Signed Quiver Flag Convolution And Klr Ext Realization

For $a:L_{\eta }\to L_{\nu }[n]$ and $f:L_{\nu }\to {\omega }_E^G[m]$, set $a\cdot f=f[n]\circ a$. Associativity of composition makes ${⨁}_{\nu }{\mathrm{Ext}}_G(L_{\nu },{\omega }_E^G)$ a left module for the opposite-composition product. Adjunction and dualizing transitivity [F2] give ${\mathrm{Ext}}_G^n(L_{\nu },{\omega }_E^G)=H_{d_{\nu }-n}^{\mathrm{B}\mathrm{M},G}(X_{\nu };k)=H_G^{d_{\nu }+n}(X_{\nu };k)$, giving the stated shift; the diagonal and Chern cap classes act by identity and cup product. Fiber scaling contracts the vector bundle $X_{\nu }$ equivariantly onto $F_{\nu }=G\mathrm{/}{B}_{\nu }$. Here $B_{\nu }$ is the product of triangular stabilizers, and $B_{\nu }\mathrm{/}T$ is its unitriangular group, homeomorphic to a complex affine space through its matrix entries. The Borel maps $EG\mathrm{/}T\to EG\mathrm{/}{B}_{\nu }$ have that contractible fiber, so pullback identifies $H_G(F_{\nu };k)$ with $H_T(\mathrm{p}\mathrm{t};k)$. This comparison can be made on the finite resolutions [F10], whose fiber cohomology is $k$ in degree zero: the pullback is an isomorphism by the stalkwise fiber computation and [F2]. For $T=({\mathbb{C}}^{\ast }{)}^m$, the rank-one resolutions in [F10] have bases $({\mathbb{C}\mathbb{P}}^{N-1}{)}^m$. Repeated [F5] and [F6] give the basis of products of tautological powers with each exponent less than $N$; all degrees stabilize as $N$ increases [F7, F10]. Thus the stable ring is $k[x_1,\dots ,x_m]$, with the flag quotient lines corresponding to the individual torus characters. This proves the polynomial model without asserting a polynomial extension of $H_G(\mathrm{p}\mathrm{t};k)$.

Relative positions can be computed by simultaneous echelon reduction: for two full flags the integers $\dim (F^a\cap F^{\mathrm{\prime }b})$ determine a permutation matrix, its entry at $(a,b)$ being the second difference of these integers. Splitting the intersections successively chooses one line for each nonzero entry. Grading requires each such line to have the same color in both flags, exactly the condition $w\nu =\eta$. Adapted bases then show that all flag pairs with that matrix form one $G$-orbit. Their stabilizer $P$ consists of matrices triangular in both orders: its diagonal is $T$, and its permitted off-diagonal entries form a contractible unipotent radical; $P\mathrm{/}T$ is an affine space. Over this orbit the stable-arrow condition is the associated vector bundle with fiber $U_F\cap U_{F^{\mathrm{\prime }}}$, since stability is a set of linear zero-entry conditions in each adapted basis. The projection of the orbit to the first flag has the affine echelon cell given by the off-diagonal entries allowed by the second order modulo those allowed by both. Thus the stratum is smooth, its equivariant cohomology is $H_P(\mathrm{p}\mathrm{t};k)=H_T(\mathrm{p}\mathrm{t};k)=k[x_1,\dots ,x_m]$ by the contractible-fiber argument of step 1.1, and its equivariant BM module is that ring shifted by twice the stratum's complex dimension. Its generator is its positive fundamental class, and the first flag's quotient Chern classes give the displayed generators. Each arrow entry has weight ${\chi }_b-{\chi }_a$ [F8], with its full multiplicity.

The conditions $\dim (F^a\cap F^{\mathrm{\prime }b})\ge q$ are closed rank conditions on the flag pair (vanishing of minors of $F^a\to V\mathrm{/}{F}^{\mathrm{\prime }b}$). They give the Bruhat lower unions of the echelon permutation strata; pulling them back to the incidence space gives closed unions of its strata. Choose a linear extension of that finite order. The localization sequence [F2] for each successive closed union has even-degree quotient by step 2.1. Starting with the empty union, induction shows that odd groups vanish and each even-degree sequence is short exact. The closure class of the newly adjoined stratum restricts to its fundamental class. First-coordinate Chern multiplication lifts every polynomial multiple of that generator; these lifts split the sequence as modules over the first-coordinate ring. Consequently the closure classes form a free polynomial basis in each corner. Closure classes here are indexed by the relative position $w$ as well as the corner, even if several permutations have $w\nu =\eta$.

