Publ. I.R.M.A. Strasbourg, 1992, 498/S–27 Actes 28^{e} Séminaire Lotharingien, p. 5-39

SCHUR FUNCTIONS : THEME AND VARIATIONS

BY

I. G. MACDONALD

Introduction and theme

In this article we shall survey various generalizations, analogues and deformations of Schur functions — some old, some new — that have been proposed at various times. We shall present these as a sequence of variations on a theme and (unlike e.g. Bourbaki) we shall proceed from the particular to the general. Thus Variations 1 and 2 are included in Variation 3 ; Variations 4 and 5 are particular cases of Variation 6 ; and in their turn Variations 6, 7 and 8 (in part) are included in Variation 9.

To introduce our theme, we recall [M_{1}, Ch. I, § 3] that the Schur function s_{λ}(x_{1}, …, x_{n}) (where x_{1}, …, x_{n} are independent indeterminates and λ = (λ_{1}, …, λ_{n}) is a partition of length ≤ n) may be defined as the quotient of two alternants :

(0.1)  s_{λ}(x_{1}, …, x_{n}) = \frac{det(x_{i}^{λ_{j}+n−j})_{1≤i,j≤n}}{det(x_{i}^{n−j})_{1≤i,j≤n}} .

The denominator on the right-hand side is the Vandermonde determinant, equal to the product ∏_{i<j}(x_{i} − x_{j}).

When λ = (r), s_{λ} is the complete symmetric function h_{r}, and when λ = (1^{r}), s_{λ} is the elementary symmetric function e_{r}. In terms of the h’s, the Schur function s_{λ} (in any number of variables) is given by the Jacobi-Trudi formula

(0.2)  s_{λ} = det(h_{λ_{i}−i+j})_{1≤i,j≤n}.

Dually, in terms of the elementary symmetric functions, s_{λ} is given by the Nägelsbach-Kostka formula

(0.3)  s_{λ} = det(e_{λ′_{i}−i+j})_{1≤i,j≤m}

I.G. MACDONALD

in which λ′ = (λ′_{1}, …, λ′_{m}) is the conjugate [M_{1}, Ch. I, § 1] of the partition λ.

There are (at least) two other determinantal formulas for s_{λ} : one in terms of “hooks” due to Giambelli, and the other in terms of “ribbons” discovered quite recently by Lascoux and Pragacz [LP_{2}]. If λ = (α_{1}, …, α_{p} | β_{1}, …, β_{p}) in Frobenius notation [M_{1}, Ch. I, § 1], Giambelli’s formula is

(0.4)  s_{λ} = det(s_{(α_{i}|β_{j})})_{1≤i,j≤p}.

To state the formula of Lascoux and Pragacz, let

λ^{(i,j)} = (α_{1}, …, α̂_{i}, …, α_{p} | β_{1}, …, β̂_{j}, …, β_{p})

for 1 ≤ i, j ≤ p, where the circumflexes indicate deletion of the symbols they cover ; and let

[α_{i} | β_{j}] = [α_{i} | β_{j}]_{λ} = λ − λ^{(i,j)}.

In particular, [α_{1} | β_{1}] is the rim or border of λ, and [α_{i} | β_{j}] is that part of the border consisting of the squares (h, k) such that h ≥ i and k ≥ j. With this notation explained, the “ribbon formula” is

(0.5)  s_{λ} = det(s_{[α_{i}|β_{j}]})_{1≤i,j≤p}.

Finally, we recall [M_{1}, Ch. I, § 5] the expression of a Schur function as a sum of monomials : namely

(0.6)  s_{λ} = ∑_{T} x^{T}

summed over all column-strict tableaux T of shape λ, where x^{T} = ∏_{s∈λ} x_{T (s)}. (Throughout this article, we shall find it convenient to think of

a tableau T as a mapping from (the shape of) λ into the positive integers, so that T (s) is the integer occupying the square s ∈ λ.)

All these formulas, with the exception of the original definition (0.1), have their extensions to skew Schur functions s_{λ/μ}. In place of (0.2) we have

(0.7)  s_{λ/μ} = det(h_{λ_{i}−μ_{j}−i+j}),

and in place of (0.3) we have

(0.8)  s_{λ/μ} = det(e_{λ′_{i}−μ′_{j}−i+j})

SCHUR FUNCTIONS : THEME AND VARIATIONS

where λ′, μ′ are the partitions conjugate to λ, μ respectively. For the skew versions of (0.4) and (0.5) we refer to [LP_{1}], [LP_{2}]. Finally, in place of (0.6) we have

(0.9)  s_{λ/μ} = ∑_{T} x^{T}

where now T runs over column-strict tableaux of shape λ − μ [M_{1}, Ch. I, § 5].

To complete this introduction we should mention the Cauchy identity

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

and its dual version

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

where λ′ is the conjugate of λ.

If we replace each y_{j} by y_{j}^{−1} and then multiply by a suitable power of y_{1}y_{2} …, (0.11) takes the equivalent form (when the number of variables x_{i}, y_{j} is finite)

(0.11′)  ∏_{{1≤i≤n \atop 1≤j≤m}}(x_{i} + y_{j}) = ∑_{λ} s_{λ}(x)s_{λ′}(y)

summed over partitions λ = (λ_{1}, …, λ_{n}) such that λ_{1} ≤ m, where λ̂ = (λ̂_{1}, …, λ̂_{n}) is the complementary partition defined by λ̂_{i} = m − λ_{n+1−i}, and λ̂′ is the conjugate of λ̂.

The left-hand side of (0.10) may be regarded as defining a scalar product ⟨f, g⟩ on the ring of symmetric functions, as follows. For each r ≥ 1 let p_{r} denote the r th power sum ∑ x_{i}^{r}, and for each partition λ = (λ_{1}, λ_{2}, …) let p_{λ} denote the product p_{λ_{1}}p_{λ_{2}} … The p_{λ} form a ℚ-basis of the ring of symmetric functions (in infinitely many variables, cf. [M_{1}, Ch. I]) with rational coefficients, and the scalar product may be defined by

(0.12)  ⟨p_{λ}, p_{μ}⟩ = δ_{λμ}z_{λ}

where δ_{λμ} is the Kronecker delta, and

z_{λ} = ∏_{i≥1} i^{m_{i}}.m_{i}!,

m_{i} = m_{i}(λ) being the number of parts λ_{j} of λ equal to i, for each i ≥ 1.

The Cauchy formula (0.10) is now equivalent to the statement that the Schur functions s_{λ} form an orthonormal basis of the ring of symmetric functions, i.e.,

(0.13)  ⟨s_{λ}, s_{μ}⟩ = δ_{λμ}.

Also, from this point of view, the skew Schur function s_{λ/μ} may be defined to be s_{μ}^{⊥}(s_{λ}), where s_{μ}^{⊥} is the adjoint of multiplication by s_{μ}, so that ⟨s_{μ}^{⊥}f, g⟩ = ⟨f, s_{μ}g⟩ for any symmetric functions f, g.

1st Variation : Hall-Littlewood symmetric functions

Let x_{1}, …, x_{n}, t be independent variables and let λ = (λ_{1}, …, λ_{n}) be a partition of length ≤ n. The Hall-Littlewood symmetric function indexed by λ [M_{1}, Ch. III] is defined by

(1.1)  P_{λ}(x_{1}, …, x_{n}; t) = \frac{1}{v_{λ}(t)} ∑_{w∈𝔖_{n}} w(x_{1}^{λ_{1}} … x_{n}^{λ_{n}} ∏_{i<j} \frac{x_{i} − tx_{j}}{x_{i} − x_{j}} )

in which v_{λ}(t) ∈ ℤ[t] is a polynomial (with constant term equal to 1) chosen so that the leading monomial in P_{λ} is x^{λ} = x_{1}^{λ_{1}} … x_{n}^{λ_{n}}. When t = 0, the right-hand side of (1.1) is just the expansion of the determinant det(x_{i}^{λ_{j}+n−j}), divided by the Vandermonde determinant, so that when t = 0 the formula (1.1) reduces to the definition (0.1) of the Schur function.

None of determinantal formulas (0.2) – (0.5) have counterparts for the Hall-Littlewood functions (so far as I am aware). In place of (0.6) we have

(1.2)  P_{λ}(x; t) = ∑_{T} ψ_{T} (t)x^{T}

summed over column-strict tableaux T of shape λ, where ψ_{T} (t) ∈ ℤ[t] is a polynomial given explicitly in [M_{1}, Ch. III, § 5].

Finally, in place of the Cauchy identity (0.10) we have

(1.3)  ∏_{i,j} \frac{1 − tx_{i}y_{j}}{1 − x_{i}y_{j}} = ∑_{λ} b_{λ}(t)P_{λ}(x; t)P_{λ}(y; t).

As in the case of the Schur functions, this identity may be interpreted as saying that the symmetric functions P_{λ}(x; t) are pairwise orthogonal with respect to the scalar product defined in terms of the power-sum products by

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

For more details, and in particular for the definition of the polynomials b_{λ}(t) featuring in the right-hand side of (1.3), we refer to [M_{1}, Ch. III].

