THE RING OF MULTISYMMETRIC FUNCTIONS

Francesco Vaccarino

Abstract. Let R be a commutative ring and let n, m be two positive integers. Let A_{R}(n, m) be the polynomial ring in the commuting independent variables x_{i}(j) with i = 1, …, m; j = 1, …, n and coefficients in R. The symmetric group on n letters S_{n} acts on A_{R}(n, m) by means of σ(x_{i}(j)) = x_{i}(σ(j)) for all σ ∈ S_{n} and i = 1, …, m; j = 1, …, n. Let us denote by A_{R}(n, m)^{S_{n}} the rings of invariants for this action: its elements are usually called multisymmetric functions and they are the usual symmetric functions when m = 1. In this paper we will give a presentation in terms of generators and relations that holds for any R and any n, m answering in this way to a classical question. I would like to thank M.Brion, C.De Concini and C.Procesi, in alphabetical order, for useful discussions.

Introduction

Let R be a commutative ring and let n, m be two positive integers. Let A_{R}(n, m) be the polynomial ring in the commuting independent variables x_{i}(j) with i = 1, …, m; j = 1, …, n and coefficients in R. The symmetric group on n letters S_{n} acts on A_{R}(n, m) by means of σ(x_{i}(j)) = x_{i}(σ(j)) for all σ ∈ S_{n} and i = 1, …, m; j = 1, …, n. Let us denote by A_{R}(n, m)^{S_{n}} the rings of invariants for this action: its elements are usually called multisymmetric functions and they are the usual symmetric functions when m = 1. When m = 1 then A_{R}(n, 1) ≅ R[x_{1}, x_{2}, …, x_{n}] and R[x_{1}, x_{2}, …, x_{n}]^{S_{n}} is freely generated by the elementary symmetric functions e_{1}, …, e_{n} that are given by the equality

∑_{k=0}^{n} t^{k}e_{k} := ∏_{i=1}^{n}(1 + tx_{i})  (0.1)

2000 Mathematics Subject Classification. 05E05, 13A50, 20C30. Key words and phrases. Characteristic free invariant theory, symmetric functions, repre-

sentations of symmetric groups.

Typeset by AMS-TEX

where e_{0} = 1 and t is a commuting independent variable (see [4]). Furthermore one has

e_{k}(x_{1}, …, x_{n}) = ∑_{i_{1}<i_{2}<···<i_{k}≤n} x_{i_{1}}x_{i_{2}} ⋯ x_{i_{k}}  (0.2)

Let A_{R}(m) := R[y_{1}, …, y_{m}], where y_{1}, …, y_{m} are commuting independent variables, let f ∈ A_{R}(m) and define

f(j) := f(x_{1}(j), …, x_{m}(j)) for 1 ≤ j ≤ n  (0.3)

Notice that f(j) ∈ A_{R}(n, m) for all 1 ≤ j ≤ n. Define e_{k}(f) := e_{k}(f(1), f(2), …, f(n)) i.e.

∑_{k=0}^{n} t^{k}e_{k}(f) := ∏_{i=1}^{n}(1 + tf(i))  (0.4)

Let M_{m} be the set of monomials in A_{R}(m) and M_{m}^{+} the set of those of positive degree. Let μ ∈ M_{m} and let ∂_{i}(μ) denote the degree of μ in y_{i}, for all i = 1, …, m. We set

∂(μ) := (∂_{1}(μ), …, ∂_{m}(μ))  (0.5)

for its multidegree. The total degree of μ is

l(μ) := ∑_{i} ∂_{i}(μ).  (0.6)

These degrees can be extended to A_{R}(n, m) by ∂(x_{i}(j)) = ∂(y_{i}) as S_{n}-module. If f is homogeneous of total degree l then e_{k}(f) has total degree kl (for all k and n). A monomial μ ∈ M_{m}^{+} is called primitive it is not a power of another one. We denote by 𝔐_{m}^{+} the set of primitive monomials. We are now able to state the first part of our result.

Generators. The ring of multisymmetric functions A_{R}(n, m)^{S_{n}} is generated by the e_{k}(μ) with μ ∈ 𝔐_{m}^{+}, k = 1, … n and l(e_{k}(μ)) ≤ max(n, n(m − 1)). If n = p^{s} is a power of a prime and R = ℤ or p · 1_{R} = 0 then at least one generator has degree equal to max(n, n(m − 1)). If R ⊃ ℚ then A_{R}(n, m)^{S_{n}} can also be generated by the e_{1}(μ) with μ ∈ M_{m}^{+} and l(μ) ≤ n.

