Maximal KLR tails give a unique induction head and absolute simple modules

(Nil-Hecke tail and its matrix units.) The constant-colour block $N{H}_r=R(r{\alpha }_i)$ acts on its polynomial column over $C_r=k[x_1,\dots ,x_r{]}^{S_r}$. The Artin monomial basis and perfect symmetric-variable-linear trace of Nil-Hecke idempotents and divided-power KLR projectives give integral matrix units and identify the full algebra with graded endomorphisms of that free rank-$r!$ column, hence with matrices over $C_r$ carrying the Artin degree shifts. Reducing the positive-degree central elementary symmetric functions to zero gives the graded matrix quotient over $k$; its column simple $L(i^r)$ is absolutely simple, and the divided idempotent cuts out a one-dimensional image. Its full-algebra polynomial-column graded projective cover has degree-zero endomorphism ring $k$, although its entire endomorphism ring is $C_r$. If $T$ is a finite-dimensional graded simple over $R(\mu ){\otimes }_{k}N{H}_r$, a positive-degree central operator has graded kernel and image. A nonzero such operator would therefore be invertible by graded simplicity, which is impossible on the finite nonzero degree support of $T$. Thus the positive tail center annihilates $T$, and it factors through $R(\mu ){\otimes }_k{\mathrm{Mat}}_{r!}(k)$ with the indicated shifts. Matrix units identify $T$ with $N⊠L(i^r)$ for a graded simple $R(\mu )$-module $N$, unique up to shift. This uses the augmentation matrix quotient for finite simples, not a false identification of the full nil-Hecke algebra with matrices over the ground field.

(Maximal restriction.) Let $S$ be a finite-dimensional graded simple with $r={ϵ}_i(S)>0$ maximal. The finite-dimensional module ${\mathrm{\Delta }}_i^{r}S$ is nonzero by the definition of ${ϵ}_i$ and has a simple submodule; by step 1.1 it is of the form $N⊠L(i^r)$ for a graded simple $R(\beta -r{\alpha }_i)$-module $N$. If $N$ carried a word ending in $i$, then concatenating any nonzero nil-Hecke tail word would produce a nonzero vector in $S$ of tail $r+1$, contradicting maximality, so ${ϵ}_i(N)=0$. By the left adjunction of [F2] the inclusion $N⊠L(i^r)\hookrightarrow {\mathrm{\Delta }}_i^{r}S$ corresponds to a nonzero map $M:=\mathrm{I}\mathrm{n}\mathrm{d}(N⊠L(i^r))\to S$, which is surjective since $S$ is simple. In the Mackey filtration of ${\mathrm{\Delta }}_i^{r}M$ every shuffle bringing $t>0$ colours $i$ from the first block into the final tail contains a factor ${\mathrm{\Delta }}_i^{t}N$ and is zero; the only surviving filtration quotient is $N⊠L(i^r)$ with zero crossing shift, so ${\mathrm{\Delta }}_i^{r}M=N⊠L(i^r)$ exactly. This is a one-surviving-quotient computation, not a semisimplicity assumption.

