Degenerate Specht decomposition numbers are canonical basis coefficients at one

The two charts are mutually inverse. By The dual-canonical, canonical and quasi-canonical Fock bases the vectors $L_B$ and $M_A$ are two $\mathbb{Z}[s,s^{-1}]$-bases of the finite weight spaces of the Fock module, so the transition matrices $R(s)=(p_{A,B}(-s))$ with $L_B={\sum }_A{R}_{A,B}(s)M_A$ and $D(s)=(d_{A,B})$ with $M_B={\sum }_A{d}_{A,B}(s)L_A$ satisfy $R(s)D(s)=D(s)R(s)=1$; both are triangular with diagonal entries $1$ on each weight block, and the quasi-canonical vector is $P_A={\sum }_B{d}_{A,B}(s)M_B$. The polynomial entries and the chart conventions are those of The dual-canonical, canonical and quasi-canonical Fock bases.

The type-$A$ polynomial dictionary. The explicit transition coefficients are given by the type-$A$ formulas of Parabolic Kazhdan-Lusztig polynomials give the dual-canonical transition matrix: for the right place action, $w_A$ is the shortest sorting element on the RIGHT of the entry stabilizer $K$, $d(A)=w_A^{-1}$ is shortest on the LEFT, $p_{A,B}(s)=s^{\mathrm{\ell }(w_B)-\mathrm{\ell }(w_A)}{\sum }_{z\in K}(-1{)}^{\mathrm{\ell }(z)}{P}_{w_{A}z,w_B}(s^{-2})$, and the mixed singular/parabolic composition coefficient is the finite Levi cancellation sum ${\sum }_C(-1{)}^{\mathrm{\ell }(B,C)}{P}_{d(C)w_0,d(A)w_0}(1)$. Here $J$ permutes positions within columns and $K$ permutes equal entries of the sorted tuple, so their roles differ even when the subgroups coincide: for two height-one columns with entries $0,1$, both are trivial.

The O-side multiplicity dictionary. On the category-O side the standard module $N(B)$ has a composition series whose multiplicities with respect to the simples $L(A)$ are read off in the same chart: the coefficient of $M_B$ in the expansion of $P_A$ is the multiplicity of $L(A)$ in $N(B)$ at parameter one, in the source's reciprocal convention, by Parabolic Kazhdan-Lusztig polynomials give the dual-canonical transition matrix and The highest weight Fock submodule carries the quantum-group global bases; combined with step 1.1 this gives $[N(B):L(A)]=d_{A,B}(1)$ for all $A,B$, with $d_{A,B}$ unitriangular, and $[N(B):L(B)]=1$ for every $B\in {\mathrm{C}\mathrm{o}\mathrm{l}}^{\mathrm{\Lambda }}$ in the full square matrix.

Transfer to the Hecke side. By [F2] the Schur quotient is exact, sends $N(B)$ to $S(B)$, $L(A)$ to $D(A)$ for standard $A$ and kills the off-standard simples; applying it to a composition series of $N(B)$ and using [F1] gives $[S(B):D(A)]=[N(B):L(A)]=d_{A,B}(1)$ for $A\in {\mathrm{S}\mathrm{t}\mathrm{d}}^{\mathrm{\Lambda }}$ and every $B\in {\mathrm{C}\mathrm{o}\mathrm{l}}^{\mathrm{\Lambda }}$, the sum being finite on each weight block. This proves (i).

Shape of the matrix. The full matrix $D=(d_{A,B})$ is unitriangular on ${\mathrm{C}\mathrm{o}\mathrm{l}}^{\mathrm{\Lambda }}$ by step 1.1; restricting rows to the simple labels $A\in {\mathrm{S}\mathrm{t}\mathrm{d}}^{\mathrm{\Lambda }}$ (all of which occur, and which index the simples by The degenerate Ariki categorification theorem) leaves a rectangular matrix, whose square submatrix on the standard columns is again unitriangular with diagonal entries $1$. The supported weight blocks and labels agree under the actual arbitrary-$I$ cut maps of the $G_0$ theorem, so this same coefficient calculation applies to every admitted consecutive interval. Off standard columns the equation $D(B)=0$ says precisely that $D(B)$ is not a simple module label, so the identity $[S(B):D(B)]=1$ is asserted only for $B\in {\mathrm{S}\mathrm{t}\mathrm{d}}^{\mathrm{\Lambda }}$. This proves (ii). The statement is not that the rectangular matrix is invertible, and no off-standard simple is created.

The polynomial coefficient is not a transposed inverse. In the coefficient convention used in the proof, the decomposition array has rows the simple labels and columns the Specht labels, and its entries are the coefficients of $D(s)$ at $s=1$; since $R(s)=D(s{)}^{-1}$, its transpose is $D(s{)}^{-t}$, which is in general different from the $L$-in-$M$ chart $R(s)$. On the two-factor space with labels $01<10$ one has $L_{10}=M_{10}-s{M}_{01}$, so $R_{01,10}=-s$ is the negative-entry chart, whereas its inverse has $D_{01,10}=s$, while the true decomposition entry is $d_{01,10}(1)=1$: $S(10)$ is the simple $D(01)$ and not a formal negative combination. Hence with simple rows and Specht columns the decomposition matrix is the standard-row restriction of $D$; with Specht rows and simple columns it is its transpose. Neither convention uses the transpose of $R$. This completes (iii) together with step 1.2, which keeps the roles of the entry stabilizer $K$ and column Levi $J$ separate and gives the finite Levi cancellation values at parameter one; all statements are ungraded.