Let us now find the relations between these generators that is the second part of our result. Let again m = 1. The action of S_{n} on A_{R}(n, 1) ≅ R[x_{1}, x_{2}, …, x_{n}] preserves the usual degree. We denote by Λ_{R,n}^{k} the group of invariants of degree k, and

Λ_{n,R} := R[x_{1}, x_{2}, …, x_{n}]^{S_{n}} = ⊕_{k≥0}Λ_{n,R}^{k}.

Let q_{n} : R[x_{1}, x_{2}, …, x_{n}] ⟶ R[x_{1}, x_{2}, …, x_{n−1}] be given by x_{n} ↦ 0 and x_{i} ↦ x_{i}, for i = 1, …, n − 1. This map sends Λ_{n,R}^{k} to Λ_{n−1,R}^{k}. Denote by Λ_{R}^{k} the limit of the inverse system obtained in this way. The ring Λ_{R} := ⊕_{k≥0}Λ_{R}^{k} is called the ring of symmetric functions (over R). It can be shown [4] that Λ_{R} is a free polynomial ring freely generated by the (limit of the) e_{k}, that are given by

∑_{k=0}^{∞} t^{k}e_{k} := ∏_{i=1}^{∞}(1 + tx_{i}).  (0.7)

Furthermore the kernel of the natural projection π_{n} : Λ_{R} ⟶ Λ_{n,R} is generated by the e_{n+k} with k ≥ n. In a similar way we build a limit of multisymmetric functions. For any a ∈ ℕ^{m} we set A_{R}(n, m, a) for the linear span of the monomials of multidegree a. One has

A_{R}(n, m) = ⨁_{a∈ℕ^{m}} A_{R}(n, m, a).  (0.8)

Let h be an integer with h > n. Let π_{n}^{h} : A_{R}(h, m) ⟶ A_{R}(n, m) be given by

π_{n}^{h}(x_{i}(j)) = \{ \begin{matrix}0 & if j > n \\ x_{i}(j) & if j ≤ n \end{matrix}  for all i.  (0.9)

then we prove that

π_{n}^{h}(A_{R}(h, m)^{S_{h}}) = A_{R}(n, m)^{S_{n}}  (0.10)

For any a ∈ ℕ^{m} set

A_{R}(∞, m, a) := lim_{←} A_{R}(n, m, a)^{S_{n}}  (0.11)

where the projective limit is taken with respect to n over the projective system (A_{R}(n, m, a)^{S_{n}}, π_{n,a}^{n+1}) with π_{n,a}^{n+1} : A_{R}(h, m, a) → A_{R}(n, m, a) the restriction of π_{n}^{n+1}. Set

A_{R}(∞, m) := ⨁_{a∈ℕ^{m}} A_{R}(∞, m, a).  (0.12)

We set, by abuse of notation,

e_{j}(f) := e_{j}((f(1), …, f(k), …)) ∈ A_{R}(∞, m)  (0.13)

with j ∈ ℕ and f ∈ A(m)^{+}, the augmentation ideal, i.e.

∑_{k=0}^{∞} t^{k}e_{k}(f) := ∏_{i=1}^{∞}(1 + tf(i)).  (0.14)

We can now state the second part of our main result.

Relations.

(1) The ring R ⊗ A_{ℤ}(∞, m) is a polynomial ring freely generated by the (limit of) the 1_{R} ⊗ e_{k}(μ) with μ ∈ 𝔐_{m}^{+}, k ∈ ℕ. The the kernel of the natural projection

π_{n} : R ⊗ A_{ℤ}(∞, m) ⟶ A_{R}(n, m)^{S_{n}}

is generated by the elements

e_{n+k}(f) = 0 with k ≥ 1 and f ∈ A_{R}(m)^{+}.

(2) If R ⊃ ℚ then A_{R}(∞, m) can be freely generated by the e_{1}(μ) with μ ∈ M_{m}^{+}. The the kernel of the natural projection is then generated by e_{n+1}(f) with f ∈ A_{R}(m)^{+} .

