Cyclotomic Schur weight Gram determinants are products of monic semistandard norms

Generic diagonalizability of $A_{\nu }$. The finite row-coset identity gives $x_{\nu }^2=P_{\nu }(z)x_{\nu }$, and every factor of $P_{\nu }(q_{H}t)={\prod }_R{\prod }_{j=1}^{\mathrm{\mid }R\mathrm{\mid }}(1+q_{H}t+\cdots +(q_{H}t{)}^{j-1})$ is a nonzero polynomial over any $F$, so $P_{\nu }\mathrm{\ne }0$ in $K$ and $x_{\nu }\mathrm{/}{P}_{\nu }$ is a selfadjoint idempotent. The polynomial $u_{\nu }$ is symmetric separately on each initial component interval, hence commutes with every row generator and with $x_{\nu }$, and both are selfadjoint. Distinct generic JM words provide a basis of simultaneous eigenvectors ([F4]); on the eigenline of $s$, the element $u_{\nu }$ has eigenvalue $d_s={\prod }_{k=2}^{\mathrm{\ell }}{\prod }_{r=1}^{B_{k-1}}(\mathrm{r}\mathrm{e}\mathrm{s}(s^{-1}(r))-U_k)$, so $u_{\nu }$ is diagonalizable, and since $x_{\nu }$ commutes with $u_{\nu }$, each $u_{\nu }$-eigenspace splits into the image and kernel of the idempotent $x_{\nu }\mathrm{/}{P}_{\nu }$. Hence $A_{\nu }=x_{\nu }{u}_{\nu }$ is diagonalizable with nonzero eigenvalues $\alpha =P_{\nu }{d}_s\mathrm{\ne }0$. Put $E=\mathrm{i}\mathrm{m}{A}_{\nu }$; then $V_{\lambda }=E\oplus \ker A_{\nu }$ is an orthogonal decomposition by selfadjointness, $A_{\nu }{\mathrm{\mid }}_E$ is invertible and selfadjoint, and the supplied semistandard basis [F5] identifies the $\lambda$-cell layer of $m_{\nu }{H}_n$ with the span of the tensors $w_T\otimes v_s$, $w_T={\sum }_{s:\nu (s)=T}{v}_s$, so on the generic simple block [F7] the $(w_T{)}_T$ are a basis of $E$ and $\dim E=K_{(\lambda ,\nu )}$.

The inverse-operator Gram formula. The cell quotient of the membership identity of [F5], followed by orthogonal projection to $E$ and pairing with $w_T\in E$ (which annihilates $\ker A_{\nu }$), gives $m_{(t^{\lambda },T)}\rho (h_U)=v_0{b}_{\lambda }(w_T,(A_{\nu }{\mathrm{\mid }}_E{)}^{-1}{w}_U)b_{\lambda }(v_0,-)$. On the Schur side the composite $M^{\lambda }\to M^{\nu }\to M^{\lambda }$ sending $m_{\lambda }$ to $m_{(t^{\lambda },T)}{h}_U$ is $G_{\lambda \nu }(T,U)m_{\lambda }$ modulo higher shapes by the cellular law [F6], and higher Schur maps evaluate to higher Murphy shapes and therefore vanish on this block. Comparing the resulting rank-one operators, and observing that changing $h_U$ changes it only by a map into $\ker A_{\nu }$, proves $G_{\lambda \nu }(T,U)=b_{\lambda }(w_T,(A_{\nu }{\mathrm{\mid }}_E{)}^{-1}{w}_U)$ as an identity over the ground field. This formula uses the inverse operator rather than identifying the Schur form with a restriction of the Specht form.

