JACOBIANS OF SYMMETRIC POLYNOMIALS

Alain Lascoux and Piotr Pragacz

Abstract

We give the Jacobian of any family of complete symmetric functions, or of power sums, in a finite number of variables.

Changes of bases of the ring of symmetric functions are usually performed using a canonical scalar product on this ring ([2], Ch.I). On the other hand, Cayley [1], Sylvester [6] and Mac Mahon [3] were rather led to characterize invariants like, e.g., the discriminant, by using differential calculus on symmetric polynomials. The aim of the present note is to write explicitly the Jacobians of different families of symmetric polynomials in n indeterminates 𝕏 = {x_{1}, …, x_{n}}, with respect to the indeterminates, or to the elementary symmetric functions.

Define

λ_{z}(𝕏) := ∏_{i=1}^{n}(1 + zx_{i}) = ∑_{j=0}^{n} z^{j} e_{j}(𝕏) , σ_{z}(𝕏) := 1/λ_{−z}(𝕏) = ∑_{j=0}^{∞} z^{j}h_{j}(𝕏) ,

and

∑_{j=1}^{∞} z^{j} p_{j}(𝕏)/j := log(σ_{z}(𝕏)) ,

the e_{j} = e_{j}(𝕏) being the elementary symmetric functions, the h_{j} = h_{j}(𝕏) being the complete functions, the p_{j} = p_{j}(𝕏) being the power sums.

More generally, for any k ∈ ℂ, let

σ_{z}(k 𝕏) = (σ_{z}(𝕏))^{k} = ∑_{j=0}^{∞} z^{j}h_{j}(k 𝕏) = exp(∑_{j=1}^{∞} z^{j} p_{j}(k𝕏)/j) .

For any n-tuple f_{1}, …, f_{n} in the ring of symmetric polynomials in 𝕏, one has two Jacobians , expressed by (n × n)-determinants :

J(f_{1}, …, f_{n}) := |\frac{∂}{∂_{x_{i}}} (f_{j})|_{1≤i,j≤n} and J_{e}(f_{1}, …, f_{n}) := |\frac{∂}{∂_{e_{i}}} (f_{j})|_{1≤i,j≤n} .

Since

\frac{∂}{∂_{x_{i}}} (e_{j}) = ∑_{r=0}^{j−1}(−1)^{r}x_{i}^{r} e_{j−1−r} ,

by subtracting suitable combinations of columns in J(e_{1}, …, e_{n}), we get

J(e_{1}, …, e_{n}) = |∑_{r=0}^{j−1}(−1)^{r}x_{i}^{r} e_{j−1−r}|_{1≤i,j≤n}

= |(−1)^{j−1}x_{i}^{j−1}|_{1≤i,j≤n} = ∏_{1≤i<j≤n}(x_{i} − x_{j}) =: Δ(𝕏) = Δ ,

the Vandermonde determinant. Hence, for any n-tuple of symmetric polynomials, we have the factorization

J(f_{1}, …, f_{n}) = J_{e}(f_{1}, …, f_{n}) · Δ .  (1)

The following proposition gives the expressions of some J_{e} in terms of Schur functions s_{ν}(k𝕏) which are defined as the determinants

|h_{ν_{i}−i+j}(k𝕏)|_{1≤i,j≤n}

(cf. ([2], I.3.4). This definition remains valid when ν = (ν_{1}, …, ν_{n}) is not a partition. If k = 1 and ν_{i} ≥ −(n − i) for i = 1, …, n, then the expression

|x_{i}^{ν_{j}+n−j}|_{1≤i,j≤n}/Δ

(cf. ([2], I.3.1)) in terms of the powers of the x_{i}’s, gives s_{ν}(𝕏).

Proposition 1 For μ = (μ_{1}, …, μ_{n}) ∈ (ℕ^{∗})^{n}, we have

J_{e}(p_{μ_{1}}(k𝕏), …, p_{μ_{n}}(k𝕏)) = k^{n}μ_{1} ⋯ μ_{n} s_{(μ_{1}−n,..., μ_{n}−1)}(𝕏)  (2)

and for μ ∈ ℕ^{n},

J_{e}(h_{μ_{1}}(k𝕏), …, h_{μ_{n}}(k𝕏)) = k^{n}s_{(μ_{1}−n,..., μ_{n}−1)}((k+1) 𝕏) .  (3)

(If μ_{1} > ⋯ > μ_{n} ≥ 1, then μ_{1} − n ≥ ⋯ ≥ μ_{n} − 1 is a partition.)