1. Generators

In order to prove the theorem on generators stated in the introduction we need to introduce some machinery. Let k ∈ ℕ, f_{1} …, f_{k} ∈ A_{R}(m) and t_{1}, …, t_{k} be commuting independent variables, define elements e_{(α_{1},...,α_{k})}(f_{1}, …, f_{k}) ∈ A_{R}(n, m)^{S_{n}} by

∑ ; ∑ α_{j}≤n t_{1}^{α_{1}} ⋯ t_{k}^{α_{k}}e_{(α_{1},...,α_{k})}(f_{1}, …, f_{k}) := ∏_{i=1}^{n}(1 + t_{1}f_{1}(i) + ⋯ + t_{k}f_{k}(i)) (1.1)

Example 1.2. Let n = 3 and f, g ∈ A_{R}(m) then

e_{(2,1)}(f, g) = f(1)f(2)g(3) + f(1)g(2)f(3) + g(1)f(2)f(3).

If n = 4 then

e_{(2,1)}(f, g) =f(1)f(2)g(3) + f(1)g(2)f(3) + g(1)f(2)f(3)+
f(1)f(2)g(4) + f(1)g(2)f(4) + g(1)f(2)f(4)+
f(1)f(3)g(4) + f(1)g(3)f(4) + g(1)f(3)f(4)+
f(2)f(3)g(4) + f(2)g(3)f(4) + g(2)f(3)f(4)

Let k = m and f_{j} = y_{j} for j = 1, …, m then the e_{(α_{1},...,α_{m})}(y_{1}, …, y_{m}) with ∑ α_{j} ≤ n are the well known elementary multisymmetric function that generates A_{R}(n, m)^{S_{n}} when R ⊃ ℚ see [3] or [6] and are given by

∏_{i=1}^{n}(1 + ∑_{j=1}^{m} t_{j}x_{j}(i)) = ∑ ; ∑ α_{j}≤n t_{1}^{α_{1}} ⋯ t_{m}^{α_{m}}e_{(α_{1},...,α_{m})}(y_{1}, …, y_{m}).  (1.3)

It can be easily seen that e_{(α_{1},...,α_{k})}(f_{1}, …, f_{k}) is the orbit sum (under the considered action of S_{n}) of

f_{1}(1)f_{1}(2) ⋯ f_{1}(α_{1})f_{2}(α_{1} + 1) ⋯ f_{2}(α_{1} + α_{2}) ⋯ f_{k}(α_{k})  (1.4)

It is clear that the e_{(α_{1},...,α_{k})}(f_{1}, …, f_{k}) = e_{(α_{τ} (1),...,α_{τ} (k))}(f_{τ(1)}, …, f_{τ(k)}) for all τ ∈ S_{k}; furthermore if two entries are equal, say f_{1} = f_{2}, then

e_{(α_{1},...,α_{k})}(f_{1}, …, f_{k}) = \frac{(α_{1} + α_{2})!}{α_{1}!α_{2}!}e_{(α_{1}+α_{2},...,α_{k})}(f_{1}, f_{3} …, f_{k}).  (1.5)

If the arguments f_{i} of e_{(α_{1},...,α_{k})}(f_{1}, …, f_{k}) are all distinct we say it is reduced. Let ℕ^{(M_{m}^{+})} be the set of functions M_{m}^{+} ⟶ ℕ with finite support, we set

| α |:= ∑_{μ∈M_{m}^{+}} α(μ)  (1.6)

We introduce the elements e_{α}(\overline{μ}) ∈ A_{R}(n, m)^{S_{n}} with | α |≤ n by

∑_{|α|≤n} t^{α}e_{α}(\overline{μ}) = ∏_{i=1}^{n}(1 + ∑_{μ∈M_{m}^{+}} t_{μ}μ(i))  (1.7)

where t_{μ} are commuting independent variables and t^{α} := ∏_{μ∈M_{m}^{+}} t_{μ}^{α(μ)} for all α ∈ ℕ^{(M_{m}^{+})}. If α ∈ ℕ^{(M_{m}^{+})} is such that α(μ) = k and α(ν) = 0 for some μ ∈ M_{m}^{+} and for all ν ∈ M_{m}^{+} with ν ≠ μ we see that e_{α}(\overline{μ}) = e_{k}(μ) (see introduction) i.e. the k-th elementary symmetric function evaluated at (μ(1), μ(2), …, μ(n)).

