Quiver full-flag incidence varieties and signed geometric KLR generators

Assume the Axiom of Choice (The Axiom of Choice) as inherited from the equivariant constructible, flag-variety and line-bundle suppliers below. Fix a finite loopless quiver $\mathrm{\Gamma }$ with vertex set $I$, with $h_{ij}\ge 0$ arrows $i\to j$ for $i\mathrm{\ne }j$ (parallel arrows and pairs of opposite arrows are allowed, and there are no loops), a dimension vector $\beta \in {\mathbb{N}}^I$, and a field $k$ of characteristic zero. Put $V={⨁}_i{V}_i$ with ${\dim }_{\mathbb{C}}{V}_i={\beta }_i$, $G={\prod }_i\mathrm{G}\mathrm{L}(V_i)$ and $E={⨁}_{h:i\to j}\mathrm{Hom}(V_i,V_j)$.

Incidence varieties. For a word $\nu =({\nu }_1,\dots ,{\nu }_m)$ with content $\beta$ (so $m={\sum }_i{\beta }_i$) let $F_{\nu }$ be the variety of increasing full $I$-graded complete flags $0=F^0\subset \cdots \subset F^m=V$ whose successive quotient $F^r\mathrm{/}{F}^{r-1}$ is concentrated in color ${\nu }_r$, and set $X_{\nu }=\{(x,F)\in E\times F_{\nu }:x_h(F_i^r)\subseteq F_j^r\text{ for all }h:i\to j\text{ and all }r\},{\pi }_{\nu }(x,F)=x\mathrm{.}$ Because $\mathrm{\Gamma }$ is loopless the stability condition is equivalent to $x_h(F_i^r)\subseteq F_j^{r-1}$ for all $r$, so on a split flag exactly the matrix entries $\mathrm{Hom}(D_a,D_b)$ with $a>b$ are allowed; accordingly $d_{\nu }={\sum }_i\frac{{\beta }_i({\beta }_i-1)}{2}+{\sum }_{a>b}{h}_{{\nu }_a{\nu }_b}={\dim }_{\mathbb{C}}{F}_{\nu }+\mathrm{rank}(X_{\nu }\to F_{\nu })={\dim }_{\mathbb{C}}{X}_{\nu },$ and $L_{\nu }={\pi }_{\nu !}k[d_{\nu }]\in D_{G,c}^b(E;k)$ lies in the equivariant constructible derived category of Equivariant constructible derived category of complex varieties. Define $Z_{\eta ,\nu }=X_{\eta }{\times }_E{X}_{\nu }$ with first coordinate the categorical source, and write $\mathfrak{A}={⨁}_{\eta ,\nu }{\mathrm{Ext}}_G(L_{\eta },L_{\nu })$ for the convolution algebra, so that $e_{\eta }\mathfrak{A}{e}_{\nu }={\mathrm{Ext}}_G(L_{\eta },L_{\nu })$; for homogeneous $a:L_{\eta }\to L_{\nu }[n]$ and $b:L_{\nu }\to L_{\rho }[m]$ the algebra product is the opposite-composition product $a\ast b=b[n]\circ a$, and $\mathrm{Ext}$ degree $n$ is the convolution component of BM degree $d_{\eta }+d_{\nu }-n$ (Equivariant Borel–Moore homology and oriented correspondence convolution).

Geometric generators. In the equivariant BM convolution algebra, with the standard positive complex orientation, define the idempotents $e_{\nu }=[{\mathrm{\Delta }}_{X_{\nu }}]$; the dots $x_r{e}_{\nu }=({\mathrm{\Delta }}_{X_{\nu }}{)}_{\ast }(x_r\cap [X_{\nu }])$ for $x_r=c_1^G(F^r\mathrm{/}{F}^{r-1})$ (First Chern class of tensor, dual, and conjugate lines); and, for the simple transposition $s_r$, the standard fundamental class $[C_{s_r\nu ,\nu }]$ of the closure of the relative-position-$s_r$ stratum in $Z_{s_r\nu ,\nu }$, together with ${\sigma }_r{e}_{\nu }=(-1{)}^{d_{s_r\nu }-{\dim }_{\mathbb{C}}{C}_{s_r\nu ,\nu }}[C_{s_r\nu ,\nu }]$ and ${\tau }_r{e}_{\nu }={\sigma }_r{e}_{\nu }$ if ${\nu }_r={\nu }_{r+1}$, while ${\tau }_r{e}_{\nu }=(-1{)}^{h_{{\nu }_r{\nu }_{r+1}}}{\sigma }_r{e}_{\nu }$ if ${\nu }_r\mathrm{\ne }{\nu }_{r+1}$. This is a generator normalization, not a change of the BM orientation: for distinct colors ${\sigma }_r=(-1{)}^{h_{ij}}[C{]}_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{d}\mathrm{a}\mathrm{r}\mathrm{d}}$ and ${\tau }_r=[C{]}_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{d}\mathrm{a}\mathrm{r}\mathrm{d}}$, while for equal colors ${\sigma }_r={\tau }_r=-[C{]}_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{d}\mathrm{a}\mathrm{r}\mathrm{d}}$.