2nd Variation : Jack symmetric functions

These are symmetric functions P_{λ}^{(α)}(x) depending on a parameter α, but unlike the Hall-Littlewood functions (Variation 1) there is no closed formula such as (1.1) that can serve as definition. The simplest (and original) definition is the following : analogously to (0.12) and (1.4), we define a scalar product by

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

where l(λ) is the length of the partition λ, that is to say the number of non zero parts λ_{i}. For each positive integer n, arrange the partitions of n in lexicographical order (so that (1^{n}) comes first and (n) comes last). Then the P_{λ}^{(α)}(x) are uniquely determined by the two requirements

(2.2)  P_{λ}^{(α)}(x) = x^{λ} + lower terms

where x^{λ} denotes the monomial x_{1}^{λ_{1}}x_{2}^{λ_{2}} …, and by “lower terms” is meant a sum of monomials x^{β} corresponding to sequences β = (β_{1}, β_{2}, …) that precede λ in the lexicographical order ; and

(2.3)  ⟨P_{λ}^{(α)}, P_{μ}^{(α)}⟩^{(α)} = 0  if λ ≠ μ.

The two conditions mean that the P_{λ}^{(α)} may be constructed from the monomial symmetric functions by the Gram-Schmidt process, starting (for partitions of n) with P_{(1^{n})} = e_{n}, the n th elementary symmetric function.

Since the scalar product (2.1) reduces to (0.12) when α = 1, it follows that P_{λ}^{(α)} = s_{λ} when α = 1.

In view of the definition (2.1) of the scalar product, the orthogonality property (2.3) is equivalent to the following generalization of the Cauchy identity (0.10) :

(2.4)  ∏_{i,j}(1 − x_{i}y_{j})^{−1/α} = ∑_{λ} c_{λ}(α)P_{λ}^{(α)}(x)P_{λ}^{(α)}(y)

where the c_{λ}(α) are rational functions of the parameter α which have been calculated explicitly by Stanley [S] — note, however, that his normalization of the Jack symmetric functions is different from ours.

As in the case of the Hall-Littlewood symmetric functions, none of the determinantal formulas (0.2) – (0.5) generalize, so far as is known, to the present situation. In place of (0.6) there is an explicit expression for P_{λ}^{(α)}(x) as a weighted sum of monomials, namely

(2.5)  P_{λ}^{(α)}(x) = ∑_{T} f_{T} (α)x^{T}

summed over column-strict tableaux T of shape λ, where f_{T} (α) is a rational function of α, computed explicitly by Stanley [S], to whom we refer for more details.

Finally, the dual Cauchy formula (0.11) generalizes as follows :

(2.6)  ∏_{i,j}(1 + x_{i}y_{j}) = ∑_{λ} P_{λ}^{(α)}(x)P_{λ′}^{(1/α)}(y)

where as before λ′ is the conjugate of λ.

3rd Variation

Our third variation is a family of symmetric functions P_{λ}(x; q, t), indexed as usual by partitions λ, and depending on two parameters q and t. They include the two previous variations (the Hall-Littlewood symmetric functions and the Jack symmetric functions) as particular cases (see below). Since I have given an extended account of these symmetric functions at a previous Séminaire Lotharingien [M_{2}], I shall be brief here and refer to loc. cit. for all details. The functions may be most simply defined along the same lines as in Variation 2 : we define a new scalar product on the ring of symmetric functions by

(3.1)  ⟨p_{λ}, p_{μ}⟩_{q,t} = δ_{λ,μ}z_{λ} ∏_{i≥1} \frac{1 − q^{λ_{i}}}{1 − t^{λ_{i}}} ,

and then the symmetric functions P_{λ}(x; q, t) are uniquely determined by the two requirements

(3.2) P_{λ}(x; q, t) = x^{λ} + lower terms,

(3.3)  ⟨P_{λ}, P_{μ}⟩_{q,t} = 0  if λ ≠ μ.

If we set q = t^{α} and then let t → 1, in the limit the scalar product (3.1) becomes that defined in (2.1), from which it follows that the Jack symmetric function P_{λ}^{(α)}(x) is the limit of P_{λ}(x; t^{α}, t) as t → 1. Again, if we set q = 0 the scalar product (3.1) reduces to (1.4), and it follows that P_{λ}(x; 0, t) is the Hall-Littlewood symmetric function P_{λ}(x; t). Finally, if q = t then (3.1) reduces to the original scalar product (0.12), and correspondingly P_{λ}(x; q, q) is the Schur function s_{λ}(x).

In view of the definition (3.1) of the scalar product, the orthogonality condition (3.3) is equivalent to the following extension of the Cauchy identity (0.10) :

(3.4)  ∏_{i,j} \frac{(tx_{i}y_{j}; q)_{∞}}{(x_{i}y_{j}; q)_{∞}} = ∑_{λ} b_{λ}(q, t)P_{λ}(x; q, t)P_{λ}(y; q, t).

On the left-hand side of (3.4) we have used the standard notation

(x; q)_{∞} = ∏_{i≥0}(1 − xq^{i}).

On the right-hand side, b_{λ}(q, t) is a rational function of q and t, given explicitly in [M_{2}, § 5].

As in the previous two variations, none of the determinantal formulas for Schur functions quoted in the introduction appear to generalize to the present situation. However, the formula (0.6) for s_{λ} as a sum of monomials does generalize : namely we have

(3.5)  P_{λ}(x; q, t) = ∑_{T} φ_{T} (q, t)x^{T}

where φ_{T} (q, t) is a rational function of q and t, again given explicit expression in [M_{2}, § 5].

Finally, the dual Cauchy formula (0.11) generalizes as follows [M_{2}, § 5] :

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

4 th Variation : factorial Schur functions

Let z = (z_{1}, …, z_{n}) be a sequence of independent variables. For each pair of partitions λ, μ Biedenharn and Louck have defined a skew factorial Schur function t_{λ/μ}(z) in [BL_{1}]. Their original definition (loc. cit.) was couched in terms of Gelfand patterns, and in the equivalent language of tableaux it reads as follows. If T : λ − μ → [1, n] is a column-strict tableau of shape λ − μ, containing only the integers 1, 2, …, n, let

(4.1)  z^{(T)} = ∏_{s∈λ−μ}(z_{T (s)} − T^{∗}(s) + 1),

where T^{∗}(i, j) = T (i, j) + j − i (so that T^{∗} is a row-strict tableau of shape λ − μ). Then t_{λ/μ}(z) is defined by

(4.2)  t_{λ/μ}(z) = ∑_{T} z^{(T)}

summed over all column-strict tableaux T : λ − μ → [1, n].

When μ = 0 they write t_{λ} in place of t_{λ/0}.

It is not particularly obvious from this definition that t_{λ/μ}(z) is in fact a (non-homogeneous) symmetric polynomial in z_{1}, …, z_{n}, and Biedenharn and Louck had some trouble (see [BL_{1}] pp. 407–412) in establishing this fact directly from their definition (4.2).

Some time ago I noticed that it followed rather simply from one of their results (Th. 5 of [BL_{2}]) that an alternative definition of t_{λ}(z) could be given which brought out its analogy with the Schur function s_{λ} : namely (for λ = (λ_{1}, …, λ_{n}) a partition of length ≤ n)

(4.3) t_{λ}(z) = det(z_{i}^{(λ_{j}+n−j)}) / det(z_{i}^{(n−j)}),

where z^{(r)} is the “falling factorial”

(4.4)  z^{(r)} = z(z − 1) … (z − r + 1)  (r ≥ 0).

Note that since z^{(r)} is a monic polynomial in z of degree r, the denominator in (4.3) is just the Vandermonde determinant :

det(z_{i}^{(n−j)}) = det(z_{i}^{n−j}) = ∏_{i<j}(z_{i} − z_{j}).

Hence t_{λ} as defined by (4.3) is the quotient of a skew-symmetric polynomial in z_{1}, …, z_{n} by the Vandermonde determinant, and is therefore a (non-homogeneous) symmetric polynomial in the z_{i}. Moreover, it is clear from (4.3) that t_{λ}(z) is of the form

t_{λ}(z) = s_{λ}(z) + terms of lower degree,

and hence that the t_{λ}(z), as λ runs through the partitions of length ≤ n, form a ℤ-basis of the ring Λ_{n} of symmetric polynomials in z_{1}, …, z_{n}.

In [CL], Chen & Louck show that t_{λ} (and more generally t_{λ/μ}) satisfies a determinantal identity analogous to (0.2) and (0.7). Namely if

w_{r}(z) = t_{(r)}(z)

for all r ≥ 0 (and w_{r}(z) = 0 when r < 0) then we have (loc. cit., Th. 5.1)

(4.5)  t_{λ/μ}(z) = det(w_{λ_{i}−μ_{j}−i+j}(z − μ_{j} + j − 1))

where in general z + r denotes the sequence (z_{1} + r, …, z_{n} + r).

The other determinantal formulas quoted in the introduction all have their analogues for factorial Schur functions. If we define

f_{r}(z) = t_{(1^{r})}(z) (0 ≤ r ≤ n)

(and f_{r}(z) = 0 for r < 0 and r > n), so that the f_{r} are the analogues of the elementary symmetric functions, then we have

(4.6)  t_{λ/μ}(z) = det(f_{λ′_{i}−μ′_{j}−i+j}(z + μ′_{j} − j + 1)).

We shall not stop to prove (4.6) here, nor the hook and ribbon formulas

(4.7)  t_{λ}(z) = det(t_{(α_{i}|β_{j})}(z))_{1≤i,j≤r}
= det(t_{[α_{i} | β_{j}]}(z))_{1≤i,j≤r}

(where λ = (α_{1}, …, α_{r} |β_{1}, …, β_{r}) in Frobenius notation, and for the explanation of the notation [α_{i} |β_{j}] we refer to (0.5)), since they are special cases of the corresponding results in Variation 6, which in their turn are contained in Variation 9. In this development we take (4.3) and (4.5) as definitions of t_{λ} and t_{λ/μ} respectively, and deduce (4.2) from them (see (6.16) below), very much in the spirit of [M_{1}], Chapter I, § 5.

