This is a reposted old post from 2024. This post is about symmetric polynomials. A symmetric polynomial is

such that for any permutation $\sigma\in S_n$, we have $\sigma f=f$ where $\sigma$ acts by permuting the variables. The symmetric polynomials form a subring of $\mathbb Z[x_1,\dots,x_n]$ which we denote as $\Lambda_n$. One example of symmetric polynomials is the Newton power sums $p_k=x_1^k+\cdots+x_n^k$ for $k\ge 1$. Another example is elementary symmetric polynomials $e_k=\sum_{1\le i_1<\cdots< i_k\le n}x_{i_1}\cdots x_{i_k}$ for $1\le k\le n$. They arise in Vieta's relations for a polynomial equation. Obviously $\Lambda_n$ is a graded ring in degree. For a monomial $u=ax^{k_1}_1\cdots x^{k_n}_n$, we define its weight as $\mathrm{wt}(u)=k_1+2k_2+\cdots+nk_n$, and for a polynomial $f$ define $\mathrm{wt}(f)$ as the largest weight occuring among its monomials. We note that the weight of a polynomial is the same as the degree of $f(e_1,\dots,e_n)$. One of the first results in symmetric polynomials is that the elementary symmetric polynomials forms a generating set of $\Lambda_n$ as a (graded) algebra over $\mathbb Z$. This is our first theorem.

Another set of generators is the complete symmetric polynomial $h_k=\sum_{1\le i_1\le \cdots\le i_k\le n}x_{i_1}\cdots x_{i_k}$ for $1\le k\le n$. Consider the generating functions $E(t)=\sum_{k=0}^\infty e_kt^k$ and $H(t)=\sum_{k=0}^\infty h_k t^k$. One can show that

Therefore one has $E(t)H(-t)=1$. By multiplying out coefficients this gives a relation between $e_k$ and $h_k$. When the polynomials are over $\mathbb Q$ or any field, we have another set of generators which is the Newton power sums $p_k$. For a partition $\lambda$, we define $e_{\lambda}=e_{\lambda_1}\cdots e_{\lambda_m}$, and similarly for $h_{\lambda},p_{\lambda}$ and so on. Define the symmetric monomial basis $m_{\lambda}=\sum_{\sigma\in S(\lambda)}x^{\lambda_1}_{\sigma(1)}\cdots x^{\lambda_n}_{\sigma(n)}$ where $S(\lambda)=S_{n}/\mathrm{Stab}_{S_n}(\lambda)$, then this is an integral basis as well.

A polynomial is skew-symmetric if $\sigma f=\mathrm{sgn}(\sigma)f$ for all permutations $\sigma$. The skew-symmetric polynomials $\mathrm{Skew}(n)$ is a module over $\Lambda_n$. Define the alteration of a polynomial $\mathrm{Alt}(f)=\sum_{\sigma\in S_n}\mathrm{sgn}(\sigma) \sigma f$. Let $a_{\alpha}=\mathrm{Alt}(x_1^{\alpha_1}\cdots x_n^{\alpha_n})$ where $\alpha$ is a partition without repetitions, they form a basis for $\mathrm{Skew}(n)$. Let $\delta_n=(n-1,n-2,\dots,1)$. For a partition $\lambda=(\lambda_1,\dots,\lambda_n)$, define the Schur polynomial $s_{\lambda}=a_{\lambda+\delta_n}/a_{\delta}$, they also form a basis for symmetric functions. Here are some identities for transition of basis

There are similar transition determinental identities between $h_k,e_k,p_k$. Note that $s_\lambda s_\mu=\sum_\nu c_{\lambda \mu}^v s_v$ where $c_{\lambda \mu}^v$ is called the structure constant. This constant is usually difficult to find. To find them we need the following theorem.

Littlewood showed that Schur polynomials can be represented by sum indexed by semistandard Young tableaus $s_\lambda\left(x_1, \ldots, x_n\right)=\sum_{T \in \operatorname{SSYT}_\lambda(n)} \mathbf{x}^T .$ Define $\Lambda=\displaystyle\lim_{\longleftarrow}\Lambda_n$ the ring of symmetric functions. Let $\omega:\Lambda\rightarrow\Lambda$ be $\omega(h_{\lambda})=e_{\lambda}$, then $\omega^2=\mathrm{id}$. Define also bilinear form $\langle \cdot,\cdot\rangle:\Lambda^2\rightarrow\mathbb R$ extended by $\langle m_{\lambda},h_{\mu}\rangle=\delta_{\lambda\mu}$. Wrt this form, the Schur polynomials form an orthonormal basis with $\omega$ an isometry of it. This is the Hall inner product.

One application of symmetric polynomials is Lagrange's solution to depressed quartic equation. Let $\alpha_i$ be the roots of $x^4+a_2x^2+a_3x+a_4=0$. Let $f_1=(\alpha_1+\alpha_2)(\alpha_3+\alpha_4)$ and $f_2,f_3$ the other two in its orbit, then for any symmetric polynomial $s$, we have $s(f_1,f_2,f_3)$ is symmetric. Thus $u_{i}=e_i(f_1,f_2,f_3)$ are symmetric for $i=1,2,3$. Then $u_i$ can be written as a polynomial in terms of $a_i$ by Vieta. Since $x^3-u_1x^2+u_2x+u_3$ has coefficients polynomials in $a_i$, we can use the cubic formula to solve for $f_i$ in terms of $a_i$. Using $f_1=(\alpha_1+\alpha_2)(\alpha_3+\alpha_4)=-(\alpha_1+\alpha_2)^2$ since $\alpha_1+\alpha_2+\alpha_3+\alpha_4=0$, and likewise for $f_2,f_3$, we can solve for $\alpha_i$ in terms of $a_i$. For further connection between Galois theory and symmetric functions, read this.