(Unique head and simplicity.) Every nonzero quotient $Q$ of $M$ has nonzero defining corner: if $eQ=0$ with $e=e(\beta -r{\alpha }_i,r{\alpha }_i)$, then the composite $Ae{\otimes }_B(N⊠L(i^r))\to M\to Q$ is zero by the tensor/Res adjunction, hence $Q=0$. The finite graded semisimple head of $M$ therefore has exactly one simple summand, occurring once, because its exact $e$-restriction is a quotient of the simple $eM=N⊠L(i^r)$; this head is $S$, and $eM\to eS$ is an isomorphism, so exactness gives $e\ker (M\to S)=0$ and every other composition factor of $M$ has smaller tail. Underlying and absolute simplicity of $S$: induct on the height of $\beta$. For positive height choose $i$ with $r>0$; by step 2.1 and induction $N$ and $L(i^r)$ are absolutely simple, so $eS=N⊠L(i^r)$ has endomorphism ring $k$; if $U\le S$ is any ungraded submodule with $eU\mathrm{\ne }0$, then $eU=eS$ and $U=S$ because $S$ is generated by $eS$; if $eU=0$ and $U\mathrm{\ne }0$, filter $S$ by $F_{d}S={⨁}_{j\le d}{S}_j$ and $U$ by $F_{d}U=U\cap F_{d}S$. Every homogeneous algebra element of degree $a$ sends $F_{d}U$ into $F_{d+a}U$, so $\mathrm{gr}U$ embeds as a nonzero graded submodule of $\mathrm{gr}S=S$. Since $e$ has degree $0$ and annihilates $U$, it annihilates $\mathrm{gr}U$; graded simplicity gives $\mathrm{gr}U=S$, contradicting $eS\mathrm{\ne }0$. Thus $S$ is simple as an ungraded module, and every $R$-endomorphism of $S$ is determined on $eS$, giving ${\mathrm{End}}_R(S)=k$. The finite-density matrix argument then shows absolute simplicity over every field: any $m$ linearly independent vectors can be sent to arbitrary targets by one algebra element, by induction on $m$ and the commutant $k$ clause, so the action realizes all $k$-linear matrices and remains simple after field extension.

(Triangular divided-string basis.) Choose a cyclic order of the finite colour support and recursively strip maximal $i$-tails, producing for each simple shift orbit a finite tuple $Y_b$ of ordered divided powers; two simples with the same tuple agree by the unique-head recursion of step 3.1. Form $P(Y_b)$ from the reversed ordered normalized divided blocks. Its homogeneous Hom to $S_c$ is the corresponding product of divided idempotents acting on $S_c$; comparing tuples lexicographically, if $Y_b>Y_c$ at the first differing entry then the required tail exceeds the maximal tail of the successively stripped simple $S_c$, and the corner is zero. When the tuples agree, every maximal restriction is the simple tensor computed in step 2.1 and each nil-Hecke divided idempotent reduces the tail to one dimension, so by induction the diagonal Hom is a single Laurent monomial with coefficient $1$. Shifting each $S_b$ makes the diagonal $1$. The finite matrix is triangular over $A=\mathbb{Z}[q,q^{-1}]$ with diagonal $1$; Krull-Schmidt and the projective-cover/simple Hom pairing (now with $\mathrm{End}(S)=k$ from step 3.1) identify it with the matrix expressing the projective classes in the cover basis, and successive integral triangular elimination proves that the selected classes $[P(Y_b)]$ are an $A$-basis of $K_0(R(\beta ))$; the whole family of divided words spans because the recursion covers every simple, while redundancy of non-selected words is allowed. This completes the integral surjectivity argument without a PBW rank or flat-family shortcut.

(First-color version.) For maximal FIRST-color blocks use the defining idempotent $e(r{\alpha }_i,\mu )$ and $M=\mathrm{I}\mathrm{n}\mathrm{d}(L(i^r)⊠N)$ with ${ϵ}_i^L(N)=0$; a simple submodule of the maximal first-block restriction ${\mathrm{\Delta }}_i^{(L,r)}S$ has the form $L(i^r)⊠N$ by the same nil-Hecke matrix units of step 1.1, and any first-colour support of $N$ would extend the maximal first block, so it vanishes. In the Mackey restriction of ${\mathrm{\Delta }}_i^{(L,r)}M$ every nontrivial shuffle bringing a positive number $t$ of $i$-colours from $N$ into the first block contains ${\mathrm{\Delta }}_i^{(L,t)}N$ and is zero, so the sole uncrossed quotient is $L(i^r)⊠N$ with shift $0$; the defining-corner generation and exact-head argument of steps 3.1 and 3.1 are unchanged. Thus the first-block assertions are proved directly, with no unsigned star-crystal preservation assumed.