Graded Krull-Schmidt and shift-orbit bases for finitely generated KLR modules

(Central polynomial subalgebra and freeness.) Let $C_{\beta }$ be the color-symmetric polynomial subalgebra: on the summand $e(\nu )R(\beta )e(\nu )$ it is the polynomial ring in the elementary symmetric functions of the dots of each color separately. It is central: unequal crossings permute dot positions, and equal-color divided differences ${\mathrm{\partial }}_a$ commute with symmetric polynomials in $x_a,x_{a+1}$ by the Leibniz rule ${\mathrm{\partial }}_a(fg)={\mathrm{\partial }}_a(f)g+s_a(f){\mathrm{\partial }}_a(g)$ applied to $s_a$-invariant $f$. For a color occurring $m$ times, start with $A_0=k[E_1,\dots ,E_m]$ and $f_0(T)=T^m-E_1{T}^{m-1}+\cdots +(-1{)}^m{E}_m$. Successively put $A_r=A_{r-1}[X_r]\mathrm{/}(f_{r-1}(X_r))$ and $f_r(T)=f_{r-1}(T)\mathrm{/}(T-X_r)$ for $1\le r\le m$; the division has zero remainder in $A_r[T]$ and $f_r$ is monic of degree $m-r$. Monic division makes $A_r$ free of rank $m-r+1$ over $A_{r-1}$, with basis $1,X_r,\dots ,X_r^{m-r}$. In $A_m$ the identity $f_0(T)={\prod }_{r=1}^m(T-X_r)$ identifies $E_j$ with the elementary symmetric polynomial in the $X_r$. Thus $k[x_1,\dots ,x_m]\to A_m$, $x_r\mapsto X_r$, is surjective, and its inverse sends $X_r\mapsto x_r$ and $E_j$ to that elementary symmetric polynomial: each successive root relation holds under this substitution. Consequently $k[x_1,\dots ,x_m]$ is free of rank $m!$ over its symmetric coefficient subring, on the Artin monomials $x_1^{a_1}\cdots x_m^{a_m}$ with $0\le a_r\le m-r$; only the first extension, not the full polynomial ring, has rank $m$. Combining with the PBW basis [F1], whose dot monomials factor over the color blocks, gives that $R(\beta )$ is a finite free $C_{\beta }$-module, of rank $(\mathrm{\mid }\beta \mathrm{\mid }!{)}^2$ per weight counting the permutation basis; only centrality and this freeness are used below.

(Krull-Schmidt and splitting.) Every finitely generated graded module is bounded below and locally finite by [F1]. Choose finitely many homogeneous generators of degrees $d_j$; evaluation on the generators injects ${\mathrm{End}}^0(M)$ into the finite product of the finitely many finite-dimensional pieces $M_{d_j}$, so every degree-zero endomorphism algebra is finite-dimensional, and a nontrivial homogeneous idempotent splits $M$ into two nonzero summands with strictly smaller endomorphism algebras; splitting terminates. For an indecomposable $M$, every endomorphism $f$ satisfies a polynomial minimal equation because ${\mathrm{End}}^0(M)$ is finite-dimensional; factoring it as $T^{r}g(T)$ with $g(0)\mathrm{\ne }0$, Bezout gives a polynomial projector separating the nilpotent part from the invertible part, so indecomposability forces $f$ or $1-f$ to be invertible. Hence the nonunits form the unique maximal ideal, and the usual comparison of two decompositions $⨁p_i{i}_j{p}_j=\mathrm{i}\mathrm{d}$ produces a matching summand and cancellation, proving uniqueness in the degree-zero category; the shifts are recorded as part of each summand.

(Simples and covers.) For a graded simple $S$ and a positive-degree central $c\in C_{\beta }^{+}$, the kernel and image of $c$ are graded submodules; if $c$ acted nontrivially it would be injective and hence surjective, contradicting the least nonzero degree of the bounded-below module $S$; so $C_{\beta }^{+}S=0$ and $S$ is finite-dimensional because $\bar{R}=R(\beta )\mathrm{/}{C}_{\beta }^{+}R(\beta )$ is finite-dimensional by step 1.1. The finite-dimensional basis theorem [F3] applies to $\bar{R}$: it has finitely many graded simple shift orbits, and the same then holds for $R(\beta )$, since any simple $R(\beta )$-module is one. Projective covers are then lifted to $R(\beta )$: choose a finite shifted-free $R$-module $F$ whose reduction $\bar{F}$ surjects onto the projective cover $\bar{Q}\twoheadrightarrow \bar{S}$. That surjection splits by projectivity of $\bar{Q}$, giving an idempotent of ${\mathrm{End}}_{\bar{R}}^0(\bar{F})$; entrywise reduction ${\mathrm{End}}_R^0(F)\to {\mathrm{End}}_{\bar{R}}^0(\bar{F})$ is surjective; the reduction kernel in ${\mathrm{End}}_R^0(F)$ is nilpotent because its entries have positive central degree and a long product has degree exceeding every fixed difference of generator degrees minus the lower bound of $F$, so the projector onto $\bar{Q}$ is lifted by the finite nilpotent-ideal idempotent-lifting iteration $e\mapsto e-(2e-1)(e^2-e)$; the image $Q$ of the lifted projector is projective, its kernel is graded superfluous (after reduction by $C^{+}$ it is superfluous for $\bar{Q}$, and a bounded-below graded quotient $X=C^{+}X$ vanishes by its least nonzero degree), so $Q$ is an indecomposable projective cover, and every indecomposable projective is such a cover up to shift.

(Orbit bases.) Finite decompositions into indecomposables give that $K_0(R(\beta ))$ is free on the shift orbits of indecomposable finite graded projectives, with $q[P]=[P\{1\}]$, and finite length of finite-dimensional modules gives that $G_0(R(\beta ))$ is free on the simple shift orbits; both are $\mathbb{Z}[q,q^{-1}]$-modules in the sense of Graded Grothendieck groups, shift action, and Cartan map, the shift action is free because a nonzero finite-dimensional graded module is not isomorphic to a nonzero shift of itself (compare its least and greatest degrees), and the finite-dimensional theorem [F3] gives the same statement for the quotient categories. For a projective cover $P_b$ and simple $S_c$ with their selected shifts, $\mathrm{Hom}(P_b,S_c)=0$ unless $b=c$ at the selected shift, when it equals $\mathrm{End}(S_b)$ by the uniqueness of the cover, whose degree-zero endomorphism ring is a division $k$-algebra by Schur's argument; finite-dimensionality alone does not identify it with $k$; this is the diagonal-entry computation of the Hom pairing of Projective–module Hom pairing on class generators. The stronger statement that the diagonal scalar rings equal $k$ over every field, and the associated unscaled dual-basis conclusion, are not inferred here from finite-dimensional algebra alone and are proved separately later on this page; nothing above uses them.