Self-dual KLR projectives have positive first-block divided-string leading terms

(Self-dual simples.) The maximal-tail supplier [F1] gives a triangular divided-idempotent/simple Hom matrix with diagonal $1$ over $A=\mathbb{Z}[q,q^{-1}]$, which remains invertible at $q=1$. Ordinary dimensions of divided-word corners are determined by ordinary word dimensions: in each constant-colour block the nil-Hecke matrix presentation makes the polynomial column simple of rank $r!$ and the divided idempotent a rank-one matrix projection, so $\dim (e_{\text{div}}M)=\dim (e_{\text{expanded}}M)\mathrm{/}{\prod }_{r}r!$ for every finite module $M$, including nonsemisimple central quotients. Hence ordinary word characters separate simple shift orbits. The vector duality $D(M{)}_d={\mathrm{Hom}}_k(M_{-d},k)$ preserves simple modules and fixes ordinary word dimensions because $\psi$ fixes the word idempotents, so $DS\cong S\{s\}$ for some integer $s$.

(Evenness and normalization.) Choose a total colour order and set $p(\nu )={\sum }_{a<b,{\nu }_a>{\nu }_b}{b}_{{\nu }_a{\nu }_b}<mrow><mi\; mathvariant="normal">m</mi><mi\; mathvariant="normal">o</mi><mi\; mathvariant="normal">d</mi></mrow>2$. Each adjacent unequal swap changes $p$ by its crossing degree modulo $2$, and equal crossings and dots have even degree, so the PBW basis puts $e(\eta )Re(\nu )$ in degrees congruent to $p(\eta )-p(\nu )$; a graded module therefore splits into the two submodules on which $d-p(\nu )$ is fixed modulo $2$ on $e(\nu )M_d$, and a simple uses one of them. Since $-d-p(\nu )\equiv d-p(\nu )(\mathrm{m}\mathrm{o}\mathrm{d}2)$, $DS$ has the same sector as $S$, so the shift $s$ above is even; replacing $S$ by $S\{s\mathrm{/}2\}$ gives $DS=S$, and uniqueness follows because a nonzero finite-dimensional graded module is not isomorphic to a nonzero shift of itself (compare its finite least and greatest degrees).

(Self-dual covers and induction.) For a finite projective $P$ and finite module $M$, evaluation gives $\mathrm{Hom}(P^{\mathrm{\#}},DM)=D\mathrm{Hom}(P,M)$ with ordinary homogeneous degrees: verify on $P=Re(\nu )\{a\}$, where both sides are the dual of the finite-dimensional space $e(\nu )M\{-a\}$ (Nil-Hecke idempotents and divided-power KLR projectives), and restrict to a finite summand. The duality ${}^{\mathrm{\#}}$ is an involution on finite projectives, so $P_b^{\mathrm{\#}}$ is the same normalized indecomposable projective cover of $D{S}_b=S_b$, giving $P_b^{\mathrm{\#}}=P_b$; a nonzero projective is bounded below, so the normalization is unique. If $M$ is a finite projective over the separated block algebra $B$, then ${\mathrm{Hom}}_R(Re{\otimes }_{B}M,R)={\mathrm{Hom}}_B(M,eR)={\mathrm{Hom}}_B(M,B){\otimes }_{B}eR$ by evaluation on finite free modules and summands, and twisting by $\psi$ identifies this with $Re{\otimes }_B{M}^{\mathrm{\#}}$ without reversing the horizontal block order; for $M=P_a⊠P_{i,r}$ this proves that $\mathrm{I}\mathrm{n}\mathrm{d}M$ is self-dual. Restriction commutes with $D$ because $\psi$ fixes the defining block idempotent, so the maximal FIRST-block restriction of the normalized $S_b$ is exactly the normalized $L(i^r)⊠S_a$ with no residual shift: that tensor is $D$-self-dual and any extra shift would contradict finite support.

(Leading coefficient $1$.) Use the exact tensor/Res adjunction of Generic KLR induction and restriction adjunctions preserve projectives and the maximal-tail lemma [F1] to compute $\mathrm{Hom}(\mathrm{I}\mathrm{n}\mathrm{d}(P_{i,r}⊠P_a),S_c)$: it vanishes if ${ϵ}_i^L(S_c)<r$; if ${ϵ}_i^L(S_c)=r$, the maximal restriction is $L(i^r)⊠S_{a^{\mathrm{\prime }}}$ and the Hom is zero unless $a^{\mathrm{\prime }}=a$, when it is exactly $k$ in degree $0$; the unique-head lemma makes $c=b$ in that case. Since Hom from a projective detects the multiplicity polynomial of each summand in the decomposition, the induced projective has exactly $[P_b]$ with coefficient $1$ at tail $r$ and all other summands at strictly higher tail, giving the displayed expansion. Self-duality of the induction and of each normalized projective forces every multiplicity polynomial to be bar invariant, and finite decompositions give nonnegative coefficients.

(Integral string ideals.) For every $n$, induction with a divided block of length $r\ge n$ has no summands of tail $<n$ by the same Hom computation of step 4.1, so $\gamma (I_{(i,\ge n)})$ is contained in the span of the corresponding covers. Conversely descend on the finite possible tail $r$ (bounded by ${\beta }_i$): the coefficient-$1$ expansion expresses every $[P_b]$ of tail $r$ as $[P_{i,r}][P_a]$ minus classes of strictly higher tail, which lie in $I_{(i,\ge r)}$ already; this proves the reverse inclusion integrally, using the sum of all divided images and not a Gaussian-unit identification with a single ordinary power ideal.

(Signs.) Suppose $[P_b]=\gamma ({\mathrm{s}\mathrm{i}\mathrm{g}\mathrm{n}}_{b}G(b^{\mathrm{\prime }}))$ has been established by a separate norm/geometric argument. The ideal equality of step 5.1 and the adapted LEFT-string lower-global basis [F4] identify the tail partitions of these signed vectors with the LEFT string partitions; at the exact LEFT-string quotient the lower-global recursion has coefficient $+1$ for $F_i^{(r)}G(a^{\mathrm{\prime }})$ when $a^{\mathrm{\prime }}$ has LEFT string $0$. Induction on root height fixes ${\mathrm{s}\mathrm{i}\mathrm{g}\mathrm{n}}_a=+1$, and comparing the projective coefficient-$1$ expansion with this quotient fixes ${\mathrm{s}\mathrm{i}\mathrm{g}\mathrm{n}}_b=+1$; the unit/vacuum starts the induction. The reversal supplier used elsewhere promises only signed star preservation, so an unsigned right recursion is deliberately not inferred here. The Axiom of Choice enters only through the inherited quantum-half/global suppliers [F4] and The KLR category categorifies the negative quantum half (The Axiom of Choice); no additional simultaneous module-basis choice is used, and the finite duality, parity, Frobenius and first-block multiplicity arguments above are choice-free.