The chain basis and the unitriangular transition. Let $R_1<\cdots <R_m=n$ be the ends of the consecutive nonempty rows of the type tableau of $\nu$. A semistandard $T$ determines the chain $\mathit{\lambda }(T)=({\lambda }_T[1],\dots ,{\lambda }_T[m])$ of prefix shapes filled by its first $j$ type rows; every standard refinement in the fibre of $T$ has prefix shape ${\lambda }_T[j]$ on its first $R_j$ entries, and conversely the chain determines $T$, so chains and semistandard tableaux of type $\nu$ correspond bijectively. The elementary symmetric polynomials in $X_1,\dots ,X_{R_j}$, for every row end $R_j$, commute with every row generator (a type row lies wholly inside or wholly outside the prefix and Bernstein commutation fixes symmetric polynomials), hence with $m_{\nu }$, and thus preserve $E$; they act on the chain-$T$ layer by the elementary symmetric functions of its prefix residue multiset. Distinct multipartitions have distinct residue multisets over $K$ (distinct $U_a$ separate components, and within a component the content-diagonal multiplicities recover the partition), so these operators separate the chains. Ordering $T$ by a linear extension of dominance of the chains, the action on $E$ is triangular in the chain order with diagonal the chain scalar; commuting diagonalizable operators are jointly diagonalized, and the projector $E_T$ onto the chain-$T$ eigenspace is $1$ on the fibre of $T$ and $0$ on all other fibres. Hence $g_T=w_T{E}_T=w_T+{\sum }_{U:\mathit{\lambda }(U)>\mathit{\lambda }(T)}{a}_{TU}{w}_U,$ the $g_T$ form a basis of $E$, the transition matrix from $(w_T)$ to $(g_T)$ is upper unitriangular with determinant exactly $1$, and $b_{\lambda }(g_T,g_U)=0$ for $T\mathrm{\ne }U$ because differing chains are separated by a selfadjoint row-prefix operator.

Spectral-block determinant. Decompose $E={⨁}_{\alpha \mathrm{\ne }0}{E}_{\alpha }$ into $A_{\nu }$-eigenspaces; they are mutually orthogonal with nondegenerate restricted forms. Choose bases ${\mathcal{B}}_{\alpha }$ of the $E_{\alpha }$, let $H_{\alpha }$ be the $b_{\lambda }$-Gram matrix of ${\mathcal{B}}_{\alpha }$, and let $X$ be the coordinate matrix whose columns are the $w_T$ in the concatenated basis. Step 2.1 gives $G_{\lambda \nu }=X^{\mathsf{T}}{\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}}_{\alpha }({\alpha }^{-1}{H}_{\alpha })X$, hence by [F8] $\det G_{\lambda \nu }=(\det X{)}^2{\prod }_{\alpha \mathrm{\ne }0}{\alpha }^{-\dim E_{\alpha }}\det H_{\alpha }\mathrm{.}$ A change of the bases ${\mathcal{B}}_{\alpha }$ multiplies each $\det H_{\alpha }$ by $(\det R_{\alpha }{)}^2$ and divides $\det X$ by ${\prod }_{\alpha }\det R_{\alpha }$, so the expression is independent of the arbitrary choices; individual terms may have negative valuation while the total is integral because the Schur Gram matrix is integral. No diagonalization of a symmetric form is used, so characteristic $2$ causes no extra issue.

Normalization by the minimal refinement. Let $t_T$ be the dominance-minimal standard refinement of $T$: in each type-row class place the next increasing interval into its nodes in reverse physical-component order, then reverse physical-row order, then left to right. The eigenvalue of $u_{\nu }$ on $f_{t_T}$ is $u_T={\prod }_x{\prod }_{b>a_T(x)}(r_x-U_b)$, which is nonzero in the stated generic fields because the flag condition [F3] places the node $x$ in a physical component strictly earlier than $b$, so $r_x$ and $U_b$ are distinct monomials. Since $u_{\nu }$ commutes with $x_{\nu }$, we have $f_{t_T}{x}_{\nu }=u_T^{-1}{f}_{t_T}{m}_{\nu }\in E$, and this vector lies in the chain-$T$ eigenspace, hence is a scalar multiple of $g_T$; the scalar is $c_T=[f_{t_T}](f_{t_T}{x}_{\nu })$, and comparing the coefficient of $v_{t_T}$ — which equals $1$ in $g_T$, while no strictly smaller refinement lies in that chain — gives $g_T=c_T^{-1}{f}_{t_T}{x}_{\nu }$ with the nonzero value of $c_T$ computed below.