5 th Variation : α-paired factorial Schur functions

Let z = (z_{1}, …, z_{n}) again be a sequence of independent variables, and let α be another variable (or parameter). In parallel with the factorial Schur functions (Variation 4) Biedenharn and Louck [BL_{1}] have defined α-paired factorial Schur functions T_{λ/μ}(α; z). As in the previous case, their definition was couched in terms of Gelfand patterns, and in the equivalent language of tableaux it reads as follows. Let

\overline{z}_{i} = −α − z_{i} (1 ≤ i ≤ n)

and for each column-strict tableau T : λ − μ → [1, n] let

(5.1)  (α : z)^{(T)} = ∏_{s∈λ−μ}(z_{T (s)} − T^{∗}(s) + 1)(\overline{z}_{T (s)} − T^{∗}(s) + 1)

where (as in § 4) T^{∗} is the row-strict tableau associated with T (i.e., T^{∗}(i, j) = T (i, j) + j − i). Then

(5.2)  T_{λ/μ}(α; z) = ∑_{T}(α : z)^{(T)}

summed over all column-strict tableaux T : λ − μ → [1, n]. (When μ = 0, they write T_{λ} in place of T_{λ/0}.)

Chen and Louck remark ([CL], p. 18) that “it is quite surprising that the α-paired factorial Schur function enjoys all the properties of the ordinary factorial Schur function.” The reason for this, we believe, lies in the fact that both these classes of symmetric functions are special cases of those to be defined in our 6 th Variation. In the present situation the falling factorial z^{(r)} is replaced by

z^{(r)} \overline{z}^{(r)} = ∏_{i=0}^{r−1}(z − i)(\overline{z} − i)

where \overline{z} = −α − z; and since

(z − i)(\overline{z} − i) = z\overline{z} + αi + i^{2}

it follows that we may write

z^{(r)} \overline{z}^{(r)} = ∏_{i=1}^{r}(x + a_{i})

where x = z\overline{z} and a_{i} = α(i − 1) + (i − 1)^{2}. In Variation 6 below

the building blocks are the products (x|a)^{r} = ∏_{i=1}^{r}(x + a_{i}) defined by an arbitrary sequence a_{1}, a_{2}, …

We may then take as an alternative definition of T_{λ}(α; z), where λ is a partition of length ≤ n,

(5.3)  T_{λ}(α; z) = \frac{det(z_{i}^{(λ_{j}+n−j)} \overline{z}_{i}^{(λ_{j}+n−j)})}{det(z_{i}^{(n−j)} \overline{z}_{i}^{(n−j)})}

([CL], Th. 6.2) ; all the determinantal formulas (Jacobi-Trudi etc.) together with the tableau definition (5.2) are consequences of (5.3), as we shall show in a more general context in the next section.

6 th Variation

Let R be any commutative ring and let a = (a_{n})_{n∈ℤ} be any (doubly infinite) sequence of elements of R. For each r ∈ ℤ we define τ^{r}a to be the sequence whose nth term is a_{n+r} :

(τ^{r}a)_{n} = a_{n+r}.

Let

(x|a)^{r} = (x + a_{1}) … (x + a_{r})

for each r ≥ 0. Clearly we have

(6.1) (x|a)^{r+s} = (x|a)^{r} (x|τ^{r}a)^{s}

for all r, s ≥ 0.

Now let x = (x_{1}, …, x_{n}) be a sequence of independent indeterminates over R, and for each α = (α_{1}, …, α_{r}) ∈ ℕ^{n} define

(6.2)  A_{α}(x|a) = det((x_{i} |a)^{α_{j}})_{1≤i,j≤n}.

In particular, when α = δ = (n − 1, n − 2, …, 1, 0), since (x_{i} |a)^{n−j} is a monic polynomial in x_{i} of degree (n − j), it follows that

(6.3)  A_{δ}(x|a) = det(x_{i}^{n−j}) = ∏_{i<j}(x_{i} − x_{j})

is the Vandermonde determinant Δ(x), independent of the sequence a. Since A_{α}(x|a) is a skew symmetric polynomial in x_{1}, …, x_{n}, it is therefore divisible by A_{δ}(x|a) in R[x_{1}, …, x_{n}]. Moreover, the determinant A_{α}(x|a) clearly vanishes if any two of the α_{i} are equal, and hence (up to sign) we may assume that α_{1} > ⋯ > α_{n} ≥ 0, i.e., that α = λ + δ where λ = (λ_{1}, …, λ_{n}) is a partition of length ≤ n. It follows therefore that

(6.4)  s_{λ}(x|a) = A_{λ+δ}(x|a) / A_{δ}(x|a)

is a symmetric (but not homogeneous) polynomial in x_{1}, …, x_{n} with coefficients in R. Moreover it is clear from the definitions that

A_{λ+δ}(x|a) = a_{λ+δ}(x) + lower terms,

in the notation of [M_{1}], ch. I, and hence that

s_{λ}(x|a) = s_{λ}(x) + terms of lower degree.

Hence the s_{λ}(x|a) form an R-basis of the ring Λ_{n,R} = R[x_{1}, …, x_{n}]^{𝔖_{n}}.

These polynomials s_{λ}(x|a), and their skew analogues s_{λ/μ}(x|a) to be defined later, form our 6th Variation. They include Variations 4 and 5 as special cases : for Variation 4 we take R = ℤ, x_{i} = z_{i} and a_{n} = 1 − n for all n ∈ ℤ; for Variation 5 we take R = ℤ[α], x_{i} = z_{i}\overline{z}_{i} and a_{n} = (n − 1)α + (n − 1)^{2}. The Schur functions themselves are given by the zero sequence : a_{n} = 0 for all n ∈ ℤ. When λ = (r) we shall write

h_{r}(x|a) = s_{(r)}(x|a) (r ≥ 0)

with the usual convention that h_{r}(x|a) = 0 if r < 0 ; and when λ = (1^{r}) (0 ≤ r ≤ n) we shall write

e_{r}(x|a) = s_{(1^{r})}(x|a) (0 ≤ r ≤ n)

with the convention that e_{r}(x|a) = 0 if r < 0 or r > n.

Let t be another indeterminate and let

f(t) = ∏_{i=1}^{n}(t − x_{i}).

From (6.3) it follows that

f(t) = A_{δ_{n+1}}(t, x_{1}, …, x_{n} |a) / A_{δ_{n}}(x_{1}, …, x_{n} |a).

By expanding the determinant A_{δ_{n+1}} along the top row we shall obtain

(6.5)  f(t) = ∑_{r=0}^{n}(−1)^{r}e_{r}(x|a)(t|a)^{n−r}.

Let E(x|a), H(x|a) be the (infinite) matrices

\begin{aligned}H(x|a) & = (h_{j−i}(x|τ^{i+1}a))_{i,j∈ℤ}, \\ E(x|a) & = ((−1)^{j−i}e_{j−i}(x|τ^{j}a))_{i,j∈ℤ}. \end{aligned}

Both are upper unitriangular, and they are related by

(6.6)  E(x|a) = H(x|a)^{−1}.

Proof. — We have to show that

∑_{j}(−1)^{k−j}e_{k−j}(x|τ^{k}a) h_{j−i}(x|τ^{i+1}a) = δ_{ik}

for all i, k. This is clear if i ≥ k, so we may assume i < k. Since f(x_{i}) = 0 it follows from (6.5) that

∑_{r=0}^{n}(−1)^{r}e_{r}(x|a) (x_{i} |a)^{n−r} = 0

and hence, replacing a by τ^{s−1}a and multiplying by (x_{i} |a)^{s−1}, that

(1)  ∑_{r=0}^{n}(−1)^{r}e_{r}(x|τ^{s−1}a) (x_{i} |a)^{n−r+s−1} = 0

for all s > 0 and 1 ≤ i ≤ n. Now it is clear, from expanding the determinant A_{(m)+δ}(x|a) down the first column, that h_{m}(x|a) is of the form

(2)  h_{m}(x|a) = ∑_{i=1}^{n}(x_{i} |a)^{m+n−1}u_{i}(x)

with coefficients u_{i}(x) rational functions of x_{1}, …, x_{n} independent of m. (In fact, it is easily seen that u_{i}(x) = 1/f′(x_{i}).)

From (1) and (2) it follows that

∑_{r=0}^{n}(−1)^{r}e_{r}(x|τ^{s−1}a) h_{s−r}(x|a) = 0

for each s > 0. Putting s = k − i and replacing a by τ^{i+1}a we obtain

∑_{i≤j≤k}(−1)^{k−j}e_{k−j}(x|τ^{k}a) h_{j−i}(x|τ^{i+1}a) = 0,

as required.