Firstly, since p_{j}(k𝕏) = kp_{j}(𝕏), we have

\frac{∂}{∂_{x_{i}}} (p_{j}(k𝕏)) = k \frac{∂}{∂_{x_{i}}} (p_{j}(𝕏)) = kj x_{i}^{j−1} ,

and

J(p_{μ_{1}}(k𝕏), …, p_{μ_{n}}(k𝕏)) = k^{n}μ_{1} ⋯ μ_{n} · |x_{i}^{μ_{j}−1}|_{1≤i,j≤n} .

Using the expression of a Schur function in terms of the powers of the x_{i}’s, one gets equation (2).

Secondly, using the Leibniz rule for differentiation of the product, we have

\frac{∂}{∂_{e_{i}}} (σ_{z}(k 𝕏)) = \frac{∂}{∂_{e_{i}}} (1/λ_{−z}(𝕏)^{k}) = −k(−z)^{i}/(λ_{−z}(𝕏))^{k+1} = −k(−z)^{i}σ_{z}((k+1) 𝕏) .

(4)

Thus the expression of a Schur function in terms of complete functions entails equation (3). The proposition has been proved.

For example, we have

(n!)^{−1} J(p_{1}, p_{2}, …, p_{n}) = J(h_{1}, h_{2}, …, h_{n}) = (−1)^{n(n−1)/2}Δ ,

and for n = 3,

J_{e}(h_{5}(𝕏), h_{3}(𝕏), h_{2}(𝕏)) = s_{2,1,1}(2𝕏) .

Remark 1 In analogy to (4), we record

\frac{∂}{∂_{e_{i}}} (λ_{z}(k 𝕏)) = kz^{i}λ_{z}((k−1) 𝕏) .  (5)

Remark 2 For any three partitions μ, ν, η of the same weight d, let (μ, ν, η) be the multiplicity of the trivial representation of the symmetric group 𝔖_{d} in the tensor product of the three irreducible representations of 𝔖_{d} labeled by the partitions μ, ν, η ([2], I.7). Then, for any k ∈ ℂ, any partition μ of d, one has the expansion

s_{μ}(k𝕏) = ∑_{ν,η}(μ, ν, η)s_{η}(k)s_{ν}(𝕏) ,

sum over all pair of partitions of weight d, s_{η}(k) meaning the specialization of the Schur function s_{η} under h_{j} → \binom{k+j−1}{j}, j ≥ 0.

For example, for n = 3, d = 4, one has the following expression for s_{2,1,1}(2𝕏):

(s_{4}(2) + s_{3,1}(2) + s_{2,2}(2))s_{2,1,1}(𝕏) + (s_{3,1}(2))s_{2,2}(𝕏) + (s_{3,1}(2) + s_{2,2}(2))s_{3,1}(𝕏)

= 9s_{2,1,1}(𝕏) + 3s_{2,2}(𝕏) + 4s_{3,1}(𝕏)

(in that case, all coefficients ((2, 1, 1), ν, η) are 0 or 1).

Remark 3 Scott [5], followed by Muir [4], p.148, wrongly asserted that the Jacobians of p_{μ_{1}}(𝕏), …, p_{μ_{n}}(𝕏) and h_{μ_{1}}(𝕏), …, h_{μ_{n}}(𝕏) differ by a scalar factor. The error was to deduce \frac{∂}{∂_{p_{j}}} (h_{j}) = 1/j from the Newton relation:

jh_{j} = ∑_{r=0}^{j−1} p_{j−r} h_{r} ,

the power sums appearing in this formula not being algebraically independent when j > n. One “raison d’être” of the present note was to correct this error.

References

[1] A. Cayley, On symmetric functions and seminvariants , Amer. J. of Math. 15 (1893) 1–74.

[2] I. G. Macdonald, Symmetric functions and Hall polynomials, Oxford University Press, Oxford, 2nd edition (1995).

[3] Capt. McMahon, Operators in the theory of seminvariants, Quart. J. Math. 20 (1885) 362–365.

[4] T. Muir, History of the Theory of Determinants, Vol. III, Dover reprint (1960).

[5] R. F. Scott, On some alternating functions of n variables, Messenger of Math. 11 (1882), 98-103.

[6] J.J. Sylvester, Note on Capt. Mac Mahon’s transformationn of the theory of invariants, Messenger of Math. 14 (1884) 163–165.

A.L.: CNRS, Institut Gaspard Monge, Université de Marne-la-Vallée, F-77454 Marne-la-Vallée Cedex, France Alain.Lascoux@univ-mlv.fr

P.P.: Institute of Mathematics, Polish Academy of Sciences, Śniadeckich 8, PL-00950 Warsaw, Poland pragacz@impan.gov.pl