Initial norm and adjacent-swap recurrence. For the initial tableau $f_{t^{\lambda }}=v_{t^{\lambda }}$, and the cellular Gram entry is the coefficient of $m_{\lambda }$ in $m_{\lambda }^2$ modulo higher shapes, so with $P_{\lambda }(z)={\prod }_{\text{rows }r}[\mathrm{\mid }r\mathrm{\mid }{]}_z!$ and the row generators acting on $v_{t^{\lambda }}$ by $z$ and $u_{\lambda }$ by its residue product, ${\gamma }_{t^{\lambda }}=P_{\lambda }(z){\prod }_{a=1}^{\mathrm{\ell }}{\prod }_{x\in [{\lambda }^{(a)}]}{\prod }_{b>a}(U_a{z}^{\mathrm{content}(x)}-U_b)\mathrm{.}$ If $t$ strictly dominates $s=t(i,i+1)$ with $s$ standard and $A=r_t(i)$, $B=r_t(i+1)$, then $f_t{T}_i$ is supported on $f_t,f_s$, with diagonal coefficients $a_t=(z-1)B\mathrm{/}(B-A)$, $a_s=(z-1)A\mathrm{/}(A-B)$ satisfying $a_t+a_s=z-1$, off-diagonal coefficient $1$ in the monic normalization, and reverse coefficient $b_s=z+(z-1)a_t-a_t^2=(z-a_t)(1+a_t)$; selfadjointness and orthogonality give the recurrence ${\gamma }_s={\gamma }_t\frac{(zB-A)(B-zA)}{(B-A{)}^2}={\gamma }_{t}R(A,B),$ whose denominator is nonzero because a permitted standard swap concerns incomparable nodes.

The monic Schur norm and the determinant product. Selfadjointness and $x_{\nu }^2=P_{\nu }{x}_{\nu }$ give $b_{\lambda }(f_{t_T}{x}_{\nu },f_{t_T}{x}_{\nu })=P_{\nu }{b}_{\lambda }(f_{t_T},f_{t_T}{x}_{\nu })=P_{\nu }{c}_T{\gamma }_{t_T}$, and $A_{\nu }{g}_T=P_{\nu }{u}_T{g}_T$ because $A_{\nu }$ acts as $P_{\nu }{u}_T$ on the chain-$T$ eigenspace. Using step 2.1 in the $g$-basis (the transition from $(w_T)$ has determinant $1$) together with the orthogonality of step 2.2, $b_{\lambda }(g_T,(A_{\nu }{\mathrm{\mid }}_E{)}^{-1}{g}_T)=\frac{b_{\lambda }(g_T,g_T)}{P_{\nu }{u}_T}=\frac{{\gamma }_{t_T}}{c_T{u}_T},$ and the off-diagonal entries vanish, so $\det G_{\lambda \nu }={\prod }_{T\in {\mathrm{SStd}}_{\nu }(\lambda )}\frac{{\gamma }_{t_T}}{c_T{u}_T}\mathrm{.}$