Proposition 1.8. The (ordered) set

B_{n,m,R} := \{e_{α}(\overline{μ}) : | α |≤ n\}

is a R-basis of A_{R}(n, m)^{S_{n}}. The (ordered) set

B_{n,m,α,R} := \{e_{α}(\overline{μ}) : | α |≤ n and ∂(e_{α})(\overline{μ}) = β\}

is a R-basis of A_{R}(n, m, β)^{S_{n}}, for all β ∈ ℕ^{m}.

Proof. The e_{α}(\overline{μ}) are complete system of representative (for the action of S_{n}) of the orbit sums of the products

{μ_{1}(1)μ_{2}(2) ⋯ μ_{n}(n) : μ_{i} ∈ M_{m} , i = 1, …, n}

furthermore ∂(e_{α})(\overline{μ}) = ∑_{μ∈M_{m}^{+}} α_{μ}∂(μ).

Let us calculate the product between two elements as above.

Proposition 1.9 - Product Formula. Let k, h ∈ ℕ, f_{1} …, f_{k}, g_{1}, …, g_{h} ∈ A_{R}(m) and t_{1}, …, t_{k}, s_{1}, …, s_{h} be commuting independent variables, set

e_{α}(\overline{f}) := e_{(α_{1},...,α_{k})}(f_{1}, …, f_{k}) and e_{β}(\overline{g}) := e_{(β_{1},...,β_{h})}(g_{1}, …, g_{h})

then

e_{α}(\overline{f})e_{β}(\overline{g}) = ∑_{γ} e_{γ}(f_{1}, …, f_{k}, g_{1}, …, g_{h}, f_{1}g_{1}, f_{1}g_{2}, …, f_{k}g_{1}, …, f_{k}g_{h})

where γ := (γ_{10}, …, γ_{k0}, γ_{01}, …, γ_{0h}, γ_{11}, γ_{12}, …, γ_{k1}, …, γ_{kh}) is such that

\{ \begin{matrix}γ_{ij} ∈ ℕ \\ | γ |≤ n \\ ∑_{j=0}^{h} γ_{ij} = α_{i} for i = 1, …, k \\ ∑_{i=0}^{k} γ_{ij} = β_{j} for j = 1, …, h. \end{matrix}

Proof. The result follows from

∑ ; ∑ α_{j}, ∑ β_{l}≤n ∏_{j=1}^{k} ∏_{l=1}^{h} t_{j}^{α_{j}}s_{l}^{β_{l}}e_{α}(\overline{f})e_{β}(\overline{g}) =

( ∑ ; ∑ α_{j}≤n ∏_{j=1}^{k} t_{j}^{α_{j}}e_{α}(\overline{f}))( ∑ ; ∑ β_{l}≤n ∏_{l=1}^{h} s_{l}^{β_{l}}e_{β}(\overline{g})) =

∏_{i=1}^{n}(1 + ∑_{j=1}^{k} t_{j}f_{j}(i)) ∏_{i=1}^{n}(1 + ∑_{l=1}^{h} s_{l}g_{l}(i)) =

∏_{i=1}^{n}(1 + ∑_{j=1}^{k} t_{j}f_{j}(i) + ∑_{l=1}^{h} s_{l}g_{l}(i) + ∑_{j≤l} t_{j}s_{l}f_{j}(i)g_{l}(i)) =

∑_{γ}(∏_{a=1}^{k} t_{a}^{γ_{a0}} ∏_{b=1}^{h} s_{b}^{γ_{0b}} ∏_{a=1}^{k} ∏_{b=1}^{h}(t_{a}s_{b})^{γ_{ab}}e_{γ}(\overline{f}, \overline{g}, \overline{fg}))

where \overline{fg} := (f_{1}g_{1}, f_{1}g_{2}, …, f_{k}g_{1}, …, f_{k}g_{h}) and γ satisfies the condition of the statement.

Example 1.10. Let us calculate in A_{R}(2, 3)^{S_{2}}

e_{(1,1)}(a, b)e_{2}(c) = ∑_{0≤k,h≤1} e_{(1−k,1−h,2−k−h,h,k)}(a, b, c, ac, bc) = e_{(1,1)}(ac, bc)

since 1 − k + 1 − h + 2 − k − h + h + k = 4 − k − h ≤ 2.

