Semistandard weight dimensions detect virtual Weyl classes

Exactness of a weight projection. Fix an idempotent $p=p_{\nu }\in S$. For a short exact sequence $0\to A\to B\stackrel{f}{\to }C\to 0$ of right $S$-modules, the sequence $0\to Ap\to Bp\to Cp\to 0$ is exact as vector spaces. Injectivity and the kernel identity follow from the underlying sequence and $a=ap$ for $a\in Ap$. For surjectivity, lift $c\in C$ to $b\in B$; then $f(bp)=cp$. Right multiplication by $p$ is a linear projection, and preserves every $S$-submodule, although it need not be an $S$-linear endomorphism when $p$ is not central. Taking dimensions proves additivity. Only the canonical partition weight projections are used.

Weight functionals on composition series. Apply step 1.1 to every short exact sequence of a finite composition series of $N$. Its factors are the simple modules $L^{\mu }$, so ${\dim }_F(N{p}_{\nu })={\sum }_{\mu }[N:L^{\mu }]{\dim }_F(L^{\mu }{p}_{\nu })\mathrm{.}$ The defining relations of the Grothendieck group similarly give $[N]={\sum }_{\mu }[N:L^{\mu }][L^{\mu }]$. This argument requires exactness on vector spaces, not the false assertion that every noncentral weight projection lies in ${\mathrm{End}}_S(N)$.

Invertibility of the weight matrix: put $X_{\mu \nu }:={\dim }_F(L^{\mu }{\phi }_{T^{\nu }{T}^{\nu }})$ for $\mu ,\nu \in {\mathrm{\Lambda }}^{+}(d,n)$. Feeding a composition series of $W^{\lambda }$ into step 2.1 gives $K_{\lambda \nu }={\dim }_F(W^{\lambda }{\phi }_{T^{\nu }{T}^{\nu }})={\sum }_{\mu }{d}_{\lambda \mu }{X}_{\mu \nu },$ that is, $K=dX$ with $d=(d_{\lambda \mu })$ and $K=(K_{\lambda \nu })$. Both $d$ and $K$ are unitriangular over $\mathbb{Z}$ by [F1] and [F2] respectively, hence invertible over $\mathbb{Z}$, and therefore $X=d^{-1}K$ is invertible over $\mathbb{Z}$.

Conclusion: write $c_{\mu }:=[M:L^{\mu }]$ and $b_{\mu }:=c_{\mu }-{\sum }_{\lambda }{a}_{\lambda }{d}_{\lambda \mu }\in \mathbb{Z}$. By step 2.1 applied to $M$ and to the identity ${\dim }_F(W^{\lambda }{\phi }_{T^{\nu }{T}^{\nu }})=K_{\lambda \nu }$, for every $\nu$ $0={\dim }_F(M{\phi }_{T^{\nu }{T}^{\nu }})-{\sum }_{\lambda }{a}_{\lambda }{\dim }_F(W^{\lambda }{\phi }_{T^{\nu }{T}^{\nu }})={\sum }_{\mu }{c}_{\mu }{X}_{\mu \nu }-{\sum }_{\lambda }{a}_{\lambda }{\sum }_{\mu }{d}_{\lambda \mu }{X}_{\mu \nu }={\sum }_{\mu }{b}_{\mu }{X}_{\mu \nu }\mathrm{.}$ Since $X$ is invertible over $\mathbb{Z}$ by step 3.1, $b_{\mu }=0$ for all $\mu$. Hence $c_{\mu }={\sum }_{\lambda }{a}_{\lambda }{d}_{\lambda \mu }$, and step 2.1 together with the defining relations [F3] gives $[M]={\sum }_{\mu }{c}_{\mu }[L^{\mu }]={\sum }_{\lambda }{a}_{\lambda }{\sum }_{\mu }{d}_{\lambda \mu }[L^{\mu }]={\sum }_{\lambda }{a}_{\lambda }[W^{\lambda }]\mathrm{.}$

Consequence: if two finite-dimensional right $S$-modules $M,N$ satisfy ${\dim }_F(M{\phi }_{T^{\nu }{T}^{\nu }})={\dim }_F(N{\phi }_{T^{\nu }{T}^{\nu }})$ for all $\nu \in {\mathrm{\Lambda }}^{+}(d,n)$, then applying the argument of step 4.1 to the difference of their multiplicity vectors shows ${\sum }_{\mu }([M:L^{\mu }]-[N:L^{\mu }])X_{\mu \nu }=0$ for all $\nu$, so the invertibility of $X$ gives $[M:L^{\mu }]=[N:L^{\mu }]$ for all $\mu$ and hence $[M]=[N]$ in $G_0(S\text{-mod})$.