The explicit symmetrizer value of $c_T$. We use [F2] only for the row parabolic, localized at differences of variables in the same type-row interval. This restriction is essential: differences between arbitrary JM entries need not be invertible on a Specht module. The chain-$T$ subspace of step 2.2 is stable under the row parabolic; its JM eigenwords are precisely the standard refinements of $T$, so the entries in each type row occupy a horizontal strip in each physical component. Two nodes of a horizontal strip in one component have distinct contents: for upper row $i$ and lower row $j$, every strip node of row $i$ lies strictly right of every strip node of row $j$, making the contents strictly ordered. For different components the content difference is at most $n-2$ (each node has absolute content bounded by its component size minus one), so it cannot cancel the nonzero multiple $n(a-b)$ in the deformation; in the universal field the independent $U_a$ separate them. Thus every same-row variable difference acts invertibly on this entire chain subspace. The finite row Bernstein relations therefore extend its action to this row-local algebra. The crossed-product coefficient calculation below is an identity in that row-local algebra and can be evaluated here, without extending the action to the localization at all variable differences. For a row interval $I$, put $x_I={\sum }_{w\in S_I}{T}_w$, write $x_I={\sum }_w{a}_{w}w$; the row-coset identity $T_i{x}_I=z{x}_I$ gives $a_{s_{i}w}=s_i(a_w)$ by coefficient comparison with $T_i=c_{i,i+1}+d_{i,i+1}{s}_i$, and the coefficient of the longest permutation comes from the product of the $d_{ij}=(X_j-z{X}_i)\mathrm{/}(X_j-X_i)$ over the positive roots of $I$, each once. Hence $a_1=w_0\left({\prod }_{i<j\in I}\frac{X_j-z{X}_i}{X_j-X_i}\right)={\prod }_{i<j\in I}\frac{z{X}_j-X_i}{X_j-X_i}=C_I(X)$. Disjoint row intervals commute, so $x_{\nu }={\prod }_I{x}_I$ has identity coefficient ${\prod }_I{C}_I(X)$; expanding $x_I=C_I(X)+{\sum }_{w\mathrm{\ne }1}{b}_w(X){\mathrm{\Phi }}_w$ and using that every ${\mathrm{\Phi }}_w$ with $w\mathrm{\ne }1$ changes the distinct row entries and hence the JM word, the diagonal coefficient of $f_{t_T}{x}_{\nu }$ is exactly $C_{t_T}$ evaluated at the residues of $t_T$. Therefore $c_T=C_{t_T}={\prod }_{i<j\text{ same type row}}\frac{z{r}_{t_T}(j)-r_{t_T}(i)}{r_{t_T}(j)-r_{t_T}(i)}$. Every factor is nonzero in the stated generic fields: in one physical component two nodes of a horizontal strip have distinct contents, and in the deformation two nodes of different components have content difference at most $n-2$, which cannot cancel the exponent $n(a-b)$.

The inversion-set formula and the explicit product. Every standard tableau is a linear extension of the node order; bubble-sorting it toward the initial extension swaps adjacent inverted (hence incomparable) nodes exactly once and never swaps comparable ones, so reversing those swaps and multiplying the recurrence of step 3.3 gives ${\gamma }_{t_T}={\gamma }_{t^{\lambda }}{\prod }_{\begin{array}{c}x<y\\ t_T(x)>t_T(y)\end{array}}R(r_x,r_y),$ and $R$ is symmetric in its arguments. Dividing by $C_{t_T}$ and $u_T$ and using ${\gamma }_{t^{\lambda }}\mathrm{/}{u}_T=P_{\lambda }(z)B_T$ with $B_T={\prod }_x{\prod }_{a(x)<b\le a_T(x)}(r_x-U_b)$ (a literal nested-product cancellation by the flag condition), together with the telescoping row identity ${\prod }_{i<j}C(A_i,A_j)=[r{]}_z!$ for residues $A_j=A_1{z}^{j-1}$ in a row of length $r$, the four pair classes — same physical row with same or different labels, different physical rows with same or different labels — give exactly $\frac{{\gamma }_{t_T}}{C_{t_T}{u}_T}=B_T{\prod }_{(x,y)\in \mathcal{C}(T)}C(r_x,r_y){\prod }_{(x,y)\in \mathcal{I}(T)}R(r_x,r_y),$ where $\mathcal{C}(T)$ collects pairs $x<y$ that are in one physical row with different labels or in different physical rows with the same label, and $\mathcal{I}(T)$ collects pairs $x<y$ in different physical rows with $L_T(x)>L_T(y)$. Multiplying over $T$ and collecting like node pairs yields $\det G_{\lambda \nu }={\prod }_T{B}_T{\prod }_{x<y}C(r_x,r_y{)}^{c_{xy}}R(r_x,r_y{)}^{i_{xy}},c_{xy}=\mathrm{\#}\{T:(x,y)\in \mathcal{C}(T)\},i_{xy}=\mathrm{\#}\{T:(x,y)\in \mathcal{I}(T)\}\mathrm{.}$ All factors are nonzero in the two generic regimes by step 4.2, the empty weight contributes determinant $1$, and no factor $P_{\nu }$ or $c_T$ is inverted in the $\mathcal{O}$-lattice; this proves both displayed determinant formulas.