Generic cyclotomic Schur deformation standards have nondegenerate forms

Word separation over the universal fraction field. Assign to each standard multitableau the sequence of joint Jucys–Murphy words: the diagonal values $U_a{z}^{j-i}$ at the nodes carrying the labels $1,\dots ,n$ ([F3]). If two standard multitableaux first differ at step $r$, then their restrictions to the boxes $1,\dots ,r-1$ coincide, and the two added nodes are two different addable nodes of that common prefix shape. If the two nodes lie in one physical component, their contents $j-i$ are different (two addable nodes of a partition have different contents), and $z$ is transcendental over the base field, so $z^{j-i}\mathrm{\ne }{z}^{j^{\mathrm{\prime }}-i^{\mathrm{\prime }}}$. If they lie in different components $a\mathrm{\ne }b$, the values $U_a{z}^{j-i}$ and $U_b{z}^{j^{\mathrm{\prime }}-i^{\mathrm{\prime }}}$ are distinct in $K$, because $z,U_1,\dots ,U_{\mathrm{\ell }}$ are algebraically independent monomials and the components differ. Hence the joint word determines the tableau, different standard multitableaux -- even of different shapes -- have different joint words, and distinct shapes have disjoint sets of word characters.

Generic simplicity and split semisimplicity. By [F3] each $S_{\lambda }$ has a simultaneous triangular filtration whose successive characters are its own words of step 1.1. Hence every composition factor $D_{\mu }$ of the nonzero module $S_{\lambda }$ has a word occurring in $S_{\mu }$ and a word occurring in $S_{\lambda }$; disjointness of shape word sets forces $\mu =\lambda$. In particular $D_{\lambda }\mathrm{\ne }0$, and the cellular diagonal multiplicity $[S_{\lambda }:D_{\lambda }]=1$ of [F4] forces $S_{\lambda }=D_{\lambda }$ to be simple, with the distinct absolutely simple modules $S_{\lambda }$ as the only composition factors. The cellular basis gives ${\dim }_K{H}_K={\sum }_{\lambda }({\dim }_K{S}_{\lambda }{)}^2$. The semisimple quotient is a product of full matrix algebras on the exhaustive absolutely simple $S_{\lambda }$: to see this, the rank-one cell action on each simple gives every matrix unit, and the distinct simple labels give the distinct simple factors. Its dimension is the same sum, so the radical has dimension zero. Thus $H_K$ is split semisimple; no splitting of the cellular filtration was assumed.

Word separation for the Lyle–Mathas deformation. Fix $N\ge 1$, nonzero $q_H,Q_a$ in $F$, and set $z=q_{H}t$, $U_a=Q_a{t}^{Na}$ in $F(t)$. For two standard multitableaux that first differ at a step $r\le N$, the common prefix of size $r-1$ has components of sizes summing to $r-1$; the two added nodes lie in components $a,b$ and each added content $c=j-i$ is bounded in absolute value by the size of its own component, so $\mathrm{\mid }c-c^{\mathrm{\prime }}\mathrm{\mid }<r\le N$. A collision $U_a{z}^c=U_b{z}^{c^{\mathrm{\prime }}}$ with $a\mathrm{\ne }b$ would force the equality $Na+c=Nb+c^{\mathrm{\prime }}$ of $t$-adic exponents, i.e. the nonzero multiple $N(a-b)$ of $N$ would equal $c^{\mathrm{\prime }}-c$ with $\mathrm{\mid }{c}^{\mathrm{\prime }}-c\mathrm{\mid }<N$, impossible. If $a=b$, then $z$ is transcendental over $F$, so $z^c=z^{c^{\mathrm{\prime }}}$ forces $c=c^{\mathrm{\prime }}$. Thus the joint words separate all rank-$d$ standards in $F(t)$ as in step 1.1, and the argument of step 2.1 proves that the rank-$d$ Specht modules are simple and the corresponding Hecke algebra is split semisimple over $F(t)$, with no affine-center or block-classification input.

Nondegenerate Schur standards. By [F6], over any field with unit parameters, $[{\mathrm{\Delta }}_{\lambda }:L_{\mu }]=[S_{\lambda }:D_{\mu }]$ whenever $D_{\mu }\mathrm{\ne }0$, and every ${\mathrm{\Delta }}_{\lambda }$ is nonzero. Over $K$, step 2.1 makes every $S_{\lambda }$ simple; hence $D_{\mu }=0$ never occurs, $[{\mathrm{\Delta }}_{\lambda }:L_{\lambda }]=[S_{\lambda }:S_{\lambda }]=1$, and since all $L_{\mu }$ are distinct the multiplicity list forces ${\mathrm{\Delta }}_{\lambda }=L_{\lambda }$ to be simple; the diagonal cell-form entry of the standard of shape and type $\lambda$ is $1$, so the cellular form is nonzero and, being invariant on a simple module, is nondegenerate. The same argument applies over $F(t)$ to the rank-$d$ standards using step 3.1, and the construction of [F5] and the multiplicities of [F6] are identities over the respective ground fields, so the charged finite-$e$ specializations used by the Lyle–Mathas deformation are included. No valuation of a weight-Gram determinant is computed.