_{1} ⋯ x_{k} can be factored out of the determinant and the divided differences may be reintroduced;

lhs = x_{1} ⋯ x_{k}|∂_{k−1} ⋯ ∂_{1}F_{j+k−1}(t, X_{1}), …, ∂_{k−1}F_{j+k−1}(t, X_{k−1}), F_{j+k−1}(t, X_{k})| . A.20

Transformation on columns of this determinant using relation A.15 completes the proof.

Proof of Lemma 4.6 This identity, as it is a consequence of a sucession of transformations performed on the columns, can be proven by induction on k given that

G_{j}(t^{r}, X_{l}) = χ_{(1|l−k)}^{(k)} G_{j}(t^{r}, X_{k}) − c(k) G_{j}(t^{r}, X_{k−1}) ,  A.21

which is a direct consequence of lemma 1.1. If k = 1, G_{j+1}(t^{0}, X_{0}) = 0 implies that

G_{j+1}(1, X_{l}) = χ_{(1|l−1)}^{(1)} G_{j+1}(1, X_{1}) ,  A.22

which is the case k = 1 of the lemma. Assuming that the lemma holds for a k−1 determinant, we get

\begin{matrix}|G_{j+1}(t^{k−1}, X_{l})G_{j+2}(t^{k−2}, X_{l}) … G_{j+k}(1, X_{l})| \\ = χ_{(1|l−1)}^{(1)} ⋯ χ_{(1|l−k+1)}^{(k−1)}|G_{j+1}(t^{k−1}, X_{l})G_{j+2}(t^{k−2}, X_{k−1}) … G_{j+k}(1, X_{1})| . \end{matrix} A.23

The first column becomes χ_{(1|l−k)}^{(k)}G_{j+1}(t^{k−1}, X_{k}) − c(k) G_{j+1}(t^{k−1}, X_{k−1}), by relation A.21, producing a sum of two determinants. As the first of these is exactly the desired result, the second must be shown to vanish. Specifically, it must be that

χ_{(1|l−1)}^{(1)} ⋯ χ_{(1|l−k+1)}^{(k−1)}|G_{j+1}(t^{k−1}, X_{k−1})G_{j+2}(t^{k−2}, X_{k−1}) … G_{j+k}(1, X_{1})| = 0 .  A.24

The left hand side of this expression may be transformed first by using property 1.14 to refactor χ_{(1|l−1)}^{(1)} ⋯ χ_{(1|l−k+1)}^{(k−1)} into the form χ_{(k−1|n−k+1)}χ_{(1|k−2)}^{(1)} ⋯ χ_{(1|1)}^{(k−2)}, and then by using the induction hypothesis, where l = k − 1;

\begin{matrix}lhs = & χ_{(k−1|n−k+1)}χ_{(1|k−2)}^{(1)} ⋯ χ_{(1|1)}^{(k−2)}|G_{j+1}(t^{k−1}, X_{k−1})G_{j+2}(t^{k−2}, X_{k−1}) … G_{j+k}(1, X_{1})| \\ = & χ_{(k−1|n−k+1)}|G_{j+1}(t^{k−1}, X_{k−1})G_{j+2}(t^{k−2}, X_{k−1}) … G_{j+k}(1, X_{k−1})| . \end{matrix}

A.25

We now have χ_{(k−1|n−k+1)} applied to a determinant with k columns that satisfies the special case of x_{k} = 0 in Lemma 4.5. Thus, the determinant vanishes, proving the lemma.

Acknowledgments

The author, J. Morse, expresses thanks to Adriano Garsia for providing the (NSF) support that made possible this joint work. L. Lapointe is supported through an NSERC postdoctoral fellowship. We have extensively used the algebraic combinatorics environment, ACE, Maple library [V].

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