Next, we have analogues of the Jacobi-Trudi and Nägelsbach-Kostka formulas (0.2), (0.3) :

(6.7) If λ is a partition of length ≤ n, then

\begin{aligned}s_{λ}(x|a) & = det(h_{λ_{i}−i+j}(x | τ^{1−j}a)) \\ & = det(e_{λ′_{i}−i+j}(x|τ^{j−1}a)). \end{aligned}

Proof. — Let α = (α_{1}, …, α_{n}) ∈ ℕ^{n}. From equation (2) above we have

\begin{aligned}h_{α_{i}−n+j}(x|τ^{1−j}a) & = ∑_{k=1}^{n}(x_{k} |τ^{1−j}a)^{α_{i}+j−1}u_{k}(x) \\ & = ∑_{k=1}^{n}(x_{k} |a)^{α_{i}} (x_{k} |τ^{1−j}a)^{j−1}u_{k}(x) \end{aligned}

by (6.1). This shows that the matrix H_{α} = (h_{α_{i}−n+j}(x|τ^{1−j}a))_{i,j} is the product of the matrices ((x_{k} |a)^{α_{i}})_{i,k} and B = ((x_{k} |τ^{1−j}a)^{j−1}u_{k}(x))_{k,j}. On taking determinants it follows that

det(H_{α}) = A_{α} det(B).

In particular, when α = δ, the matrix H_{δ} = (h_{j−i}(x|τ^{1−j}a)) is unitriangular and hence has determinant equal to 1. It follows that A_{δ} det(B) = 1 and hence that

det(H_{α}) = A_{α}(x|a) / A_{δ}(x|a),

for all α ∈ ℕ^{n}. Taking α = λ + δ, we obtain the first of the formulas (6.7). The second formula, involving the e’s, is then deduced from it and (6.6), exactly as in the case of Schur functions ([M_{1}], ch. I, (2.9)).

Remark. — A consequence of (6.7) is that the determinant

det(h_{λ_{i}−i+j}(x|τ^{1−j}a)),

which appears to involve not only a_{1}, a_{2}, … but also a_{0}, a_{−1}, …, a_{2−l(λ)}, is in fact independent of the latter.

More generally, if λ and μ are partitions we define

(6.8)  s_{λ/μ}(x|a) = det(h_{λ_{i}−μ_{j}−i+j}(x|τ^{μ_{j}−j+1}a))

and then it follows as above from (6.6) that

(6.9)  s_{λ/μ}(x|a) = det(e_{λ′_{i}−μ′_{j}−i+j}(x|τ^{−μ′_{j}+j−1}a)).

Moreover,

(6.10)  s_{λ/μ}(x|a) = 0 unless 0 ≤ λ′_{i} − μ′_{i} ≤ n for all i.

The proof is the same as for Schur functions : [M_{1}] ch. I, § 5.

The hook and ribbon formulas (0.4), (0.5) remain valid in the present context : if λ = (α_{1}, …, α_{p} |β_{1}, …, β_{p}) in Frobenius notation, then

(6.11)  s_{λ}(x|a) = det(s_{(α_{i} | β_{j})}(x|a))_{1≤i,j≤p}
= det(s_{[α_{i} | β_{j}]}(x|a))_{1≤i,j≤p}.

This will be considered in a more general context in § 9.

Let y = (y_{1}, …, y_{m}) be another set of indeterminates, and let (x, y) denote (x_{1}, …, x_{n}, y_{1}, …, y_{m}). Then we have

(6.12) (i)  E(x, y |a) = E(y |τ^{n}a) E(x|a),
(ii)  H(x, y |a) = H(x|a) H(y |τ^{n}a).

Proof. — It is enough to prove (i), since (ii) then follows by taking inverses and invoking (6.6). From (6.5) we have

∑_{i=0}^{m+n}(−1)^{i}e_{i}(x, y |a)(t|a)^{m+n−i} = ∏_{i=1}^{n}(t − x_{i}) ∏_{j=1}^{m}(t − y_{j})

\begin{matrix}= ∑_{j=0}^{n}(−1)^{j}e_{j}(x|a)(t|a)^{n−j} ∑_{k=0}^{m}(−1)^{k}e_{k}(y |τ^{n−j}a)(t|τ^{n−j}a)^{m−k} \\ = ∑_{j,k}(−1)^{j+k}e_{j}(x|a) e_{k}(y |τ^{n−j}a) (t|a)^{m+n−j−k} \end{matrix}

by use of (6.1). Since the polynomials (t|a)^{r}, r ≥ 0 are linearly independent, we may equate coefficients to obtain

e_{i}(x, y |a) = ∑_{j+k=i} e_{j}(x|a) e_{k}(y |τ^{n−j}a).

With a change of notation this relation takes the form

(−1)^{k−i}e_{k−i}(x, y |τ^{k}a) = ∑_{j}(−1)^{k−j}e_{k−j}(x|τ^{k}a) (−1)^{j−i}e_{j−k}(y |τ^{n+j}a)

which establishes (i).

(6.13) Let λ, μ be partitions. Then

s_{λ/μ}(x, y |a) = ∑_{ν} s_{ν/μ}(x|a) s_{λ/ν}(y |τ^{n}a).

Proof. — Let r ≥ max(l(λ), l(μ)). By definition (6.8), s_{λ/μ}(x, y |a) is the r × r minor of H(x, y |a) corresponding to the row indices μ_{1} − 1, …, μ_{r} − r and the column indices λ_{1} − 1, …, λ_{r} − r, that is to say, it is the element of ⋀^{r} H(x, y |a) indexed by these sets of indices. The formula (6.13) now follows from (6.12) (ii) and the functoriality of exterior powers,^{∗} which together imply that ⋀^{r} H(x, y |a) = ⋀^{r} H(x|a). ⋀^{r} H(y |τ^{n}a).

By iterating (6.13) we obtain the following result. Let x^{(i)}, …, x^{(n)} be n sets of variables, where x^{(i)} = (x_{1}^{(1)}, …, x_{r_{i}}^{(i)}), and let λ, μ be partitions. Then

(6.14) s_{λ/μ}(x^{(i)}, …, x^{(n)} |a) = ∑_{(ν)} ∏_{i=1}^{n} s_{ν^{(i)}/ν^{(i−1)}}(x^{(i)} |τ^{r_{1}+···+r_{i−1}}a)

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

We shall apply (6.14) in the case that each x^{(i)} consists of a single variable x_{i} (so that r_{i} = 1 for 1 ≤ i ≤ n). For a single x we have s_{λ/μ}(x|a) = 0 unless λ − μ is a horizontal strip, by (6.10) ; and if λ − μ is a horizontal strip it follows from (6.8) that

\begin{aligned}s_{λ/μ}(x|a) & = ∏_{i≥1} h_{λ_{i}−μ_{i}}(x|τ^{μ_{i}−i+1}a) \\ & = ∏_{i≥1}(x|τ^{μ_{i}−i+1}a)^{λ_{i}−μ_{i}}. \end{aligned}

∗ also known as the Cauchy-Binet identity.

I.G. MACDONALD

since h_{r}(x|a) = s_{(r)}(x|a) = (x|a)^{r} in the case of a single x, from the definition (6.4). Hence (6.15) For a single x we have

s_{λ/μ}(x|a) = ∏_{s∈λ−μ}(x + a_{c(s)+1})

if λ − μ is a horizontal strip, and s_{λ/μ}(x|a) = 0 otherwise. (Here c(s) is the content of s, i.e., c(s) = j − i if s = (i, j).)

From (6.14) and (6.15) it now follows that if x = (x_{1}, …, x_{n})

(6.16)  s_{λ/μ}(x|a) = ∑_{T}(x|a)^{T}

summed over column strict tableaux T : λ − μ → [1, n], where

(x|a)^{T} = ∏_{s∈λ−μ}(x_{T (s)} + a_{T^{∗}(s)})

and T^{∗}(i, j) = T (i, j) + j − i (so that T^{∗} is row-strict).

When a_{i} = 1 − i for all i ∈ ℤ (Variation 4), (6.16) reduces to the definition (4.2) of the factorial Schur functions.

Finally, there is an analogue of the dual Cauchy formula : namely (with the notation of (0.11′))

(6.17)  ∏_{i=1}^{n} ∏_{j=1}^{m}(x_{i} + y_{j}) = ∑_{λ} s_{λ}(x|a) s_{λ′}(y | − a)

where −a is the sequence (−a_{n})_{n∈ℤ}.

Proof. — Consider the quotient

A_{δ_{m+n}}(x, y) / A_{δ_{n}}(x)A_{δ_{m}}(y)

which by (6.3) is equal to ∏_{i,j}(x_{i}−y_{j}). On the other hand, Laplace expansion of the determinant A_{δ_{m+n}}(x, y) gives

A_{δ_{m+n}}(x, y) = ∑_{λ⊂(m^{n})}(−1)^{|_{λ}|}A_{λ+δ_{n}}(x) A_{_{λ′}+δ_{m}}(y).

Hence we have

∏_{i,j}(x_{i} − y_{j}) = ∑_{λ⊂(m^{n})}(−1)^{|_{λ}|}s_{λ}(x|a) s_{λ′}(y |a)

and by replacing each y_{j} by −y_{j} we obtain (6.17).

Remark. — From the definition (6.1) it follows that

(x|a)^{r} = ∑_{k≥0} x^{k}e_{r−k}(a^{(r)}),

where a^{(r)} = (a_{1}, a_{2}, …, a_{r}). Hence, with x = (x_{1}, …, x_{n}),

A_{α}(x|a) = det( ∑_{β_{k}≥0} x_{i}^{β_{k}}e_{β_{k}−α_{j}}(a^{(α_{j})}))

= ∑_{β} det(x_{i}^{β_{k}}) det(e_{β_{k}−α_{j}}(a^{(α_{j})}))

summed over β = (β_{1}, …, β_{n}) ∈ ℕ^{n} such that β_{1} > β_{2} > ⋯ > β_{n}.