The intrinsic signed presentation. Define $S_{\mathrm{\Gamma }}(\beta )$ as the free unital $k$-algebra on word idempotents $e_{\nu }$, dots $x_r$ and crossings ${\tau }_r$ ($1\le r<m$) modulo the relations: $e_{\nu }{e}_{{\nu }^{\mathrm{\prime }}}={\delta }_{\nu {\nu }^{\mathrm{\prime }}}{e}_{\nu }$ with ${\sum }_{\nu }{e}_{\nu }=1$; the dots commute and $x_r{e}_{\nu }=e_{\nu }{x}_r$; ${\tau }_r{e}_{\nu }=e_{s_r\nu }{\tau }_r$; ${\tau }_r^2{e}_{\nu }=Q_{{\nu }_r{\nu }_{r+1}}(x_r,x_{r+1})e_{\nu }$, where $a_{ii}=2$, $a_{ij}=-h_{ij}-h_{ji}$ for $i\mathrm{\ne }j$, $Q_{ii}=0$ and $Q_{ij}(u,v)=(-1{)}^{h_{ij}}(u-v{)}^{h_{ij}+h_{ji}}$; far crossings commute; the distinct-color dot slides $({\tau }_r{x}_t-x_{s_r(t)}{\tau }_r)e_{\nu }=0$ for every $1\le t\le m$ when ${\nu }_r\mathrm{\ne }{\nu }_{r+1}$ (in particular ${\tau }_r{x}_r{e}_{\nu }=x_{r+1}{\tau }_r{e}_{\nu }$ and ${\tau }_r{x}_{r+1}{e}_{\nu }=x_r{\tau }_r{e}_{\nu }$); the equal-color mixed relations $({\tau }_r{x}_r-x_{r+1}{\tau }_r)e_{\nu }=e_{\nu }$ and $({\tau }_r{x}_{r+1}-x_r{\tau }_r)e_{\nu }=-e_{\nu }$ when ${\nu }_r={\nu }_{r+1}$; other dots commute with ${\tau }_r$; and, for $1\le r<m-1$, the braid relation $({\tau }_{r+1}{\tau }_r{\tau }_{r+1}-{\tau }_r{\tau }_{r+1}{\tau }_r)e_{\nu }=0$ unless ${\nu }_r={\nu }_{r+2}$, in which case it equals $[Q_{{\nu }_r{\nu }_{r+1}}(x_r,x_{r+1})-Q_{{\nu }_r{\nu }_{r+1}}(x_{r+2},x_{r+1})]e_{\nu }\mathrm{/}(x_{r+2}-x_r)$. Grade $S_{\mathrm{\Gamma }}(\beta )$ by $\mathrm{deg}{e}_{\nu }=0$, $\mathrm{deg}{x}_r{e}_{\nu }=2$, $\mathrm{deg}{\tau }_r{e}_{\nu }=-a_{{\nu }_r{\nu }_{r+1}}$; every defining relation is homogeneous for this grading, including the polynomial braid quotient: in the outer-equal-color case the two distinct-color outer crossings contribute $2(h_{ij}+h_{ji})$ and the middle equal-color crossing contributes $-2$, matching the degree $2(h_{ij}+h_{ji}-1)$ of the correction.

Well-formedness. The smooth-flag incidence lemma Stable-flag incidence spaces admit the oriented proper correspondence calculus gives that $X_{\nu }\to F_{\nu }$ is a smooth complex algebraic vector bundle, that ${\pi }_{\nu }$ is projective and hence proper, and that the displayed formula computes ${\dim }_{\mathbb{C}}{X}_{\nu }$; $F_{\nu }={\prod }_i\mathrm{Flag}(V_i)$ is smooth projective (The variety of complete flags of a finite-dimensional vector space is smooth projective). The polynomial model $H_G^{\bullet }(X_{\nu };k)=k[x_1,\dots ,x_m]$, with shifted avatar $P_{\nu }=H_G^{\bullet }(X_{\nu };k)[d_{\nu }]$ in which the unit has degree $-d_{\nu }$, and the fact that the displayed geometric classes realize the stated graded relations are discharged by Relative-position incidence strata give a faithful geometric PBW basis ↗ and Quiver-flag convolution generators satisfy the signed KLR relations ↗; these two results are recorded in justified_by rather than in deps because both depend on this definition. The listed relations define a quotient of a free algebra, so $S_{\mathrm{\Gamma }}(\beta )$ is well posed as a graded algebra, and the geometric generator normalization introduces no further choice: the Axiom of Choice is consumed exactly through the AC-declared suppliers named above, while the quiver, word and sign data are finite. This specialized presentation is internal and does not replace the generic KLR definition.