Let $\operatorname{Sym}$ be the ring of symmetric functions over $\mathbb{Q}$, equipped with the Hall [inner product](/page/Inner%20Product) $\langle \cdot, \cdot \rangle$. For every pair of partitions $\lambda$ and $\mu$, let $p_\lambda$ and $p_\mu$ denote the corresponding power sum symmetric functions, and let \begin{align*} z_\lambda = \prod_{i \geq 1} i^{m_i(\lambda)} m_i(\lambda)!, \end{align*} where $m_i(\lambda)$ is the multiplicity of the part $i$ in $\lambda$. Then \begin{align*} \langle p_\lambda, p_\mu \rangle = z_\lambda \delta_{\lambda,\mu}. \end{align*} Moreover, for each $n \in \mathbb{N}$, the restriction of $\langle \cdot, \cdot \rangle$ to the homogeneous component $\operatorname{Sym}^n$ is nondegenerate.

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 $\{s_\mu\}$ form an [orthonormal basis](/page/Orthonormal%20Basis). For a partition $\lambda$, let \begin{align*} M_{s_\lambda}: \Lambda &\to \Lambda \\ f &\mapsto s_\lambda f \end{align*} be multiplication by $s_\lambda$, and let \begin{align*} s_\lambda^\perp: \Lambda &\to \Lambda \end{align*} denote the adjoint of $M_{s_\lambda}$ with respect to $(\cdot,\cdot)_\Lambda$. Then, for every pair of partitions $\lambda$ and $\nu$, \begin{align*} s_\lambda^\perp(s_\nu)=s_{\nu/\lambda}. \end{align*} Here $s_{\nu/\lambda}$ denotes the skew Schur function, with the convention that $s_{\nu/\lambda}=0$ when the Young…

Let $\Lambda_{\mathbb{Z}}$ be the ring of symmetric functions over $\mathbb{Z}$ in the variables $x_1,x_2,\dots$. For every partition $\mu$ and every integer $r \geq 0$, \begin{align*} h_r s_\mu = \sum_{\substack{\lambda \text{ a partition}\\ \mu \subset \lambda\\ |\lambda|-|\mu|=r\\ \lambda/\mu \text{ is a horizontal strip}}} s_\lambda, \end{align*} where $h_r$ is the complete homogeneous symmetric function of degree $r$, $s_\nu$ denotes the Schur function indexed by a partition $\nu$, and $\lambda/\mu$ is a horizontal strip if it contains no two boxes in the same column.

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 $X = (x_1,x_2,\dots)$ and $Y = (y_1,y_2,\dots)$ be two independent alphabets, and let $\operatorname{Sym}_{\mathbb{Q}}[X]$ and $\operatorname{Sym}_{\mathbb{Q}}[Y]$ denote the rings of symmetric functions over $\mathbb{Q}$ in the alphabets $X$ and $Y$, respectively. Let $\langle \cdot,\cdot\rangle_X$ be the Hall [inner product](/page/Inner%20Product) in the $X$ variables, characterized by \begin{align*} \langle s_\lambda[X],s_\mu[X]\rangle_X=\delta_{\lambda\mu} \end{align*} for all integer partitions $\lambda$ and $\mu$. If the Cauchy kernel is \begin{align*} \Omega[X,Y]=\sum_{\lambda}s_\lambda[X]s_\lambda[Y], \end{align*} where the sum ranges over all integer partitions $\lambda$, then for every $f \in \operatorname{Sym}_{\mathbb{Q}}$, \begin{align*} \langle f[X],\Omega[X,Y]\rangle_X=f[Y]. \end{align*}…

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 $\lambda$ be a partition of $r$. The number $f^\lambda$ of standard Young tableaux of shape $\lambda$ is \begin{align*} f^\lambda=\frac{r!}{\prod_{u\in\lambda}h(u)}. \end{align*}

Let $\Lambda_{\mathbb{Q}}$ be the ring of symmetric functions over $\mathbb{Q}$, and let $\omega: \Lambda_{\mathbb{Q}} \to \Lambda_{\mathbb{Q}}$ be the standard algebra involution determined by \begin{align*} \omega(e_n) = h_n \end{align*} for every $n \geq 1$. Then, for every integer $n \geq 1$, \begin{align*} \omega(h_n) = e_n, \qquad \omega(p_n) = (-1)^{n-1}p_n. \end{align*} Consequently, for every partition $\lambda = (\lambda_1,\dots,\lambda_r)$, \begin{align*} \omega(p_\lambda) = (-1)^{|\lambda|-\ell(\lambda)}p_\lambda, \end{align*} where $p_\lambda := p_{\lambda_1}\cdots p_{\lambda_r}$, $|\lambda| := \lambda_1+\cdots+\lambda_r$, and $\ell(\lambda) := r$.

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 $c_{\lambda\mu}^{\nu}=0$.

Let \begin{align*} \operatorname{Sym}_{\mathbb Q}=\bigoplus_{n \geq 0}\operatorname{Sym}_{\mathbb Q,n} \end{align*} be the graded ring of symmetric functions over $\mathbb Q$, equipped with the Hall [inner product](/page/Inner%20Product) $\langle \cdot,\cdot\rangle$, whose homogeneous components of distinct degrees are orthogonal and whose restriction to each $\operatorname{Sym}_{\mathbb Q,n}$ is nondegenerate. For $f \in \operatorname{Sym}_{\mathbb Q}$, let \begin{align*} M_f:\operatorname{Sym}_{\mathbb Q} &\to \operatorname{Sym}_{\mathbb Q} \\ g &\mapsto fg \end{align*} denote multiplication by $f$, and let \begin{align*} f^\perp:\operatorname{Sym}_{\mathbb Q} &\to \operatorname{Sym}_{\mathbb Q} \end{align*} denote the adjoint of $M_f$ with respect to the Hall inner product. Then, for…