On dividing both sides by the Vandermonde determinant Δ(x) and replacing α, β by λ + δ, μ + δ respectively, we obtain

(6.18)  s_{λ}(x|a) = ∑_{μ⊂λ} s_{μ}(x) det(e_{λ_{i}−μ_{j}−i+j}(a^{(λ_{j}+n−j)})),

symmetric in the x’s but not in the a’s.

Now assume that the a’s are independent variables ; then we can let n → ∞ (which would not have been possible in the contexts of Variations 4 and 5). In the limit the right-hand side of (6.18) becomes, by virtue of (0.8),

∑_{μ⊂λ} s_{μ}(x)s_{λ′/μ′}(a)

where x = (x_{1}, x_{2}, …) and a = (a_{1}, a_{2}, …). It follows that

(6.19)  lim_{n→∞} s_{λ}(x_{1}, …, x_{n} |a) = s_{λ}(x||a),

where s_{λ}(x||a) is the “supersymmetric Schur function” defined by

s_{λ}(x||a) = det(h_{λ_{i}−i+j}(x||a))

in which h_{r}(x||a) is the coefficient of t^{r} in the power series expansion of

∏_{i≥1}(1 − tx_{i})^{−1} ∏_{j≥1}(1 + ta_{j}). Thus the limit as n → ∞ of s_{λ}(x_{1}, …, x_{n} |a)

is symmetric in the a’s as well as in the x’s. From (6.19) and (6.16) we conclude that, with the notation of (6.16),

(6.20)  s_{λ}(x||a) = ∑_{T}(x|a)^{T}

summed over all column-strict tableaux T of shape λ with positive integer entries.

For the skew functions the corresponding result reads as follows. Let x = (x_{n})_{n∈ℤ}, a = (a_{n})_{n∈ℤ} now be two doubly infinite sequences of independent variables, and let λ, μ be partitions such that λ ⊃ μ. The “skew supersymmetric Schur function” s_{λ/μ}(x||a) is defined by

s_{λ/μ}(x||a) = det(h_{λ_{i}−μ_{j}−i+j}(x||a)),

where h_{r}(x||a) is now the coefficient of t^{r} in the power series expansion of ∏_{i∈ℤ}(1 − tx_{i})^{−1} ∏_{j∈ℤ}(1 + ta_{j}). Then we have

(6.21)  s_{λ/μ}(x||a) = ∑_{T}(x|a)^{T}

summed over all column-strict tableaux T : λ − μ → ℤ. (6.20) and (6.21) were found independently by Ian Goulden and Curtis Greene.

7 th Variation

Here we shall work over a finite field F = 𝔽_{q} of cardinality q (so that q is a prime power). Let x_{1}, …, x_{n} be independent indeterminates over F, and let V ⊂ F [x_{1}, …, x_{n}] denote the F -vector space spanned by the x_{i}, so that F [x_{1}, …, x_{n}] is the symmetric algebra S(V) of V over F.

For each α = (α_{1}, …, α_{n}) ∈ ℕ^{n} we define

(7.1) A_{α} = det(x_{i}^{q^{α_{j}}} )_{1≤i,j≤n}.

If v ∈ V, v ≠ 0, so that (7.2) v = a_{1}x_{1} + ⋯ + a_{n}x_{n} with coefficients a_{i} ∈ F, not all zero, then we have

v^{q^{r}} = a_{1}x_{1}^{q^{r}} + ⋯ + a_{n}x_{n}^{q^{r}}

for all integers r ≥ 0, from which it follows that the determinant (7.1) is divisible by v in S(V). Hence if V_{0} is the subset of V consisting of all the vectors (7.2) for which the first non zero coefficient a_{i} is equal to 1, we see that A_{α} is divisible in S(V) by the product

(7.3)  P = P (x_{1}, …, x_{n}) = ∏_{v∈V_{0}} v,

which is homogeneous of degree

Card(V_{0}) = q^{n−1} + q^{n−2} + ⋯ + 1.

In particular, when α = δ_{n} = δ = (n − 1, n − 2, …, 1, 0), A_{δ} is divisible by P, and is a homogeneous polynomial of the same degree q^{n−1} + q^{n−2} + ⋯ + 1 ; moreover the leading term in each of P and A_{δ} is the monomial x_{1}^{q^{n−1}}x_{2}^{q^{n−2}} … x_{n}, and therefore

(7.4)  P = A_{δ}.

The determinant A_{α} clearly vanishes if any two of the α_{i} are equal, and hence (up to sign) we may assume that α_{1} > ⋯ > α_{n} ≥ 0, i.e., that α = λ + δ where λ = (λ_{1}, …, λ_{n}) is a partition of length ≤ n. It follows from what we have just proved that

(7.5)  S_{λ}(x_{1}, …, x_{n}) = A_{λ+δ} / A_{δ}

is a polynomial, i.e., an element of S(V), homogeneous of degree

∑_{i=1}^{n}(q^{λ_{i}} − 1)q^{n−i}.

These polynomials S_{λ} (and their skew analogues S_{λ/μ} that we shall define later) constitute our 7 th Variation. Clearly they are symmetric in x_{1}, …, x_{n}; but they are in fact invariant under a larger group, namely the group GL_{n}(F) (or GL(V)).

For if g = (g_{ij}) ∈ GL_{n}(F), we have

gx_{i} = ∑_{k=1}^{n} g_{ki}x_{k}

and therefore

(gx_{i})^{q^{r}} = ∑_{k} g_{ki}x_{k}^{q^{r}}

for all integers r ≥ 0, from which it follows that gA_{α} = (det g)A_{α} and hence that

S_{λ}(gx_{1}, …, gx_{n}) = S_{λ}(x_{1}, …, x_{n}).

Consequently S_{λ}(x_{1}, …, x_{n}) depends only on (λ and) the vector space V, and not on the particular basis x_{1}, …, x_{n} of V, and accordingly we shall write S_{λ}(V) in place of S_{λ}(x_{1}, …, x_{n}) from now on.

When λ = (r) we shall write

H_{r}(V) = S_{(r)}(V) (r ≥ 0)

with the usual convention that H_{r}(V) = 0 if r < 0 ; and when λ = (1^{r}) (0 ≤ r ≤ n) we shall write

E_{r}(V) = S_{(1^{r})}(V) (0 ≤ r ≤ n)

with the convention that E_{r}(V) = 0 if r < 0 or r > n.

A well-known theorem of Dickson states that the subalgebra of GL(V)- invariant elements of S(V) is a polynomial algebra over F, generated by the E_{r}(V) (1 ≤ r ≤ n). But by contrast with the classical situation, the S_{λ}(V) do not form an F -basis of S(V)^{GL(V)}, as one sees already in the simplest case n = 1.

Let t be another indeterminate and let

(7.6)  f_{V} (t) = ∏_{v∈V} (t + v).

From (7.3) and (7.4) it follows that

\begin{aligned}f_{V} (t) & = P (t, x_{1}, …, x_{n})/P (x_{1}, …, x_{n}) \\ & = A_{δ_{n+1}}(t, x_{1}, …, x_{n})/A_{δ_{n}}(x_{1}, …, x_{n}). \end{aligned}

By expanding the determinant A_{δ_{n+1}} along the top row, we shall obtain

(7.7)  f_{V} (t) = t^{q^{n}} − E_{1}(V)t^{q^{n−1}} + ⋯ + (−1)^{n}E_{n}(V)t.

Since (at + bu)^{q^{r}} = at^{q^{r}} + bu^{q^{r}} for all a, b ∈ F and integers r ≥ 0 (t, u being indeterminates) it follows from (7.7) that

(7.8)  f_{V} (at + bu) = af_{V} (t) + bf_{V} (u),

i.e., that f_{V} is an additive (or Ore) polynomial.

Let φ : S(V) → S(V) denote the Frobenius map, namely

φ(u) = u^{q} (u ∈ S(V)).

The mapping φ is an F -algebra endomorphism of S(V), its image being F [x_{1}^{q}, …, x_{n}^{q}]. Since we shall later encounter negative powers of φ, it is convenient to introduce

Ŝ(V) = ⋃_{r≥0} S(V)^{q^{−r}}

where S(V)^{q^{−r}} = F [x_{1}^{q^{−r}}, …, x_{n}^{q^{−r}}]. On Ŝ(V), φ is an automorphism.

Let E(V), H(V) be the (infinite) matrices

\begin{aligned}H(V) & = (φ^{i+1}H_{j−i}(V))_{i,j∈ℤ}, \\ E(V) & = ((−1)^{j−i}φ^{j}E_{j−i}(V))_{i,j∈ℤ}. \end{aligned}

Both are upper triangular, with 1’s on the diagonal. They are related by

(7.9)  E(V) = H(V)^{−1}.

Proof. — We have to show that

∑_{j}(−1)^{k−j}φ^{k}(E_{k−j})φ^{i+1}(H_{j−i}) = δ_{ik}

for all i, k. This is clear if i ≥ k. If i < k, we may argue as follows : since f_{V} (x_{i}) = 0 it follows from (7.7) that

φ^{n}(x_{i}) − E_{1}φ^{n−1}(x_{i}) + ⋯ + (−1)^{n}E_{n}x_{i} = 0

and hence that

(1) φ^{n+r−1}(x_{i}) − φ^{r−1}(E_{1})φ^{n+r−2}(x_{i})

+ ⋯ + (−1)^{n}φ^{r−1}(E_{n})φ^{r−1}(x_{i}) = 0

for all r ≥ 0 and 1 ≤ i ≤ n. On the other hand, by expanding the determinant A_{(r)+δ} down the first column, it is clear that H_{r} = H_{r}(V) is of the form

(2)  H_{r} = ∑_{i=1}^{n} u_{i}φ^{n+r−1}(x_{i})

with coefficients u_{i} ∈ F (x_{1}, …, x_{n}) independent of r. From (1) and (2) it follows that

(3)  H_{r} − φ^{r−1}(E_{1})H_{r−1} + ⋯ + (−1)^{n}φ^{r−1}(E_{n})H_{r−n} = 0

for each r ≥ 0. Putting r = k − i and operating on (3) with φ^{i+1}, we obtain

∑_{i≤j≤k}(−1)^{k−j}φ^{k}(E_{k−j})φ^{i+1}(H_{j−i}) = 0

as required.