Fix a diagonal torus and refine each orbit by echelon cells of the first flag. A cell has a unique fixed coordinate flag and a section given by an ordered product of elementary unipotent matrices. Acting by that section trivializes the orbit's affine echelon fiber and the common stable-arrow vector bundle. Torus conjugation scales each echelon coordinate by a root character, and each arrow coordinate by ${\chi }_b-{\chi }_a$. All are nonzero: flag coordinates involve distinct lines of one color, and looplessness makes arrow coordinates join lines of distinct colors. This gives affine representation cells with one fixed point. Ordering first by the closed relative-position filtration and then by the closed echelon filtration of the first flag gives a closed filtration by these cells; the same construction applies to $X_{\nu }$. On each cell the equivariant BM group is a shift of $S_T=H_T(\mathrm{p}\mathrm{t};k)$, and point pushforward multiplies its generator by the product of its nonzero weights, by [F4, F9]. Induction through the closed/open sequences therefore proves freeness over $S_T$, and proves that inverting all these weights makes fixed-point pushforward an isomorphism. The fixed locus has zero arrow coordinates and all coordinate flag pairs. Refined middle self-intersection gives $[p,q]\ast [q,t]=\mathrm{e}\mathrm{u}(T_q{X}_{\mathrm{m}\mathrm{i}\mathrm{d}\mathrm{d}\mathrm{l}\mathrm{e}})[p,t]$; other middle fixed points give disjoint supports. Every displayed Euler class is invertible in this localization.

We first justify passage from $G$ to $T$. Put $R=H_G(\mathrm{p}\mathrm{t};k)$ and $S_T=H_T(\mathrm{p}\mathrm{t};k)$. For $Y=X_{\nu }$ or $Z_{\eta ,\nu }$, the approximation of $EG{\times }_{B_{\nu }}Y\to EG{\times }_{G}Y$ is a proper flag bundle with fiber $G\mathrm{/}{B}_{\nu }$; replacing $B_{\nu }$ by $T$ does not change cohomology by its contractible quotient. On its projective-bundle tower the monomials in quotient Chern classes with successive exponents $0\le c_j<{\beta }_i-j+1$ form a fiber basis by [F5, F6]. Include the unit monomial. The maps defined by these global classes ${⨁}_{j}k[-\mathrm{\mid }{e}_j\mathrm{\mid }]\to R{p}_{\ast }k$ are isomorphisms on every stalk by proper base change, hence are isomorphisms. Applying the projection formula to the normalized dualizing complex proves the same decomposition for BM groups, not only constant-coefficient cohomology. Applying it also to $Y=\mathrm{p}\mathrm{t}$ shows that $S_T$ is free over $R$ on this basis, and gives $H_T^{\mathrm{B}\mathrm{M}}(Y)=S_T{\otimes }_R{H}_G^{\mathrm{B}\mathrm{M}}(Y)$ with the normalized dimension shifts. These comparisons commute with convolution and its module action by [F2, F4]; in particular restriction is injective and the $T$ module is generated over $S_T$ by the $G$ module. Thus if $a\in \mathfrak{A}$ kills $P$, its restriction kills the entire $T$ module. After weight localization, expand it in the fixed-pair basis of step 3.2 and apply it to each fixed-point basis vector $[q]$. The result is ${\sum }_p{c}_{pq}\mathrm{e}\mathrm{u}(T_q{X}_{\mathrm{m}\mathrm{i}\mathrm{d}\mathrm{d}\mathrm{l}\mathrm{e}})[p]=0$, so all coefficients vanish. Freeness over $S_T$ makes localization injective, and injectivity of restriction then gives $a=0$. Finite CW-type models and stabilization for all these comparisons are [F10].

Consider a reduced adjacent-transposition word for $w$. In ordinary full-flag coordinates, its elementary flag chains are a tower of projective-line bundles: at a step only the intermediate subspace between two fixed successive subspaces can vary. The open echelon chart has one parameter for each inversion of $w$; successive pivot elimination recovers those parameters uniquely from an endpoint pair in position $w$. This follows inductively when appending a length-increasing swap, which introduces a previously uncrossed pair of pivot lines. The chart is dense in the irreducible tower, so its proper endpoint image is the closure of the position-$w$ orbit. Consequently every chain, including the graded chains in our elementary supports, has endpoint position $\le w$, and an endpoint pair in position $w$ has a unique chain of this fixed reduced type. For a graded endpoint pair the recovered intermediate flags remain graded, since each pivot elimination only combines lines of the same color. To compute the top coefficient, use a split endpoint pair in this open stratum. Along a reduced chain each labeled pair crosses at most once: word length equals the number of inverted pairs, since an adjacent swap changes that number by one, and adjacent sorting realizes that lower bound. Thus, for any fixed arrow entry, its allowedness indicators $A,B,C\in \{0,1\}$ at three length-additive stages are monotone. They satisfy $\min (A,C)+B=\min (A,B)+\min (B,C)$, proving the equality of arrow-weight multisets in $(U_1\cap U_3)\oplus U_2$ and $(U_1\cap U_2)\oplus (U_2\cap U_3)$, separately for each of the $h_{ij}$ copies. For the flag coordinates, the unique-chain echelon chart cancels the intermediate flag tangent coordinates against the refined middle diagonal; its remaining coordinates are exactly those of the endpoint orbit. Taking positive tangent Euler products therefore gives $\mathrm{e}\mathrm{u}(C_u)\mathrm{e}\mathrm{u}(C_v)=\mathrm{e}\mathrm{u}(C_{uv})\mathrm{e}\mathrm{u}(X_{\mathrm{m}\mathrm{i}\mathrm{d}\mathrm{d}\mathrm{l}\mathrm{e}})$ at the open fixed pair for length-additive stages. By the fixed-pair formula of step 3.2 the top coefficient is $1$ for standard classes. The generator normalizations multiply it only by signs, so the top coefficient of ${\tau }_w{e}_{\nu }$ is $\pm 1$.

