Semistandard tableaux expand Schur characters

First fix a partition $\nu ⊢n$ with at most $r$ parts, padded by zeros to length $r$. By [F3] the weight-$\nu$ subspace $E_{\nu }$ of $V^{\otimes n}$ has as its basis the elementary tensors whose index word has content $\nu$. The group $S_n$ permutes these basis vectors by place permutations, and the action on the basis is transitive (any word with content $\nu$ is a rearrangement of $1^{{\nu }_1}\cdots r^{{\nu }_r}$), with the stabilizer of a word of content $\nu$ being the subgroup of permutations preserving the letter classes, a conjugate of the Young subgroup $S_{\nu }$; hence $E_{\nu }$ is isomorphic to the Young permutation module $M^{\nu }$ (Young subgroups, tabloids, and permutation modules).

By Young's rule [F2] the multiplicity of the simple module $S^{\lambda }$ in $E_{\nu }\cong M^{\nu }$ is the Kostka number $K_{\lambda \nu }$, which by Semistandard tableaux and Kostka numbers is the number of semistandard tableaux of shape $\lambda$ and weight $\nu$.

Compare the $S_n$-isotypic $S^{\lambda }$-component of $V^{\otimes n}$. By the Schur--Weyl decomposition [F1] and non-isomorphism of distinct Specht modules [F2], the $S^{\lambda }$-isotypic component of $V^{\otimes n}$ is $S^{\lambda }\otimes S_{\lambda }(V)$, on which $S_n$ acts on the first factor alone; therefore the $S^{\lambda }$-multiplicity in $E_{\nu }$ equals $\dim S_{\lambda }(V{)}_{\nu }$, the dimension of the weight-$\nu$ space of the $\mathrm{GL}(V)$-module $S_{\lambda }(V)$. Combined with step 2.1 this gives $\dim S_{\lambda }(V{)}_{\nu }=K_{\lambda \nu }=\mathrm{\#}\{T:T\text{ semistandard of shape }\lambda \text{ and weight }\nu \}\mathrm{.}$

For an arbitrary weight $\alpha \in {\mathbb{Z}}_{\ge 0}^r$ of total $n$, sort its entries into a partition $\nu$. A permutation matrix carries the weight-$\alpha$ space of $S_{\lambda }(V)$ isomorphically onto its weight-$\nu$ space, by conjugating the diagonal torus. The Bender--Knuth involutions of Bender--Knuth involutions permute the weights of semistandard tableaux likewise give a bijection between tableaux of weights $\alpha$ and $\nu$; use a product of adjacent transpositions sorting $\alpha$. Thus step 3.1 holds for every composition weight $\alpha$, with $K_{\lambda \alpha }$ denoting this tableau count. Summing these weight dimensions over all $\alpha$ and using the definition of the character of Schur modules and their characters gives $\mathrm{ch}{S}_{\lambda }(V)={\sum }_{\alpha }{K}_{\lambda \alpha }{x}^{\alpha }={\sum }_T{x}^{\mathrm{wt}(T)}$, summed over semistandard tableaux of shape $\lambda$ with entries in $\{1,\dots ,r\}$. By the tableau expansion of [F4] this equals $s_{\lambda }(x_1,\dots ,x_r)$; the case $\mathrm{\ell }(\lambda )>r$ is the vanishing definition $S_{\lambda }(V)=0$, matched by $s_{\lambda }(x_1,\dots ,x_r)=0$ in [F4]. In particular $\mathrm{ch}{S}_{\lambda }(V)=0$ exactly when $\mathrm{\ell }(\lambda )>r$ or $\lambda$ has no semistandard tableau with entries in $\{1,\dots ,r\}$; the latter never happens for $\mathrm{\ell }(\lambda )\le r$ (fill row $i$ with the letter $i$), so $S_{\lambda }(V)\mathrm{\ne }0$ if and only if $\mathrm{\ell }(\lambda )\le r$, and the multiplicity of each weight $\alpha$ in $S_{\lambda }(V)$ is the number of semistandard tableaux of shape $\lambda$ and weight $\alpha$.

The polynomial ${\sum }_T{x}^{\mathrm{wt}(T)}$ is symmetric in $x_1,\dots ,x_r$ by Bender--Knuth involutions permute the weights of semistandard tableaux, consistently with $s_{\lambda }$ being the bialternant quotient, whose numerator and denominator are alternating and whose quotient is therefore a symmetric polynomial.