Corollary 1.11. Let k ∈ ℕ, a_{1}, …, a_{k} ∈ A_{R}(m), α = (α_{1}, …, α_{k}) ∈ ℕ^{k} with ∑ α_{j} ≤ n, then e_{(α_{1},...,α_{k})}(a_{1}, …, a_{k}) belongs to the subring of A_{R}(n, m)^{S_{n}} generated by the e_{i}(μ) with i = 1, …, n and μ monomial in the a_{1}, …, a_{k}.

Proof. Let k, a_{1}, …, a_{k}, α be as in the statement, then e_{α_{1}}(a_{1})e_{(α_{2},...,α_{k})}(a_{2}, …, a_{k}) =

= e_{(α_{1},...,α_{k})}(a_{1}, …, a_{k}) + ∑ e_{γ}(a_{1}, …, a_{k}, a_{1}a_{2}, …, a_{1}a_{k})

where now

γ = (γ_{10}, γ_{01}, …, γ_{0h}, γ_{11}, γ_{12}, …, γ_{1h})

with h = k − 1 and ∑_{j=0}^{h} γ_{1j} = α_{1} with ∑_{j=1}^{h} γ_{1j} > 0 and γ_{0j} + γ_{1j} = α_{j} for j = 1, …, h. Thus

γ_{10} + γ_{01} + ⋯ + γ_{0h} + γ_{11} + ⋯ + γ_{1h} = ∑_{j} α_{j} − ∑_{j=1}^{h} γ_{1j} < ∑_{j} α_{j}.

Hence

e_{(α_{1},...,α_{k})}(a_{1}, …, a_{k}) =
e_{α_{1}}(a_{1})e_{(α_{2},...,α_{k})}(a_{2}, …, a_{k}) − ∑ e_{γ}(a_{1}, …, a_{k}, a_{1}a_{2}, a_{1}a_{3}, …, a_{1}a_{k})

with ∑_{j} α_{j} < ∑_{r,s} γ_{rs}. By induction on ∑_{j} α_{j} it is then possible to write e_{(α_{1},...,α_{k})}(a_{1}, …, a_{k}) as a polynomial in e_{j}(μ) with μ monomial in the a_{1}, …, a_{k}.

Example 1.12. Consider e_{(2,1)}(a, b) in A_{R}(3, m) as in example 1.2, then

e_{(2,1)}(a, b) = e_{2}(a)e_{1}(b) − e_{(1,1)}(a, ab) = e_{2}(a)e_{1}(b) − e_{1}(a)e_{1}(ab) + e_{1}(a^{2}b).

We now recall some basic facts about classical symmetric functions, for further reading on this topic see [4]. In Λ_{R} (see Introduction) we have a further operation beside the product: the plethysm. Let g, f ∈ Λ_{R}, we say that h ∈ Λ_{R} is the plethysm of g by f and we denote it by h = g · f if h is obtained by substituting the monomials appearing in f at the place of the variables in g. We have another distinguished kind of functions in Λ_{R} beside the elementary symmetric ones: the powers sums. For any r ∈ ℕ the r-th power sum is

p_{r} := ∑_{i≥1} x_{i}^{r}.

Let g ∈ Λ_{R}, then the plethysm g · p_{r} = g(x_{1}^{r}, x_{2}^{r}, …, x_{k}^{r}, …). Since the e_{i} generate Λ_{R} we have that g · p_{r} can be expressed as a polynomial in the e_{i}, we set

P_{h,k} := e_{h} · p_{k}

a polynomial in the e_{i}’s. These result are clearly valid also in Λ_{n,R} with the precaution that e_{i} = 0 for i ≥ n + 1.

Proposition 1.13. For all a ∈ A_{R}(m), and k, i ∈ ℕ, e_{i}(a^{k}) belongs to the subring of A_{R}(n, m)^{S_{n}} generated by the e_{j}(a).

Proof. Let a ∈ A(m) and consider e_{h}(a^{k}) ∈ A(n, m)^{S_{n}}, we have (see Introduction)

e_{h}(a^{k}) = e_{h}(a(1)^{k}, …, a(n)^{k}) = P_{h,k}(e_{1}(a(1), …, a(n)), …, e_{n}(a(1), …, a(n)))

and the result is proved.

