Remark

Tag 06EQ, permanent. Cite it as [Bourbaki, Tag 06EQ].

Algebra, chapter IV, § 6. Symmetric functions

— With the notation of the above proof we have

\Delta u = \sum_{i \in I} X_i \cdot D_i(u)

(IV, p. 33, Cor. 1). In other words, Euler's relation (IV, p. 8, Prop 6) extends to formal power series : if $u \in A[X]$ is homogeneous of degree $n$ , then

(21)

n \cdot u = \sum_{i \in I} X_i \cdot D_i(u) .

When I is finite, consisting of $n$ elements, we have $s_k = 0$ for $k > n$ . Then Newton's relations may be written

\begin{aligned} p_2 &= s_1 p_1 - 2 s_2 \\ p_3 &= s_1 p_2 - s_2 p_1 + 3 s_3 \\ &\ \ \cdots \cdots \cdots \cdots \cdots \cdots \cdots \\ p_{n-1} &= s_1 p_{n-2} - s_2 p_{n-3} + \cdots + (-1)^{n-1} s_{n-2} p_1 + (-1)^n (n-1) s_{n-1} \\ p_n &= s_1 p_{n-1} - s_2 p_{n-2} + \cdots + (-1)^n s_{n-1} p_1 + (-1)^{n+1} n s_n \end{aligned}

and

(22)

p_k = s_1 p_{k-1} - s_2 p_{k-2} + \cdots + (-1)^{n+1} s_n p_{k-n} \quad (\text{for } k > n) .

The first $n$ of the above relations hold for any $I$ ; we find for example

p_1 = s_1 , \quad p_2 = s_1^2 - 2 s_2 , \quad p_3 = s_1^3 - 3 s_1 s_2 + 3 s_3 .

More generally, let $S = (S_n)_{n \geq 1}$ be a family of indeterminates. Let us define the polynomials $P_d \in \mathbf{Z}[S_1, \ldots, S_d]$ recursively by $P_1 = S_1$ and

P_d = \sum_{k=1}^{d-1} (-1)^{k-1} S_k P_{d-k} + (-1)^{d+1} d S_d \quad (d \geq 2) .

Then we have the « universal formulae » $p_d = P_d(s_1, \ldots, s_d)$ holding over any ring A and family X of indeterminates. Let $P = (P_k)_{k \geq 1}$ be a sequence of indeterminates. Since $p_k$ is homogeneous of degree k in $A[[X]]$ , there exists precisely one continuous A-algebra homomorphism

\lambda_I : A[[P]] \to A[[X]]^{\mathrm{sym}}

such that $\lambda_I(P_k) = p_k$ for every integer $k \geq 1$ (IV, p. 28). If we assign to $P_k$ the weight k, then $\lambda_I$ transforms an isobaric polynomial of weight $n$ in the $P_k$ into a formal power series which is homogeneous of degree $n$ in the $X_i$ .

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

Cited by

Nothing.

Remark

Cited by

Cites