Next, we have analogues of the Jacobi-Trudi and Nägelsbach-Kostka formulas (0.2), (0.3) :

(7.10) Let λ be a partition of length ≤ n = dim V. Then

\begin{aligned}S_{λ}(V) & = det(φ^{1−j}H_{λ_{i}−i+j}(V)) \\ & = det(φ^{j−1}E_{λ′_{i}−i+j}(V)). \end{aligned}

Proof. — Let α = (α_{1}, …, α_{n}) ∈ ℕ^{n}. From equation (2) above we have

φ^{1−j}(H_{α_{i}−n+j}) = ∑_{k=1}^{n} φ^{α_{i}}(x_{k})φ^{1−j}(u_{k}) (1 ≤ i, j ≤ n)

which shows that the matrix (φ^{1−j}H_{α_{i}−n+j})_{i,j} is the product of the matrices (φ^{α_{i}}x_{k})_{i,k} and (φ^{1−j}u_{k})_{k,j}. On taking determinants it follows that

(1)  det(φ^{1−j}H_{α_{i}−n+j}) = A_{α}B

where B = det(φ^{1−j}u_{k}).

In particular, taking α = δ (so that α_{i} − n + j = j − i), the left-hand side of (1) becomes equal to 1, so that A_{δ}B = 1 and therefore

det(φ^{1−j}H_{α_{i}−n+j}) = A_{α} / A_{δ}

for all α ∈ ℕ^{n}. Taking α = λ+δ, we obtain the first of the formulas (7.10). The second formula (involving the E’s) is then deduced from it and (7.9), exactly as in the case of Schur functions ([M_{1}], Ch. I § 2).

More generally, if λ and μ are partitions we define

(7.11)  S_{λ/μ}(V) = det(φ^{μ_{j}−j+1}H_{λ_{i}−μ_{j}−i+j}(V))

and then it follows as above from (7.9) that also

(7.11′)  S_{λ/μ}(V) = det(φ^{−μ′_{j}+j−1}E_{λ′_{i}−μ′_{j}−i+j}(V)).

Moreover

(7.12)  S_{λ/μ} = 0 unless 0 ≤ λ′_{i} − μ′_{i} ≤ n for all i ≥ 1.

The proof is the same as for Schur functions ([M_{1}], Ch. I § 5).

The hook and ribbon formulas (0.4), (0.5) remain valid in the present context : if λ = (α_{1}, …, α_{p} |β_{1}, …, β_{p}) in Frobenius notation, then

\begin{aligned}S_{λ}(V) & = det(S_{(α_{i} | β_{j})}(V))_{1≤i,j≤p} \\ & = det(S_{[α_{i} | β_{j}]}(V))_{1≤i,j≤p} \end{aligned}

in the notation of (0.5). These identities are formal consequences of the definition (7.11) : see § 9 below.

Remark. — Since H_{r}(V) has degree (q^{r} − 1)q^{n−1}, it follows that the degree of φ^{μ_{j}−j+1}H_{λ_{i}−μ_{j}−i+j}(V) is

q^{μ_{j}−j+1}(q^{λ_{i}−μ_{j}−i+j} − 1)q^{n−1} = q^{λ_{i}+n−i} − q^{μ_{j}+n−j}.

Hence the determinant (7.11) (and likewise (7.12)) is isobaric, and S_{λ/μ}(V) is homogeneous of degree

(7.13)  ∑_{i=1}^{n}(q^{λ_{i}} − q^{μ_{i}})q^{n−i}.

We shall next consider analogues of the “addition formula”

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

for Schur functions ([M_{1}], Ch. I § 5). For this purpose let U be an F -vector subspace of V. From (7.8) it follows that

f_{U} : v ↦ f_{U}(v) = ∏_{u∈U}(v + u)

is an F -linear mapping of V into S(V) with kernel U. Hence f_{U}(V) is isomorphic to the quotient of V by U, and we shall write

(7.14)  f_{U}(V) = V/U

thereby reserving the notation V/U for this particular embedding of the quotient space in S(V). If U′ is a vector space complement of U in V, the elements of V/U are the products f_{U}(v) = ∏_{u∈U}(v+u) as v runs through U′,

that is to say, they are the products in S(V) of the elements of the cosets of U in V.

With this notation we have

(7.15)  f_{V} = f_{V/U} ∘ f_{U}  [i.e., f_{V} (t) = f_{V/U}(f_{U}(t))].

Proof. — We have

f_{V/U}(f_{U}(t)) = ∏_{w∈V/U}(f_{U}(t) + w)

= ∏_{u′∈U′}(f_{U}(t) + f_{U}(u′))

= ∏_{u′∈U′} f_{U}(t + u′)  [by (7.8)]

= ∏_{{u∈U \atop u′∈U′}} (t + u + u′) = f_{V} (t).

(7.16) Let T be a vector subspace of U. Then

V/U = (V/T) / (U/T)

(with equality, not merely isomorphism).

Proof. — By definition

(V/T)/(U/T) = f_{U/T} (V/T) = f_{U/T} (f_{T} (V))
= f_{U}(V) = V/U.  [by (7.15)]

(7.17) (i) E(V) = φ^{dim(V/U)}(E(U)) · E(V/U),

(ii) H(V) = H(V/U) · φ^{dim(V/U)}(H(U)).

Proof. — It is enough to prove (i), since (ii) then follows by taking inverses and using (7.9). From (7.7) and (7.15) we have (dim V = n, dim U = m)

∑_{i≥0}(−1)^{i}E_{i}(V)φ^{n−i}(t) = f_{V} (t) = f_{V/U}(f_{U}(t))

\begin{matrix}= ∑_{j≥0}(−1)^{j}E_{j}(V/U)φ^{n−m−j}(∑_{k≥0}(−1)^{k}E_{k}(U)φ^{m−k}(t)) \\ = ∑_{j,k≥0}(−1)^{j+k}E_{j}(V/U)φ^{n−m−j}(E_{k}(U))φ^{n−j−k}(t) \end{matrix}

and therefore

E_{i}(V) = ∑_{j+k=i} E_{j}(V/U)φ^{n−m−j}(E_{k}(U)).

With a change of notation this can be written in the form

E_{c−a}(V) = ∑_{a≤b≤c} φ^{n−m+b−c}(E_{b−a}(U))E_{c−b}(V/U)

or equivalently

(−1)^{c−a}φ^{c}(E_{c−a}(V))

= ∑_{b}(−1)^{b−a}φ^{n−m+b}(E_{b−a}(U))(−1)^{c−b}φ^{c}(E_{c−b}(V/U)),

proving (i).

(7.18) Let λ, μ be partitions. Then

S_{λ/μ}(V) = ∑_{ν} S_{ν/μ}(V/U) · φ^{dim(V/U)}(S_{λ/μ}(U)).

Proof. — Suppose r ≥ max(l(λ), l(μ)). By definition (7.11), S_{λ/μ}(V) is the r × r minor of H(V) corresponding to the row indices μ_{1} − 1, …, μ_{r} − r and the column indices λ_{1} − 1, …, λ_{r} − r, that is to say, it is the

element of ⋀^{r} H(V) indexed by these sets of indices. The formula (7.18) now follows from (7.17) (ii) and the functoriality of exterior powers, which

together imply that ⋀^{r} H(V) = ⋀^{r} H(V/U) · φ^{dim(V/U)} ⋀^{r} H(U).

By iteration of (7.18) (and making use of (7.16)) we obtain

(7.19) Let V_{0} > V_{1} > ⋯ > V_{r} be a chain of subspaces of V. Then

S_{λ/μ}(V_{0}/V_{r}) = ∑_{(ν)} ∏_{i=1}^{r} φ^{dim(V_{0}/V_{i−1})}S_{ν^{(i)}/ν^{(i−1)}}(V_{i−1}/V_{i})

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

As in the case of Schur functions, we shall use (7.19) to express S_{λ/μ}(V) as a sum over column-strict tableaux of shape λ − μ.

If U is any finite-dimensional subspace of F (V), let

π(U) = ∏_{{u∈U \atop u≠0}} u.

From (7.7) it follows that

(7.20)  π(U) = (−1)^{d}E_{d}(U)

if dim U = d.

If U′ is a subspace of U, then U/U′ is defined by (7.14) as a subspace of S(U) ⊂ S(V), and its elements are the products (in S(U) or S(V)) of the elements of the cosets of U′ in U. From this it follows that

(7.21)  π(U/U′) = π(U)/π(U′) = ∏_{u∈U−U′} u.

We now consider the case where U is 1-dimensional.

(7.22) Let U be a 1-dimensional subspace of S(V) and let λ, μ be partitions. Then

S_{λ/μ}(U) = (−1)^{|λ−μ|} ∏_{s∈λ−μ} φ^{c(s)}π(U)

if λ − μ is a horizontal strip, and is zero otherwise. (Here c(s) is the content of s : c(s) = j − i if s = (i, j).)