Recall that a monomial μ ∈ M_{m}^{+} is called primitive if it is not a power of another one and we denote by 𝔐_{m}^{+} the set of primitive monomials. Let we state again the first part of the main result

Theorem 1.14 - Generators. The ring of multisymmetric functions A_{R}(n, m)^{S_{n}} is generated by the e_{k}(μ) with μ ∈ 𝔐_{m}^{+}, k = 1, … n and l(e_{k}(μ)) ≤ max(n, n(m − 1)). If n = p^{s} is a power of a prime and R = ℤ or p · 1_{R} = 0 then at least one generator has degree equal to max(n, n(m − 1)). If R ⊃ ℚ then A_{R}(n, m)^{S_{n}} can also be generated by the e_{1}(μ) with μ ∈ M_{m}^{+} and l(μ) ≤ n.

Proof. The elements e_{α}(\overline{μ}) ∈ B_{n,m,R}, that form a R-basis by 1.8, can be expressed as polynomial in e_{i}(μ) with i = 1, …, n and μ ∈ M_{m}^{+}, by 1.11. If μ = ν^{k}, with ν ∈ 𝔐_{m}^{+} then e_{i}(μ) can be expressed as a polynomial in the e_{j}(ν) in A(n, m)^{S_{n}}, by 1.13. Since for all μ ∈ M_{m}^{+} there exist k ∈ ℕ and ν ∈ 𝔐_{m}^{+} such that μ = ν^{k} we have that A(n, m)^{S_{n}} is generated as commutative ring by the e_{j}(ν), with ν ∈ 𝔐_{m}^{+} and j = 1, …, n. The result then follows by the following due to Fleischmann [2]: A_{R}(n, m)^{S_{n}} can be generated by elements of total degree ℓ ≤ max(n, n(m − 1)), for any commutative ring R, with sharp bound if n = p^{s} a power of a prime and R = ℤ or p · 1_{R} = 0. If R ⊃ ℚ then the result follows from Newton’s Formulas and the well known result of H.Weyl (see [3],[6]).

2. Relations

Let now h be an integer with h > n. Let π_{n}^{h} : A_{R}(h, m) ⟶ A_{R}(n, m) be given by

π_{n}^{h}(x_{i}(j)) = \{ \begin{matrix}0 & if j > n \\ x_{i}(j) & if j ≤ n \end{matrix}  for all i.  (2.1)

this is a multigraded ring epimorphism such that

∏_{i=1}^{h}(1 + t_{1}a_{1}(i) + ⋯ + t_{k}a_{k}(i)) ↦ ∏_{i=1}^{n}(1 + t_{1}a_{1}(i) + ⋯ + t_{k}a_{k}(i))  (2.2)

where k, a_{j}, t_{j} are as usual. Hence

π_{n}^{h}(e_{α}(\overline{μ})) = \{ \begin{matrix}e_{α}(\overline{μ}) & if | α |≤ n \\ 0 & if | α |> n. \end{matrix}  (2.3)

thus, by Prop.1.8, for all a ∈ ℕ^{m} the restriction

π_{n,a}^{h} : A_{R}(h, m, a) → A_{R}(n, m, a)  (2.4)

is such that

π_{h,a}^{h}(A_{R}(h, m, a)^{S_{h}}) = A_{R}(n, m, a)^{S_{n}}.  (2.5)

and then (A_{R}(n, m, a)^{S_{n}}, π_{n,a}^{n+1}) is a projective sytem. For any a ∈ ℕ^{m} set

A_{R}(∞, m, a) := lim_{←} A_{R}(n, m, a)^{S_{n}}  (2.6)

where the projective limit is taken with respect to n over the above projective system and set

π_{n,a} : A_{R}(∞, m, a) ⟶ A_{R}(n, m, a)^{S_{n}}  (2.7)

for the natural projection. Set

A_{R}(∞, m) := ⨁_{a∈ℕ^{m}} A_{R}(∞, m, a).  (2.8)

and

π_{n} := ⊕_{a∈ℕ^{m}}π_{n,a}.  (2.9)

Similarly to the classic case (m = 1) we make an abuse of notation and set e_{α}(\overline{μ}) := lim_{←} e_{α}(\overline{μ}), for any α ∈ ℕ^{(M_{m}^{+})} where a = ∂(e_{α}(\overline{μ})). In the same way we set

e_{j}(f) := lim_{←} e_{j}(f) with j ∈ ℕ where f ∈ A_{R}(m)^{+} is homogeneous of positive multidegree so that j ∂(f) = a.

Proposition 2.10. Let a ∈ ℕ^{m}.

