For partitions $\lambda$ and $\mu$ of $n$, the Kostka number $K_{\lambda\mu}$ is zero unless $\lambda \unrhd \mu$, where $\lambda \unrhd \mu$ denotes the [dominance order](/page/Dominance%20Order), meaning $\sum_{i=1}^{k}\lambda_i \geq \sum_{i=1}^{k}\mu_i$ for every $k \geq 1$. Moreover, $K_{\lambda\lambda}=1$.

Let $\lambda=(\lambda_1,\dots,\lambda_n)$ be a partition with at most $n$ parts, where trailing parts are allowed to be $0$. Let $\delta := (n-1,n-2,\dots,0)$, and for any $n$-tuple $\alpha=(\alpha_1,\dots,\alpha_n)$ of nonnegative integers define the alternant \begin{align*} a_\alpha(x_1,\dots,x_n) := \det\left(x_i^{\alpha_j}\right)_{1\leq i,j\leq n}. \end{align*} Then the Schur polynomial in $n$ variables satisfies \begin{align*} s_\lambda(x_1,\dots,x_n)=\frac{a_{\lambda+\delta}(x_1,\dots,x_n)}{a_\delta(x_1,\dots,x_n)}. \end{align*}

Let $\Lambda$ be the ring of symmetric functions over $\mathbb{Z}$. For every partition $\mu$ and every integer $r \geq 0$, \begin{align*} e_r s_\mu = \sum_{\lambda} s_\lambda, \end{align*} where $e_r \in \Lambda$ is the $r$th elementary symmetric function, $s_\mu \in \Lambda$ is the Schur function indexed by $\mu$, and the sum ranges over all partitions $\lambda$ such that $\lambda / \mu$ is a vertical strip of size $r$, meaning that $\lambda / \mu$ has exactly $r$ boxes and contains at most one box in each row.

Let $\Lambda$ be the ring of symmetric functions over $\mathbb{Z}$, and let \begin{align*} \omega: \Lambda &\to \Lambda \end{align*} be the standard omega involution, defined by $\omega(h_r)=e_r$ for every integer $r \geq 0$, where $h_r$ and $e_r$ denote the complete homogeneous and elementary symmetric functions, respectively. For every partition $\lambda$, with conjugate partition $\lambda'$, one has \begin{align*} \omega(s_\lambda)=s_{\lambda'}. \end{align*}

Let $\Lambda_{\mathbb{Q}}$ be the ring of symmetric functions over $\mathbb{Q}$, equipped with the Hall [inner product](/page/Inner%20Product) determined on the power-sum basis by \begin{align*} \langle p_\alpha, p_\beta\rangle = \delta_{\alpha,\beta} z_\alpha, \end{align*} where $z_\alpha = \prod_{i \geq 1} i^{m_i(\alpha)} m_i(\alpha)!$ for a partition $\alpha=(1^{m_1(\alpha)}2^{m_2(\alpha)}\cdots)$. For $k \geq 1$, let $p_k^\perp:\Lambda_{\mathbb{Q}}\to\Lambda_{\mathbb{Q}}$ denote the adjoint of multiplication by $p_k$ with respect to this inner product. If $\lambda=(1^{m_1(\lambda)}2^{m_2(\lambda)}\cdots)$ is a partition, then \begin{align*} p_k^\perp p_\lambda = \begin{cases} k\,m_k(\lambda)\,p_{\lambda\setminus k}, & m_k(\lambda)>0,\\ 0, & m_k(\lambda)=0, \end{cases} \end{align*}…

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 $\lambda = (\lambda_1,\dots,\lambda_r)$ 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 $R$ be a commutative $\mathbb{Q}$-algebra, and let $A$ and $B$ be alphabets in plethystic notation. Assume that for every integer $r \geq 1$ the power sums satisfy \begin{align*} p_r[A+B] &= p_r[A] + p_r[B], \\ p_r[A-B] &= p_r[A] - p_r[B]. \end{align*} For each alphabet expression $X \in \{A,B,A+B,A-B\}$, define the complete symmetric generating series \begin{align*} H[X;t] = \sum_{n=0}^{\infty} h_n[X]t^n \in R[[t]] \end{align*} by the formal exponential identity \begin{align*} H[X;t] = \exp\left(\sum_{r=1}^{\infty}\frac{p_r[X]t^r}{r}\right). \end{align*} Then, as identities in the formal [power series](/page/Power%20Series) ring $R[[t]]$, \begin{align*} H[A+B;t] &= H[A;t]H[B;t], \end{align*} and \begin{align*} H[A-B;t] &= \frac{H[A;t]}{H[B;t]}. \end{align*}

Let $\Lambda$ be the ring of symmetric functions over $\mathbb{Z}$, and let $\{s_\alpha\}$ denote the Schur basis indexed by partitions $\alpha$. For partitions $\lambda$ and $\nu$ with $\lambda \subset \nu$, the skew Schur function $s_{\nu/\lambda}$ has the Schur expansion \begin{align*} s_{\nu/\lambda}=\sum_{\mu} c_{\lambda\mu}^{\nu}s_\mu, \end{align*} where the sum ranges over all partitions $\mu$, and the integers $c_{\lambda\mu}^{\nu}$ are the Littlewood-Richardson coefficients defined by \begin{align*} s_\lambda s_\mu=\sum_{\rho} c_{\lambda\mu}^{\rho}s_\rho. \end{align*}