The permutation form is H-contravariant and restricts to the DJ form

The form is well defined, symmetric and bilinear. By [F1] the family $\{x_d:d\in D_{\lambda }\}$ is an $R$-basis of $M(\lambda )$, so there is exactly one $R$-bilinear form $\beta$ on $M(\lambda )$ whose Gram matrix in this basis is the diagonal matrix with entries $q^{\mathrm{\ell }(d)}$, $d\in D_{\lambda }$; this form is symmetric because its Gram matrix is diagonal, and it is the permutation form of the statement. For $\lambda =\mathrm{\varnothing }$ the basis is $\{1\}$ and $\beta (1,1)=q^0=1$.

Orbits of one generator. Fix $i$ and abbreviate $x_d:=x_{\lambda }{T}_d$. Partition $D_{\lambda }$ into the set $S$ of those $d$ for which $i,i+1$ lie in the same row of $t^{\lambda }\cdot d$, and the remaining set $D$; for $d\in D$ the element $d^{\ast }:=d{s}_i$ again lies in $D_{\lambda }$ and $(d^{\ast }{)}^{\ast }=d$, by the second and third cases of [F1] applied twice. Thus $D$ is partitioned into two-element orbits $\{u,v\}$ with $v=u{s}_i$ and $\mathrm{\ell }(v)=\mathrm{\ell }(u)+1$, and the action on such an orbit is $x_u{T}_i=x_v$, $x_v{T}_i=q{x}_u+(q-1)x_v$, while $x_d{T}_i=q{x}_d$ for $d\in S$. In particular each $x_d{T}_i$ lies in the span of the orbit of $d$.

Self-adjointness of the generators. In an orbit $\{u,v\}$ with $\mathrm{\ell }(v)=\mathrm{\ell }(u)+1$, the two diagonal pairings are $\beta (x_u{T}_i,x_u)=\beta (x_u,x_u{T}_i)=0$ and $\beta (x_v{T}_i,x_v)=\beta (x_v,x_v{T}_i)=(q-1)q^{\mathrm{\ell }(v)}$. The two off-diagonal pairings both equal $q^{\mathrm{\ell }(v)}=q{q}^{\mathrm{\ell }(u)}$ by the action in step 1.2. A singleton orbit contributes $q{q}^{\mathrm{\ell }(d)}$ on its diagonal, and pairings between distinct orbit spans vanish. Thus $\beta (x{T}_i,y)=\beta (x,y{T}_i)$ on basis vectors and hence, by bilinearity, on all of $M(\lambda )$.

Words, all of $H$, and restriction. For every word $h=T_{i_1}\cdots T_{i_r}$ of generators and all $x,y\in M(\lambda )$ one has $\beta (xh,y)=\beta (x,y{h}^{\ast })$, where $h^{\ast }=T_{i_r}\cdots T_{i_1}$: this follows from step 2.1 by induction on $r$, moving one generator at a time from the first to the second argument. Since the generators span $H$ as an $R$-algebra by [F3], every $h\in H$ is a finite $R$-linear combination of such words, and by [F2] the element $h^{\ast }$ is obtained from the same expression by reversing each word; bilinearity therefore gives $\beta (xh,y)=\beta (x,y{h}^{\ast })$ for all $h\in H$. Finally $P(\lambda )$ is a right ideal contained in $M(\lambda )$ by [F1], so the same identity holds for $x,y\in P(\lambda )$; its restriction to $P(\lambda )$ is the DJ form of the statement.

Base change. Under a ring map $R\to R^{\mathrm{\prime }}$ with $q\mapsto q^{\mathrm{\prime }}\in (R^{\mathrm{\prime }}{)}^{\times }$, the distinguished basis identifies $R^{\mathrm{\prime }}{\otimes }_{R}M(\lambda )$ with $M_{R^{\mathrm{\prime }}}(\lambda )$ and the diagonal weights specialize, identifying the scalar extension of $\beta$ with ${\beta }^{\mathrm{\prime }}$. The natural map $j:R^{\mathrm{\prime }}{\otimes }_{R}P(\lambda )\to M_{R^{\mathrm{\prime }}}(\lambda )$ has image $P_{R^{\mathrm{\prime }}}(\lambda )$: the vectors $z_{\lambda }{T}_w$ span $P(\lambda )$ and their images $z_{\lambda }^{\mathrm{\prime }}{T}_w^{\mathrm{\prime }}$ span $P_{R^{\mathrm{\prime }}}(\lambda )$. For pure tensors, and then by bilinearity, ${\beta }^{\mathrm{\prime }}(j(a\otimes p),j(b\otimes r))=ab\beta (p,r)$. This proves compatibility of the restricted forms along the natural map, without inferring injectivity of a tensor map from right exactness; the later split polytabloid basis establishes that this map is also an isomorphism. Contravariance specializes by the same identity.

The permutation form is well defined, symmetric and bilinear by step 1.1, is $H$-contravariant by step 3.1, and its restriction to $P(\lambda )$ satisfies the same identity; both forms are base-change compatible by step 4.1. Only the distinguished basis, inversion counts and the algebra structure of $H$ are used; no choice principle is used.