(1) The R-module ker π_{n,a} is the linear span of

\{e_{α}(\overline{μ}) ∈ A_{R}(∞, m, a) : | α |> n\}.

(2) The R-module homomorphisms π_{n,a} : A_{R}(∞, m, a) ⟶ A_{R}(n, m, a)^{S_{n}} are onto for all n ∈ ℕ and A_{R}(∞, m, a) ≅ A_{R}(n, m, a)^{S_{n}} for all n ≥| a | .

(3) The R-module A_{R}(∞, m, a) is a free R-module with basis

\{e_{α}(\overline{μ}) : ∂(e_{α}(\overline{μ})) = a\},

(4) The R-module A_{R}(∞, m) is a free R-module with basis

\{e_{α}(\overline{μ}) : α ∈ ℕ^{(M_{m}^{+})}\},

Proof.

(1) By 2.5, for all h > n and a ∈ ℕ^{m}, the following is an exact sequence of R-modules

0 ⟶ ker π_{n,a}^{h} ⟶ A(h, m, a)^{S_{h}} ⟶^{π_{n,a}^{h}} A(n, m, a)^{S_{n}} ⟶ 0.

that splits by (2.3) and Prop.1.8 and the claim follows.

(2) If ∑_{j=1}^{m} a_{j} ≤ k then ker π_{n,a}^{k} = 0 for all k > n, indeed

∂(e_{α}(\overline{μ}) = ∑_{μ∈M_{m}^{+}} α_{μ} ∂(μ) = a ⟹| α |≤ ∑_{j=1}^{m} a_{j} ≤ n.

Hence A(h, m, a)^{S_{h}} ≅ A(∑_{j=1}^{m} a_{j}, m, a)^{S_{h}} for all h ≥ ∑_{j=1}^{m} a_{j} and the thesis follows by (2.5).

(3) It follows from (1) and (2).

(4) It follows form (3) and (2.8)

Remark 2.11. Notice that A_{R}(m)^{⊗n} ≅ A_{R}(n, m) as multigraded S_{n}-algebras by means of

f_{1} ⊗ ⋯ ⊗ f_{n} ↔ f_{1}(1)f_{2}(2) ⋯ f_{n}(n)  (2.12)

for all f_{1}, …, f_{n} ∈ A_{R}(m). Hence A_{R}(n, m)^{S_{n}} ≅ T S^{n}(A_{R}(m)), where T S^{n}( − ) denotes the symmetric tensors functor. Since T S^{n}(A_{R}(m)) ≅ R ⨂ T S^{n}(A_{ℤ}(m)) (see [1])we have

A_{R}(n, m)^{S_{n}} ≅ R ⊗ A_{ℤ}(n, m)^{S_{n}}  (2.13)

for any commutative ring R. We then work with R = ℤ and we suppress the ℤ subscript for sake of simplicity.

Remark 2.14. The ℤ-module A(∞, m) can be endowed with a structure of ℕ^{m}- graded ring such that the π_{n} are ℕ^{m}-graded ring homomorphisms: the product e_{α}(\overline{μ})e_{β}(\overline{ν}), where α, β ∈ ℕ^{(M_{m}^{+})}, is defined using the product formula of Prop.1.9 with no limits on the maximum value of | γ | with γ appearing in the summation.

Proposition 2.15. Consider the free polynomial ring

C(m) := ⨁_{a∈ℕ^{m}} C(m, a) := ℤ[e_{i,μ}]_{i∈ℕ,μ∈𝔐_{m}^{+}}

with multidegree given by ∂(e_{i,μ}) = ∂(μ)i. The multigraded ring homomorphism

σ_{m} : ℤ[e_{i,μ}]_{i∈ℕ,μ∈𝔐_{m}^{+}} ⟶ A(∞, m)

given by

σ_{m} : e_{i,μ} ↦ e_{i}(μ), for all i ∈ ℕ, μ ∈ 𝔐_{m}^{+}

is an isomorphism, i.e. A(∞, m) is freely generated as commutative ring by the e_{i}(μ) with i ∈ ℕ, μ ∈ 𝔐_{m}^{+}.

