Let $k$ be a commutative ring with identity, and let $\operatorname{Sym}(k)$ denote the ring of [symmetric functions](/page/Symmetric%20Function) over $k$. The elementary symmetric functions $e_1,e_2,e_3,\dots \in \operatorname{Sym}(k)$ are [algebraically independent](/page/Algebraic%20Independence) over $k$.

Let $d \geq 0$ be an integer, and let $\operatorname{Sym}^d$ denote the homogeneous degree-$d$ component of the ring of symmetric functions over the coefficient ring. For each partition $\lambda = (\lambda_1,\dots,\lambda_\ell)$ of $d$, define \begin{align*} e_\lambda := e_{\lambda_1}\cdots e_{\lambda_\ell}, \qquad h_\lambda := h_{\lambda_1}\cdots h_{\lambda_\ell}, \end{align*} where $e_r$ is the elementary symmetric function of degree $r$ and $h_r$ is the complete homogeneous symmetric function of degree $r$. Then both families \begin{align*} \{e_\lambda : \lambda \vdash d\}, \qquad \{h_\lambda : \lambda \vdash d\} \end{align*} are bases of $\operatorname{Sym}^d$.

Let $\lambda$ be a partition with $\ell(\lambda)\le n$. For a cell $u=(i,a)\in\lambda$, let $c(u):=a-i$ be its content and let $h(u):=\lambda_i-a+\lambda'_a-i+1$ be its hook length, where $\lambda'_a$ is the length of column $a$ in the conjugate partition. The number of semistandard Young tableaux of shape $\lambda$ with entries in $\{1,\dots,n\}$ is \begin{align*} s_\lambda(1^n)=\prod_{u\in\lambda}\frac{n+c(u)}{h(u)}, \end{align*} where $1^n$ denotes the specialization $(1,\dots,1)$ with $n$ entries.

Let $\lambda = (\lambda_1,\lambda_2,\dots)$ and $\mu = (\mu_1,\mu_2,\dots)$ be partitions with $\mu_i \leq \lambda_i$ for every $i \geq 1$, and let $\ell \in \mathbb{N}$ satisfy $\ell \geq \ell(\lambda)$. Extend both partitions by trailing zeroes, so that $\lambda_i = \mu_i = 0$ for all sufficiently large $i$. In the ring $\Lambda$ of symmetric functions, the skew Schur function $s_{\lambda/\mu}$ satisfies \begin{align*} s_{\lambda/\mu} = \det\left(h_{\lambda_i-\mu_j-i+j}\right)_{1 \leq i,j \leq \ell}, \end{align*} where $h_m$ denotes the complete homogeneous symmetric function of degree $m$, with the conventions $h_0 = 1$ and $h_m = 0$ for $m < 0$.

Let $\Lambda$ be the ring of symmetric functions over $\mathbb{Q}$ equipped with the Hall [inner product](/page/Inner%20Product) $(\cdot,\cdot)_\Lambda$, for which the Schur functions $\{s_\lambda\}$ form an [orthonormal basis](/page/Orthonormal%20Basis). For each $f \in \Lambda$, let $f^\perp: \Lambda \to \Lambda$ denote the adjoint of multiplication by $f$, so that \begin{align*} (f^\perp g, a)_\Lambda = (g, fa)_\Lambda \end{align*} for all $g,a \in \Lambda$. Then, for every partition $\nu$ and every integer $r \geq 0$, \begin{align*} h_r^\perp s_\nu &= \sum_{\lambda \subset \nu \,:\, \nu/\lambda \text{ is a horizontal } r\text{-strip}} s_\lambda, \\ e_r^\perp s_\nu &= \sum_{\lambda \subset \nu \,:\, \nu/\lambda \text{ is a vertical } r\text{-strip}} s_\lambda. \end{align*} Here $h_r$…

Let $\Lambda$ be the ring of symmetric functions over $\mathbb{Z}$ equipped with the Hall [inner product](/page/Inner%20Product) for which the Schur functions $\{s_\kappa\}$ form an [orthonormal basis](/page/Orthonormal%20Basis). For each partition $\lambda$, let $s_\lambda^\perp: \Lambda \to \Lambda$ denote the linear operator adjoint to multiplication by $s_\lambda$, so that \begin{align*} \langle s_\lambda^\perp f, g\rangle = \langle f, s_\lambda g\rangle \end{align*} for all $f,g \in \Lambda$. Then for all partitions $\lambda,\mu,\nu$, \begin{align*} s_\lambda^\perp(s_\mu s_\nu) = \sum_{\alpha,\beta} c_{\alpha\beta}^{\lambda} \bigl(s_\alpha^\perp s_\mu\bigr) \bigl(s_\beta^\perp s_\nu\bigr), \end{align*} where $c_{\alpha\beta}^{\lambda}$ denotes the Littlewood-Richardson coefficient…

Let $\Lambda_{\mathbb{R}}$ be the graded ring of symmetric functions over $\mathbb{R}$, with Schur basis $\{s_\lambda : \lambda \text{ is a partition}\}$. If $f, g \in \Lambda_{\mathbb{R}}$ are homogeneous Schur-positive symmetric functions, then their product $fg$ is Schur-positive.

Let $\lambda = (\lambda_1,\dots,\lambda_r)$ be a partition with at most $r$ nonzero parts, so that $\lambda_1 \geq \cdots \geq \lambda_r \geq 0$. In the ring of symmetric functions, let $s_\lambda$ denote the Schur function indexed by $\lambda$, and let $h_m$ denote the complete homogeneous symmetric function of degree $m$, with conventions $h_0 = 1$ and $h_m = 0$ for $m < 0$. Then \begin{align*} s_\lambda = \det\bigl(h_{\lambda_i - i + j}\bigr)_{1 \leq i,j \leq r}. \end{align*}

Let $\lambda$ be a partition. In the ring of symmetric functions, \begin{align*} s_\lambda = \sum_{\substack{\mu \text{ a partition} \\ |\mu| = |\lambda|}} K_{\lambda\mu} m_\mu, \end{align*} where $s_\lambda$ is the Schur function, $m_\mu$ is the monomial symmetric function indexed by $\mu$, and $K_{\lambda\mu}$ is the Kostka number, namely the number of semistandard Young tableaux of shape $\lambda$ and content $\mu$.

For partitions $\lambda$, $\mu$, and $\nu$, let $c_{\lambda\mu}^{\nu}$ denote the coefficient of $s_\nu$ in the product $s_\lambda s_\mu$ of Schur functions. If $\lambda \subset \nu$ and $|\nu|=|\lambda|+|\mu|$, then $c_{\lambda\mu}^{\nu}$ equals the number of Littlewood-Richardson tableaux of skew shape $\nu/\lambda$ and content $\mu$. If either condition fails, then $c_{\lambda\mu}^{\nu}=0$.