Proof. — Since dim U = 1 we have E_{r}(U) = 0 for r ≥ 2 and hence, by (7.10),

\begin{aligned}H_{r}(U) = S_{(r)}(U) & = ∏_{j=1}^{r} φ^{j−1}E_{1}(U) \\ & = (−1)^{r} ∏_{j=1}^{r} φ^{j−1}π(U) \end{aligned}

by (7.20). From (7.12), we have S_{λ/μ}(U) = 0 unless 0 ≤ λ′_{i} − μ′_{i} ≤ 1 for all i, that is to say, unless λ − μ is a horizontal strip. If on the other hand λ − μ is a horizontal strip, we have

S_{λ/μ}(U) = det(φ^{μ_{j}−j+1}H_{λ_{i}−μ_{j}−i+j}(U))

= ∏_{i≥1} φ^{μ_{i}−i+1}H_{λ_{i}−μ_{i}}(U)

= (−1)^{|λ−μ|} ∏_{i≥1} φ^{μ_{i}−i+1} ∏_{j=1}^{λ_{i}−μ_{i}} φ^{j−1}π(U)

= (−1)^{|λ−μ|} ∏_{s∈λ−μ} φ^{c(s)}π(U).

Now let

𝔙 : V = V_{0} > V_{1} > ⋯ > V_{n} = 0

be a (full) flag in V, so that V_{i} is a vector subspace of V of dimension (n−i), for each i. Let

π_{i}(𝔙) = π(V_{i−1}/V_{i}) = ∏_{v∈V_{i−1}−V_{i}} v

for 1 ≤ i ≤ n.

From (7.19) and (7.22) it follows that

\begin{aligned}S_{λ/μ} & = ∑_{(ν)} ∏_{i=1}^{n} φ^{i−1}S_{ν^{(i)}/ν^{(i−1)}}(V_{i−1}/V_{i}) \\ & = (−1)^{|λ−μ|} ∑_{(ν)} ∏_{i=1}^{n} ∏_{s∈ν^{(i)}/ν^{(i−1)}} φ^{i−1+c(s)}π_{i}(𝔙) \end{aligned}

summed over all sequences (ν) = (ν^{(0)}, ν^{(1)}, …, ν^{(n)}) of partitions such that μ = ν^{(0)} ⊂ ν^{(1)} ⊂ ⋯ ⊂ ν^{(n)} = λ and each ν^{(i)} − ν^{(i−1)} is a horizontal strip. Such sequences are in one-one correspondence with column-strict tableaux T : λ − μ → [1, n], and hence we obtain

(7.23)  S_{λ/μ}(V) = (−1)^{|λ−μ|} ∑_{T} ψ(T, 𝔙)

summed over column strict tableaux T : λ − μ → [1, n], where

ψ(T, 𝔙) = ∏_{s∈λ−μ} φ^{T^{∗}(s)−1}π_{T (s)}(𝔙),

and T^{∗}(i, j) = T (i, j) + j − i (as in (6.16)).

Remark. — The degree of ψ(T, 𝔙) is

∑_{s∈λ−μ} q^{c(s)+T (s)−1}(q^{n−T (s)+1} − q^{n−T (s)}) = ∑_{s∈λ−μ} q^{c(s)}(q^{n} − q^{n−1})

which is easily seen to be equal to ∑_{i=1}^{n}(q^{λ_{i}} − q^{μ_{i}})q^{n−i}, in agreement with (7.13).

In the formula (7.23) the flag 𝔙 is fixed and the sum is over tableaux T. Since S_{λ/μ}(V) is GL(V)-invariant, and since the number of flags 𝔙 in V is congruent to 1 modulo q, we may sum over all flags as well :

(7.23′)  S_{λ/μ}(V) = (−1)^{|λ−μ|} ∑_{T,𝔙} ψ(T, 𝔙),

It seems plausible that (when μ = 0) there should be another expression for S_{λ}(V) as a sum over flags, namely

(7.24 ?)  S_{λ}(V) = (−1)^{|λ|} ∑_{𝔙} ψ(T_{0}, 𝔙)

where T_{0} is the tableau defined by T_{0}(i, j) = i for all (i, j) ∈ λ. For this tableau we have

\begin{aligned}ψ(T_{0}, 𝔙) & = ∏_{(i,j)∈λ} ψ^{j−1}π_{i}(𝔙) \\ & = (−1)^{|λ|} ∏_{i≥1} H_{λ_{i}}(V_{i−1}/V_{i}). \end{aligned}

The formula (7.24 ?) is true for example when λ = (1^{r}) (1 ≤ r ≤ n), and in some other cases ; but I do not know whether it is true generally. It would be enough to show that, if l(λ) < n,

(7.25 ?)  S_{λ}(V) = ∑_{L} S_{λ}(V/L)

summed over all lines (i.e., 1-dimensional subspaces) L in V.

Finally, we shall indicate an analogue of the dual Cauchy formula (0.11′). Let V (resp. W) be the F -vector space spanned by x_{1}, …, x_{n} (resp. y_{1}, …, y_{m}), the x’s and y’s being independent indeterminates over F. Let

π(V, W) = ∏_{L,M}(π(L) + π(M))

where the product on the right is over all lines L in V and M in W. Then we have (with the notation of (0.11′))

(7.26)  π(V, W) = ∑_{λ} S_{λ}(V)S_{λ′}(W).

Proof. — Consider the quotient

A_{δ_{m+n}}(x, y) / A_{δ_{n}}(x)A_{δ_{m}}(y)

which by (7.4) is equal to the product

∏_{{v∈V_{0} \atop w∈W} ; w≠0} (v + w) = ∏_{{w∈V_{0} \atop w∈W_{0}}} (v^{q−1} − w^{q−1})

= ∏_{L,M}(π(L) − π(M))  (1)

(product over lines L < V and M < W). On the other hand, by Laplace expansion of the determinant A_{δ_{m+n}}(x, y), we obtain

A_{δ_{m+n}}(x, y) = ∑_{λ⊂(m^{n})}(−1)^{|_{λ}|}A_{λ+δ_{n}}(x)A_{_{λ′}+δ_{m}}(y).  (2)

From (1) and (2) it follows that

∏_{L,M}(π(L) − π(M)) = ∑_{λ⊂(m^{n})}(−1)^{|_{λ}|}S_{λ}(V)S_{λ′}(W).

Finally, to get rid of the minus signs, replace each y_{j} by ωy_{j}, where ω lies in an extension field of F and satisfies ω^{q−1} = −1.

8 th Variation : flagged Schur functions

Let x_{1}, x_{2}, … be independent variables. For all positive integers a, b, r define h_{r}(a, b) (resp. e_{r}(a, b)) to be the complete (resp. elementary) symmetric function of degree r in the variables x_{a}, x_{a+1}, …, x_{b} if a ≤ b, and to be zero if a > b; also define h_{0}(a, b) = e_{0}(a, b) = 1 for all a, b, and h_{r}(a, b) = e_{r}(a, b) = 0 for all a, b when r < 0.

Let λ, μ be partitions of length ≤ n and let a = (a_{1}, …, a_{n}), b = (b_{1}, …, b_{n}) be sequences of positive integers. The row-flagged Schur function s_{λ/μ}(a, b) with row flags a, b is defined [W] to be

(8.1)  s_{λ/μ}(a, b) = det(h_{λ_{i}−μ_{j}−i+j}(a_{j}, b_{i}))_{1≤i,j≤n}

It is zero unless λ ⊃ μ, which we assume henceforth.

Let m ≥ λ_{1} (so that μ ⊂ λ ⊂ (m^{n})) and let a′ = (a′_{1}, …, a′_{m}), b′ = (b′_{1}, …, b′_{m}) be sequences of positive integers. The notion dual to (8.1) is that of the column-flagged Schur function s′_{λ/μ}(a′, b′)^{∗} with column flags (a′, b′) :

(8.1′)  s′_{λ/μ}(a′, b′) = det(e_{λ′_{i}−μ′_{j}−i+j}(a′_{j}, b′_{i}))_{1≤i,j≤m}

where λ′, μ′ are the partitions conjugate to λ, μ respectively.

With mild restrictions on the flags these Schur functions can be expressed as sums over tableaux :

(8.2) Suppose that a_{i} ≤ a_{i+1} and b_{i} ≤ b_{i+1} whenever μ_{i} < λ_{i+1} (i.e., the sequences a, b are increasing on each connected component of λ−μ). Then

s_{λ/μ}(a, b) = ∑_{T} x^{T}

summed over column-strict tableaux T of shape λ − μ such that a_{i} ≤

T (i, j) ≤ b_{i} for all (i, j) ∈ λ − μ, where as usual x^{T} = ∏_{s∈λ−μ} x_{T (s)}.

(8.2′) Suppose that a′_{i} − μ′_{i} ≤ a′_{i+1} − μ′_{i+1} + 1 and b′_{i} − λ′_{i} ≤ b′_{i+1} − λ′_{i+1} + 1 whenever μ′_{i} < λ′_{i+1}. Then

s′_{λ/μ}(a′, b′) = ∑_{T} x^{T}

summed over column-strict tableaux T of shape λ − μ such that a′_{j} ≤ T (i, j) ≤ b′_{j} for all (i, j) ∈ λ − μ.

Both these results are proved in [W].

