The PBW (monomial) basis theorem for KLR algebras

(Dots move right.) Move the word idempotents to the right using relations (2)–(3). On each resulting component, if $x_b$ stands immediately left of ${\tau }_a$, relation (6) gives $x_b{\tau }_{a}e(\nu )={\tau }_a{x}_{s_a(b)}e(\nu )$ except for adjacent equal colours. In that case the exact rules are $x_a{\tau }_{a}e(\nu )={\tau }_a{x}_{a+1}e(\nu )+e(\nu )$ and $x_{a+1}{\tau }_{a}e(\nu )={\tau }_a{x}_{a}e(\nu )-e(\nu )$. The leading term moves one dot past one crossing, and each correction removes that dot and crossing; hence every term has fewer dot/crossing inversions. Induction on their number puts all dots to the right of all crossings.

(Crossing words reduce.) Induct on crossing length. By [F3], a nonreduced transposition word can be changed by finitely many far and braid moves into a word with an adjacent repeat; each braid correction has three fewer crossings and is handled by induction after step 1.1, and relation (4) removes the repeated pair. For a reduced word, [F3] connects it to the chosen reduced word by finitely many braid moves; their correction terms again have fewer crossings. Thus every crossing word is a sum of chosen ${\tau }_w$ times dot polynomials, and step 1.1 puts the dots on the right. This proves spanning without a decreasing-length claim for braid moves themselves.

(Independence.) Under the representation of [F1], the image of ${\tau }_w{x}^{a}e(\nu )$ in the localization at the nonzero $P_{ij}(x_a,x_b)$ and $x_a-x_b$ is triangular: expanding the operators of the chosen reduced word, its highest term is a nonzero scalar times the permutation $w$ acting on the summand, and all other terms are permutations of strictly smaller length (an unequal crossing contributes its nonzero $P$-factor together with the transposition, an equal crossing contributes $(1-s_a)\mathrm{/}(x_a-x_{a+1})=-(s_a-1)\mathrm{/}(x_a-x_{a+1})$, again transposition plus lower-length terms). A linear relation $\sum c_{w,a,\nu }{\tau }_w{x}^{a}e(\nu )=0$ in $R_n(Q)$ would therefore give, after applying $\rho$ and looking at a source/destination block and the maximal occurring length, a nontrivial linear relation ${\sum }_w{d}_{w}w=0$ among distinct permutations $w$ of the field $k(x_1,\dots ,x_n)$. Distinct field automorphisms are linearly independent: in a shortest nonzero relation choose two different automorphisms $w,w_0$ and $g$ with $w(g)\mathrm{\ne }{w}_0(g)$. Evaluation on $fg$ minus $w_0(g)$ times evaluation on $f$ removes one term and leaves the $w$ term nonzero, contradicting minimality; a one-term relation is impossible by evaluation on $1$. Hence the spanning monomials of step 2.1 have linearly independent images, and since $\rho$ is injective by [F1] applied to the fixed factorization, the monomials themselves are linearly independent; together with step 2.1 they form a basis.

(Grading bookkeeping.) The bidegree of the basis monomial ${\tau }_w{x}_1^{a_1}\cdots x_n^{a_n}e(\nu )$ is $(\nu ,w\nu )$ together with the integer $\mathrm{deg}({\tau }_w\text{ on }\nu )+{\sum }_j{a}_j\cdot 2d_{{\nu }_j}$, where $\mathrm{deg}({\tau }_w\text{ on }\nu )$ is the sum of the degrees $-b_{{\nu }_r{\nu }_{r+1}}$ of its crossings read in the successive permuted sequences; this is independent of the parameters $Q$ and of the factorization. The summand $e(\nu )R_n(Q)e({\nu }^{\mathrm{\prime }})$ therefore has the Laurent Hilbert series ${\sum }_{w:w{\nu }^{\mathrm{\prime }}=\nu }{q}^{\mathrm{deg}({\tau }_w\text{ on }{\nu }^{\mathrm{\prime }})}{\prod }_{j=1}^n(1-q^{2d_{{\nu }_j^{\mathrm{\prime }}}}{)}^{-1}$, a finite sum of rational geometric series bounded below (each $d_{{\nu }_j}>0$ and the finitely many crossing degrees are bounded below), which proves the displayed graded-dimension claim and shows that $R(\beta {)}_m=0$ for $m$ below the finite minimum of the finitely many crossing contributions over the finitely many $\nu ,w$ with $w\nu ={\nu }^{\mathrm{\prime }}$ and ${\sum }_j{\alpha }_{{\nu }_j}=\beta$.

(Local finiteness.) Fix $\beta \in Q_{+}$ and a degree $m$. The basis monomials of $R(\beta {)}_m$ involve only the finitely many sequences $\nu \in I^{\beta }$ and the finitely many $w\in S_{\mathrm{\mid }\beta \mathrm{\mid }}$; each crossing degree is one of the finitely many integers $-b_{pq}$ and each dot degree $2d_{{\nu }_j}$ is at least $2{\min }_i{d}_i>0$, so ${\sum }_j{a}_j$ is bounded, leaving finitely many exponent tuples; hence ${\dim }_{k}R(\beta {)}_m<\mathrm{\infty }$. If $I$ is finite, $I^{\beta }$ is finite for every $\mathrm{\mid }\beta \mathrm{\mid }$, so the same argument applies to $R_n(Q)$ degree by degree. The element $x_1{\tau }_1\in R(2{\alpha }_i{)}_0$ is nonzero by the basis of step 3.1 and squares to itself by the nil-Hecke dot move, so degree zero need not be one-dimensional.