Parabolic Kazhdan-Lusztig polynomials give the dual-canonical transition matrix

Coset representatives. Put $b=a(A)w_A$, the weakly increasing tuple. The permutations sorting $a(A)$ are exactly $w_{A}K$: indeed $a(A)w=b$ if and only if $b(w_A^{-1}w)=b$. Choosing $w_A$ shortest in this coset makes it minimal-right modulo $K$, and inversion makes $d(A)=w_A^{-1}$ minimal-left. The column Levi subgroup $J$ permutes positions within each column; it is generally different from $K$, which permutes the equal-entry positions of $b$.

Explicit reciprocal action and signed chart. The independent construction Finite type-A tensor Hecke bars and complementary coefficient charts gives, at $v=s^{-1}$, the actual action $M_a{H}_i=M_{a{s}_i}$ on an ascent, $s{M}_a$ on an equality, and $M_{a{s}_i}+(s-s^{-1})M_a$ on a descent. The cyclic sorted tuple is fixed by the source bar and has stabilizer character $s$, so this is the trivial-induced tensor module with its induced bar, including equal entries. Changing generators to $G_i=-H_i$ gives the algebra at $v=s$, the sign character $-s$, and standard vector $n_{d(A)}=(-1{)}^{\mathrm{\ell }(d(A))}{M}_A$. Thus the source-positive tensor fixed vector is the sign canonical vector multiplied by its leading sign. The helper's alternating-sum calculation and the inversion anti-involution of the Hecke algebra yield its coefficient $p_{A,B}(-s)$ and precisely the polynomial $p_{A,B}(s)$ stated. This resolves the bar sign by an explicit generator and standard-vector transport.

Ordinary-bar exterior embedding and complementary spherical chart. The same helper constructs the primitive exterior vector of a decreasing column as ${\sum }_C(-s{)}^{\mathrm{\ell }(B,C)}{M}_{a(C)}$. Its adjacent generator eigenvalue is $-s^{-1}$, quantum replacements preserve its coefficient-one normalization, and quasi-$R$ naturality identifies its ordinary bar with the wedge-factor bar. Product embeddings therefore have the displayed independent column sum, with leading coefficient one and positive-source corrections; triangular uniqueness gives $\iota (L_A)=L_{a(A)}$. Independently, the reverse-position chart has representative $c_A=w_{K}d(A)w_0$. The actual bilinear normalization is $(M_a,\psi M_b^{\ast })={\delta }_{ab}$; symmetry and complementary positive/negative polynomial congruences give $U^{\mathsf{T}}\bar{V}=I$. Hence the inverse tensor chart is $D_{a,b}={\bar{V}}_{b,a}=s^{\mathrm{\ell }(w_b)-\mathrm{\ell }(w_a)}{P}_{d(b)w_0,d(a)w_0}(s^{-2})$. Expanding the primitive exterior sum in this inverse chart proves the wedge inverse formula, including its signs and all repeated-entry indices.

Support and inverse matrix. Triangularity of the ordinary KL basis and the sign alternation give $p_{A,A}=1$ and $p_{A,B}=0$ unless $A\le B$; the complementary chart $d$ defined by $M_B={\sum }_A{d}_{A,B}{L}_A$ is the inverse unitriangular matrix through $p(-s)$, as in the definition of the Fock bases of [F3].

The mixed composition matrix. Step 1.3 gives $D_{a(A),a(C)}(1)=P_{d(C)w_0,d(A)w_0}(1)$ and $d_{A,B}(1)={\sum }_C(-1{)}^{\mathrm{\ell }(B,C)}{D}_{a(A),a(C)}(1)$. Apply the singular ordinary multiplicity theorem of [F3] first to the minimal-right labels $w_C,w_A$, where $d(C)=w_C^{-1}$ and $d(A)=w_A^{-1}$. Inverse-index symmetry gives $P_{w_C{w}_0,w_A{w}_0}=P_{w_{0}d(C),w_{0}d(A)}$. Conjugation by $w_0$ permutes the simple Hecke generators, commutes with bar and preserves Bruhat triangularity; uniqueness of the ordinary KL basis therefore gives $P_{w_{0}d(C),w_{0}d(A)}=P_{d(C)w_0,d(A)w_0}$ (The type-A Kazhdan–Lusztig basis has elementary triangular bar properties). Thus these are the same multiplicities for the sorted-entry stabilizer. Finite Levi Weyl characters and PBW give $\mathrm{c}\mathrm{h}M(B)={\sum }_C(-1{)}^{\mathrm{\ell }(B,C)}\mathrm{c}\mathrm{h}{M}_{\mathrm{o}\mathrm{r}\mathrm{d}}(C)$, with $C$ running over independent column permutations. Comparing coefficients of simple characters, which are linearly independent in the finite block, gives the same signed sum. Therefore $[M(B):L(A)]=d_{A,B}(1)$ with the displayed mixed parabolic matrix. The column subgroup and the equal-entry subgroup are used at separate stages, so repeated entries cause no conflation of sign and spherical families.

Conclusion. The parabolic KL polynomials give the dual-canonical transition matrix of the Fock module in the exact entry-stabilizer/column-spherical normalization stated, and the genuine parabolic composition matrix is the displayed signed sum of spherical ordinary numbers; regular and singular ambient cases are both handled locally without Koszul duality or a double-coset category equivalence.