In general (i.e., for arbitrary choices of the flags a, b) the row flagged Schur function s_{λ/μ}(a, b) will not be equal to any column-flagged s′_{λ/μ}(a′, b′). However, there is the following duality theorem :

(8.3) Let λ, μ be partitions such that μ ⊂ λ ⊂ (m^{n}) and let α, β be positive integers such that α > m and β − α ≥ m. Let

a_{i} = α + i − μ_{i} − 1,  b_{i} = β + i − λ_{i}  (1 ≤ i ≤ n),
a′_{j} = α + μ′_{j} − j + 1,  b′_{j} = β + λ′_{j} − j  (1 ≤ j ≤ m).

Then s_{λ/μ}(a, b) = s′_{λ/μ}(a′, b′).

This is a particular case of (9.6′) in the next section ; alternatively, it is not hard to verify that with these choices of a, b, a′, b′ a column-strict tableau T of shape λ − μ satisfies the row restrictions of (8.2) if and only if it satisfies the column restrictions of (8.2′).

∗ Our notation here differs from that of [W].

9 th Variation

In this, our last variation, let h_{rs} (r ≥ 1, s ∈ ℤ) be independent indeterminates over ℤ. Also, for convenience, define h_{0s} = 1 and h_{rs} = 0 for r < 0 and all s ∈ ℤ. Define an automorphism of the ring R generated by the h_{rs} by φ(h_{rs}) = h_{r,s+1} for all r, s. Thus h_{rs} = φ^{s}h_{r}, where h_{r} = h_{r0}, and we shall generally use this alternative notation.

Now define, for any two partitions λ, μ of length ≤ n,

(9.1)  s_{λ/μ} = det(φ^{μ_{j}−j+1}h_{λ_{i}−μ_{j}−i+j})_{1≤i,j≤n}

and in particular (μ = 0)

(9.1′)  s_{λ} = det(φ^{−j+1}h_{λ_{i}−i+j})_{1≤i,j≤n}

As in the case of ordinary Schur functions ([M_{1}], Chap. I, § 5) we have

(9.2)  s_{λ/μ} = 0  unless λ ⊃ μ.

The “Schur functions” defined by (9.1) include as special cases Variations 4, 5, 6, 7 and 8 (in part). Namely for Variation 4 we specialize h_{rs} = w_{r}(z − s) ; for Variation 5, h_{rs} = w_{r}(α; z − s) ; for Variation 6, h_{rs} = h_{r}(x|τ^{s}a) ; for Variation 7, h_{rs} = φ^{s}H_{r}(V) ; and for Variation 8, h_{rs} = h_{r}(α + r + s − 1, β + s).

From (9.1′) it follows that

h_{r} = s_{(r)} (r ≥ 0)

and we define

e_{r} = s_{(1^{r})}

for all r ≥ 0, and e_{r} = 0 for r < 0.

(9.3) Let I be any interval in ℤ. Then the matrices

\begin{aligned}H & = H_{I} = (φ^{1−j}h_{j−i})_{i,j∈I} \\ E & = E_{I} = ((−1)^{j−i}φ^{−i}e_{j−i})_{i,j∈I} \end{aligned}

are inverses of each other.

Proof. — Both H and E are upper unitriangular, hence so also are HE and EH. Hence it is enough to show that the (i, k) element of HE is zero whenever i, k ∈ I and i < k, i.e., that

∑_{j} φ^{1−j}(h_{j−i})(−1)^{k−j}φ^{−j}(e_{k−j}) = 0

or equivalently that

∑_{j} φ^{1+i−j}(h_{j−i})(−1)^{k−j}φ^{i−j}(e_{k−j}) = 0.

If we put k − i = r > 0, this is equivalent to

(1)  ∑_{j=0}^{r}(−1)^{j}φ^{1−j}(h_{j})φ^{−j}(e_{r−j}) = 0.

Now by definition

e_{r} = det(φ^{1−j}(h_{1−i+j}))_{1≤i,j≤r}

and expansion of this determinant along the top row gives (1), as required.

Let λ, μ be partitions such that λ ⊃ μ, and let θ = λ − μ. The function s_{θ} = s_{λ/μ} depends not only on the skew shape θ but also on its location in the lattice plane. For each (p, q) ∈ ℤ^{2} let τ_{p,q} denote the translation (i, j) ↦ (i + p, j + q). Then it follows immediately from (9.1) that

s_{τ_{0,1}(θ)} = φs_{θ}, s_{τ_{1,0}(θ)} = φ^{−1}s_{θ}

and hence that (9.4) s_{τ_{p,q}(θ)} = φ^{q−p}s_{θ}.

In particular, s_{θ} is invariant under diagonal translation (p = q).

Next let θ̂ be the result of rotating θ through 180^{o} about a point on the main diagonal. Then we have

(9.5)  sθ̂ = εs_{θ}

where ε is the involution defined by φ^{s}h_{r} ↦ φ^{1−r−s}h_{r}.

Proof. — We may assume that λ, μ are both contained in the square (n^{n}). Let λ̂, μ̂ be their respective complements in this square, so that λ̂_{i} = n − λ_{n+1−i}, μ̂_{i} = n − μ_{n+1−i}. Then we may take θ̂ = μ̂ − λ̂, so that

s_{θ} = det(φ^{_{λ_{i}}−i+1}h_{_{μ_{j}}−_{λ_{i}}+i+j})

which is easily seen to be equal to det(φ^{i−λ_{i}}h_{λ_{i}−μ_{j}−i+j}) = εs_{λ/μ}.

Let θ′ = λ′ − μ′ be the reflection of θ in the main diagonal. Then we have

(9.6)  s_{θ′} = ωs_{θ}

where ω is the involution defined by φ^{s}h_{r} ↦ φ^{−s}e_{r} for all r, s. Equivalently,

(9.6′)  s_{λ/μ} = det(φ^{−μ′_{j}+j−1}e_{λ′_{i}−μ′_{j}−i+j}).

Proof. — Since the matrices H, E (over an appropriate interval of ℤ) are unitriangular, we have det H = det E = 1 ; and since by (9.3) they are inverses of each other, it follows that each minor of H is equal to the complementary cofactor of the transpose of E. This leads to (9.6′), exactly as in the case of classical Schur functions ([M_{1}], Ch. I, (2.9)).

All the other determinant formulas for Schur functions have their analogues in the present context, and the proofs are essentially the same. First, if λ = (α_{1}, …, α_{r} |β_{1}, …, β_{r}) in Frobenius notation, we have

(9.7)  s_{λ} = det(s_{(α_{i} | β_{j})})_{1≤i,j≤r}.

Proof (cf. [M_{1}], Ch. I, § 3, Ex. 9). — If a, b ≥ 0, so that (a|b) = (a + 1, 1^{b}), the definition (9.1′) gives

s_{(a | b)} = ∑_{i=0}^{b}(−1)^{i}φ^{−i}(h_{a+i+1})φ^{−i−1}(e_{b−i})

on expansion of the determinant along the top row. If a < 0 (but b ≥ 0) we define s_{(a | b)} by this formula : this definition gives s_{(a | b)} = 0 except when a = −1 − b, in which s_{(a | b)} = (−1)^{b}.

Suppose that l(λ) ≤ n and consider the n × n matrices A, B where

A_{ij} = φ^{1−j}(h_{λ_{i}−i+j}), B_{jk} = (−1)^{j−1}φ^{−j}(e_{n+1−j−k}).

We have then (AB)_{ik} = s_{(λ_{i}−i | n−k)} for 1 ≤ i, k ≤ n; and since det A = s_{λ} and det B = 1, it follows that

s_{λ} = det(s_{(λ_{i}−i | n−k)})_{1≤i,k≤n},

which reduces to det(s_{(α_{i} | β_{j})})_{1≤i,j≤r} exactly as in loc. cit.

Next we consider the generalization of (9.7) to skew functions s_{λ/μ}. Let λ = (α_{1}, …, α_{r} |β_{1}, …, β_{r}) as before, and let μ = (γ_{1}, …, γ_{s} |ε_{1}, … ε_{s}). Then we have

(9.8)  s_{λ/μ} = (−1)^{s} det ( \begin{matrix}A & B \\ C & 0 \end{matrix} )

where A = (s_{(α_{i} | β_{j})})_{1≤i,j≤r}, B = (φ^{γ_{j}+1}h_{α_{i}−γ_{j}})_{1≤i≤r,1≤j≤s}, C = (φ^{−ε_{i}−1}e_{β_{j}−ε_{i}})_{1≤i≤s,1≤j≤r}.

The proof of (9.8) is the same as that of Lascoux and Pragacz [LP_{1}] for classical Schur functions. (Observe that φ^{γ_{j}+1}h_{α_{i}−γ_{j}} = s_{θ_{ij}}, where θ_{ij} = (α_{i} |0) − (γ_{j} |0), and likewise that φ^{−ε_{i}−1}e_{β_{j}−ε_{i}} = s_{φ_{ij}}, where s_{φ_{ij}} = (0|β_{j}) − (0|ε_{i}).)

Finally we have, with the notation of (0.5)

(9.9)  s_{λ} = det(s_{[α_{i} | β_{j}]})_{1≤i,j≤r}

exactly as in the classical case. Again, the proof is the same as in [LP_{2}] : indeed, both proofs given there apply in the present context, and we therefore omit the details. There is also a “skew” version of (9.9) in [LP_{2}] which is likewise valid in the present context : but it is rather complicated to state in full generality, and we shall therefore leave its precise formulation to the conscientious reader.

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I. G. Macdonald,

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