The horizontal Pieri rule

Given: AC, $V={\mathbb{C}}^r$, a partition $\lambda$ with $\mathrm{\ell }(\lambda )\le r$ and an integer $d\ge 0$.

For $d>0$, the one-row Specht module $S^{(d)}$ is trivial: its tabloid module has one basis element and all column stabilizers are trivial. Thus $S_{(d)}(V)\cong (V^{\otimes d}{)}^{S_d}$. This is isomorphic to the quotient symmetric power of Symmetric and exterior powers over an arbitrary field: the averaging operator $P=d{!}^{-1}{\sum }_{\sigma \in S_d}\sigma$ annihilates every coinvariance relation and induces the inverse to the quotient map restricted to invariants, because $q(Pt)=q(t)$ and $P$ fixes invariant tensors. These maps commute with $\mathrm{GL}(V)$ (Schur modules and their characters, Column antisymmetrizers, polytabloids, and Specht modules). For $d=0$, use the empty partition in place of $(d)$; its Schur module, $V^{\otimes 0}$ and ${\mathrm{Sym}}^{0}V$ are all $\mathbb{C}$.

Littlewood--Richardson rule: for partitions $\lambda ,\mu$ with $\mathrm{\ell }(\lambda ),\mathrm{\ell }(\mu )\le r$, $S_{\lambda }(V)\otimes S_{\mu }(V)\cong {⨁}_{\nu :\mathrm{\ell }(\nu )\le r}{S}_{\nu }(V{)}^{\oplus c_{\lambda \mu }^{\nu }}$, with $c_{\lambda \mu }^{\nu }$ the number of LR tableaux of shape $\nu \mathrm{/}\lambda$ and content $\mu$, and $c_{\lambda \mu }^{\nu }=0$ unless $\lambda \subseteq \nu$ and $\mathrm{\mid }\nu \mathrm{\mid }=\mathrm{\mid }\lambda \mathrm{\mid }+\mathrm{\mid }\mu \mathrm{\mid }$ (The Littlewood--Richardson tensor-product rule, Littlewood--Richardson tableaux and coefficients).

A semistandard skew tableau of shape $\nu \mathrm{/}\lambda$ and content $(d)$ has all its entries equal to $1$; weak increase along rows is automatic, and strict increase down columns forces every column of $\nu \mathrm{/}\lambda$ to contain at most one box, so such a tableau exists if and only if $\nu \mathrm{/}\lambda$ is a horizontal strip, and then it is unique; its reading word is the constant word $11\cdots 1$, a lattice word (Skew diagrams and semistandard skew tableaux, Semistandard tableaux and Kostka numbers, Littlewood--Richardson tableaux and coefficients).

The complete symmetric polynomial $h_d={\sum }_{1\le i_1\le \cdots \le i_d\le r}{x}_{i_1}\cdots x_{i_d}$ equals $s_{(d)}$ by the one-row tableau expansion (and $h_0=s_{\mathrm{\varnothing }}=1$), and the Schur polynomials $s_{\nu }$, $\mathrm{\ell }(\nu )\le r$, are linearly independent: after multiplying a finite relation by $a_{{\delta }_r}$, the coefficient of the strictly decreasing exponent vector $\nu +{\delta }_r$ is exactly that relation’s coefficient of $s_{\nu }$ (Semistandard tableaux expand Schur characters, Stable Schur functions from bialternants, The Littlewood--Richardson tensor-product rule).