Proof. Since we defined the product in A(∞, m) as in Prop.1.9 it is easy to verify, repeating the same reasoning of the previous chapter, that A(∞, m) is generated as commutative ring by the e_{i}(μ) with i ∈ ℕ, μ ∈ 𝔐_{m}^{+}. Hence σ_{m} is onto for all m ∈ ℕ. Let a ∈ ℕ^{m} and consider the restriction σ_{m,a} : C(m, a) ⟶ A(∞, m, a). It is onto as we have just seen. A ℤ-basis of C(m, a) is

{  ∏_{i∈ℕ,k∈ℕ,μ∈𝔐_{m}^{+}} e_{i,μ} : ∑_{i∈ℕ,k∈ℕ,μ∈𝔐_{m}^{+}} i k ∂(μ) = a}

on the other hand a ℤ-basis of A(∞, m, a) is

{e_{α} : ∑_{α_{μ}∈ℕ,μ∈M_{m}^{+}} α_{μ} ∂(μ) = a}.

Let μ ∈ M_{m}^{+} then there is an unique k ∈ ℕ and an unique ν ∈ 𝔐_{m}^{+} such that μ = ν^{k}, hence

∑_{α_{μ}∈ℕ,μ∈M_{m}^{+}} α_{μ} ∂(μ) = ∑_{k∈ℕ,α_{μ}∈ℕ,ν∈𝔐_{m}^{+}} α_{μ} k ∂(ν)

so that C(m, a) and A(∞, m, a) have the same (finite) ℤ-rank and thus are isomorphic.

Corollary 1.16. Let R ⊃ ℚ then R ⊗ A(∞, m) is a polynomial ring freely generated by the e_{1}(μ) with μ ∈ M_{m}^{+}.

Proof. By Prop.2.15 and Th.1.14.

Theorem 2.17 - Relations.

(1) Let R be a commutative ring: the following is a presentation of A_{R}(n, m)^{S_{n}} in terms of generators and relations in the category of ℕ^{m}-graded Ralgebras

0 ⟶< {1_{R}⊗e_{k}(f) : k > n , f ∈ A_{R}(m)^{+}} >⟶ R⊗A(∞, m) ⟶^{π_{n}} A_{R}(n, m)^{S_{n}} ⟶ 0.

(2) If R ⊃ ℚ then A_{R}(∞, m) can be freely generated by the e_{1}(μ) with μ ∈ M_{m}^{+}. The the kernel of the natural projection is then generated by e_{n+1}(f) with f ∈ A_{R}(m)^{+} .

Proof.

(1) As before we set R = ℤ and the result will follow by Remark 2.11 and Prop.2.15. By Prop.2.10 the kernel of A(∞, m) ⟶^{π_{n}} A(n, m)^{S_{n}} has basis \{e_{α}(\overline{μ}) : | α |> n\}, let V_{k} be the submodule of A(∞, m) with basis \{e_{α}(\overline{μ}) : | α |= k}. Let A_{k} be the sub-ℤ-module of ℚ ⊗ V_{k} generated by the e_{k}(f) with f ∈ A(m)^{+}. Let g : ℚ ⊗ V_{k} ⟶ ℚ be a linear form identically zero on A_{k}, then

0 = g(e_{k}(f)) = g(e_{k}( ∑_{μ∈M_{m}^{+}} λ_{μ} μ)) = ( ∑_{|α|=k}( ∏_{μ∈M_{m}^{+}} λ_{μ}^{α_{μ}}) g(e_{α}(\overline{μ}))),

for all ∑_{μ∈M_{m}^{+}} λ_{μ} μ ∈ A(m)^{+}, hence g(e_{α}(\overline{μ})) = 0 for all e_{α}(\overline{μ}) with | α |= k , thus g = 0 and A_{k} = V_{k}.

(2) If R ⊃ ℚ the result follows from Newton’s formulas (1) and Cor.2.16

References

1. N.Bourbaky, Elements of mathematics - Algebra II Chapters 4-7, Springer-Verlag, Berlin, 1988.

2. P.Fleischmann, A new degree bound for vector invariants of symmetric groups, Trans. Am. Math. Soc. 350 (1998), 1703-1712.

3. I.Gelfand,M.Kapranov, A.Zelevinsky, Discriminants, resultants and multidimensional determinants, Birkahuser, Boston, 1994.

4. I.G.Macdonald, Symmetric Functions and Hall Polynomials - second edition, Oxford mathematical monograph, 1995.

6. H.Weyl, The classical groups, Princeton University Press, Princeton N.J., 1946.

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