Separated Hecke residues give seminormal idempotents and bases

Reconstruct a tableau from its residue tuple. At a prefix of size $k-1$, two different addable corners have integer content difference of absolute value at most $k$. Since $[a{]}_q-[b{]}_q=q^b[a-b{]}_q$ and $e>n$, their q-contents differ. Starting with the empty diagram, the tuple thus recovers each added node and the final shape. For two tableaux of a fixed shape with identical entries' residues outside $i,i+1$, first reconstruct the common prefix through $i-1$, and reconstruct the common suffix by successively deleting the unique appropriate removable corner from the fixed final shape. The intervening skew diagram has two boxes; its only possible orders are the original order and its swap. This proves both separation assertions.

Diagonalize the commuting JM operators on a cell. Their triangular basis gives a filtration with one-dimensional joint characters indexed by standard tableaux. Those characters are distinct by step 1.1. For commuting triangular operators, the simultaneous generalized eigenspaces split along their distinct joint characters: the corresponding maximal ideals of the polynomial algebra are pairwise comaximal. Each joint character occurs once in this cell, so each generalized eigenspace is one-dimensional and hence an actual joint eigenline. This finite Chinese-remainder argument works over any field and needs no choice of a generic scalar or algebraic closure. The Lagrange product of the Statement is $1$ on its target tuple and $0$ on every other tuple. Applied to $m_t$, its triangular diagonal coefficient is $1$, and the other possible terms are strictly more dominant. Thus it gives the asserted $f_t$.

Compute the diagonal coefficient using the correctly ordered JM relation $T_i{L}_i=L_{i+1}{T}_i-(q-1)L_{i+1}-1.$ For $k\mathrm{\ne }i,i+1$, $T_i$ commutes with $L_k$. Consequently separation restricts $f_s{T}_i$ to the span of $f_s$ and $f_t$ if the swap is standard, or to $f_s$ alone otherwise. Write its coefficient of $f_s$ as $c_s$. The displayed relation gives $c_s([{\mathrm{content}}_s(i){]}_q-[{\mathrm{content}}_s(i+1){]}_q)=-(q-1)[{\mathrm{content}}_s(i+1){]}_q-1.$ Its denominator is $q^{{\mathrm{content}}_s(i+1)}[\rho {]}_q$, while the numerator is $-q^{{\mathrm{content}}_s(i+1)}$, so $c_s=-1\mathrm{/}[\rho {]}_q$. This does not claim that $T_i$ commutes with $L_i$.

At this stage step 2.1 establishes the projector identities on each individual cell module. The global identities in $H$ will follow below after proving that the sum of the cell modules is faithful. No finite-$e$ classification is a premise.

The transition from $m_t$ to $f_t$ is unitriangular, so these vectors form a basis. For $s\mathrm{\ne }t$, choose $k$ at which their q-contents differ. Self-adjointness $L_k^{\ast }=L_k$ and associativity give $([{\mathrm{content}}_s(k){]}_q-[{\mathrm{content}}_t(k){]}_q)⟨f_s,f_t⟩=0.$ The scalar is nonzero, proving orthogonality. The Lagrange action also gives $f_t{F}_s={\delta }_{ts}{f}_t$.

If the swap is not standard, the consecutive entries lie adjacently in one row or one column. Then $\rho =-1$ or $\rho =1$, respectively, and step 2.2 gives $c_s=q$ or $c_s=-1$. These are the stated row and column cases; the row action is $q$, not the scalar $1$ at a general parameter.

Suppose the swap is standard. In the branch $s⊳t$, the length-additive Murphy action has leading term $m_s{T}_i=m_t$; the unitriangular eigenbasis change then gives the off-diagonal coefficient ${\alpha }_i(s)=1$. Higher-dominance terms cannot alter the coefficient at $t$ in this two-dimensional eigenline span. This is the leading-term argument in Mathas's source theorem on the seminormal basis. In both branches the quadratic relation gives ${\alpha }_i(s){\alpha }_i(t)=q+(q-1)c_s-c_s^2=\frac{[1+\rho {]}_q[1-\rho {]}_q}{[\rho {]}_q[-\rho {]}_q}\mathrm{.}$ The last equality follows directly from $[-\rho {]}_q=-q^{-\rho }[\rho {]}_q$ and the quantum-integer addition identities. Thus the opposite branch has exactly the displayed coefficient. This proves the generator formula without accepting the inconsistent branch in the 1998 manuscript.

Prove nondegeneracy in the separated range locally. The canonical Murphy basis vector has form value $[\lambda ]{!}_q={\prod }_a[{\lambda }_a]{!}_q$, because $x_{\lambda }^2=[\lambda ]{!}_q{x}_{\lambda }$; its seminormal image has the same value since the canonical tableau is dominance-maximal. This scalar is nonzero when $e>n$. If $s⊳t=s(i,i+1)$ is a standard swap, put $a=1\mathrm{/}[\rho {]}_q$. Step 4.1 gives $f_t=f_s(T_i+a)$. Associativity, orthogonality and the quadratic relation give $⟨f_t,f_t⟩=(q+a)(1-a)⟨f_s,f_s⟩=q[\rho +1{]}_q[\rho -1{]}_q[\rho {]}_q^{-2}⟨f_s,f_s⟩$. Indeed expansion of $(T_i+a{)}^2$ and its $f_s$ coefficient $-a$ gives $q-a(q-1)-a^2=(q+a)(1-a)$. The swapped incomparable boxes have $2\le \mathrm{\mid }\rho \mathrm{\mid }\le n-1$: their row and column separations plus one are at most the size of the diagram. Thus all three quantum integers here are nonzero. Every standard tableau is connected to the canonical one by length-increasing swaps of adjacent incomparable entries: bubble-sort its linear extension of the box-order poset into the canonical row-order extension, then reverse the swaps. Induction gives nonzero diagonal norm at every $f_t$, so each cell form is nonsingular.

Each cell is now its nonzero simple form quotient, absolutely simple, and all these cells are pairwise distinct and exhaust the simple modules by Nonzero cell heads classify the simple modules of a cellular algebra. Thus the split semisimple quotient $H\mathrm{/}J(H)$ has dimension ${\sum }_{\lambda }\mathrm{\mid }\mathrm{Std}(\lambda ){\mathrm{\mid }}^2$. This equals $\dim H$ because the Murphy cellular basis has exactly one element for each pair of standard tableaux of the same shape (The Murphy tableau elements form a cellular basis). Hence $J(H)=0$, $H$ is split semisimple, and the direct sum of all its simple cell modules is faithful. On that faithful module the global Lagrange products of the Statement are the orthogonal projectors from step 2.1, with different shapes having disjoint tuples by step 1.1. Therefore $F_s{F}_t={\delta }_{st}{F}_t$ holds in $H$ itself, proving the global idempotents. This argument proves only the separated range and does not assert a finite-$e$ nonzero-label criterion when $e\le n$.

Steps 1.1-6.1 prove all four claims. No semisimplicity was imposed as a hypothesis: it was derived locally from nondegenerate cells and their complete finite-cell classification. The calculation concerns the intrinsic Murphy cell and keeps the signed ordinary-module dictionary separate. All products and basis changes are finite, and no choice principle is used.