The expansion of ${\tau }_w{e}_{\nu }$ in the closure-class basis is supported in positions $\le w$ and has coefficient $\pm 1$ at $w$ by step 4.2. The finite triangular matrix is invertible over the first-coordinate polynomial ring, so these products form a free polynomial basis. Expanding each polynomial coefficient in its unique dot-monomial basis gives precisely the displayed $k$-basis; no crossing relation is needed for this argument. Each basis element is a product of word idempotents, dots and elementary crossings, with ${\tau }_{\mathrm{i}\mathrm{d}}{e}_{\nu }=e_{\nu }$ for the empty crossing product, so these classes generate $\mathfrak{A}$. For $\beta =0$ there is one empty word, one point and one unit; the same argument has an empty list of variables and crossings. Only relative-position incidence strata were used. AC is inherited through [A1].

By [F2] the diagonal classes are orthogonal idempotents with sum $1$, acting as the summand projections on $P$. The dot classes act by cup product, so $x_r{e}_{\nu }=e_{\nu }{x}_r$ and all dots commute. We compute the crossings independently before using their dot slides.

For adjacent distinct colors $i={\nu }_r$, $j={\nu }_{r+1}$, forgetting the order of the two graded lines identifies the two flag bases: the intermediate graded subspaces are uniquely recovered. In adapted bases every stable-arrow condition agrees except for the entries between these lines. The common stable-arrow space sets both directions' entries to zero, so its quotient in the first-coordinate space $X_{s_r\nu }$ consists of $h_{ij}$ copies of $\mathrm{Hom}(D_{r+1},D_r)$; its dimension is $d_{s_r\nu }-h_{ij}$. At a split fixed flag its positive tangent Euler factor is $(x_r-x_{r+1}{)}^{h_{ij}}$ by [F5, F6]. The fixed-point action formula established in [F2] gives this quotient Euler factor times $s_r(f)$: the Euler factors of the common space cancel between its fundamental class and the output's fundamental class. Thus the standard class acts by $(x_r-x_{r+1}{)}^{h_{ij}}{s}_r$, ${\sigma }_r$ multiplies it by $(-1{)}^{h_{ij}}$, and ${\tau }_r$ cancels that sign. Composing with the reverse crossing gives ${\tau }_r^2{e}_{\nu }=(x_{r+1}-x_r{)}^{h_{ij}}(x_r-x_{r+1}{)}^{h_{ji}}{e}_{\nu }=Q_{ij}(x_r,x_{r+1})e_{\nu }$. Moreover $M_C{s}_r{M}_f=M_{s_{r}f}{M}_C{s}_r$ for every polynomial $f$, proving all distinct-color dot slides, including both adjacent ones; other dots are fixed by $s_r$.

If the adjacent colors agree, forgetting $F^r$ gives a projective-line bundle: all arrow stability conditions at that stage follow from the neighboring stages, since there are no loops and the other colors' subspaces do not change across these two steps. Thus $X_{\nu }$ is the pullback of the stable-arrow bundle on the partial flag base, and $C=X_{\nu }{\times }_{X_{\mathrm{p}\mathrm{a}\mathrm{r}\mathrm{t}\mathrm{i}\mathrm{a}\mathrm{l}}}{X}_{\nu }$ has dimension $d_{\nu }+1$. The standard class acts by pullback after integration along that projective line. At the fixed point whose tautological line has class $x_r$, its tangent is $\mathrm{Hom}(D_r,D_{r+1})$, of Euler class $x_{r+1}-x_r$; at the other fixed point the Euler class is reversed. The cellwise localization calculation of [F2] therefore gives $f\mathrm{/}(x_{r+1}-x_r)+s_{r}f\mathrm{/}(x_r-x_{r+1})=(s_{r}f-f)\mathrm{/}(x_r-x_{r+1})$ for the standard class. The sign $(-1{)}^{d_{\nu }-(d_{\nu }+1)}=-1$ makes ${\tau }_r=D_r=(1-s_r)\mathrm{/}(x_r-x_{r+1})$. The numerator vanishes when the variables coincide, so division is polynomial. $D_{r}f$ is symmetric in these variables, giving $D_r^2=0$; expansion gives $D_r{M}_{x_r}-M_{x_{r+1}}{D}_r=1$ and $D_r{M}_{x_{r+1}}-M_{x_r}{D}_r=-1$. Unaffected variables commute with $D_r$.

Disjoint crossings commute, since neither their variables nor their colors affect the other crossing. For three distinct colors both triple products cross the same three labeled pairs exactly once with the same oriented factors, so they coincide. Write $x=x_r$, $y=x_{r+1}$, $z=x_{r+2}$ and $D_1=(1-s_1)\mathrm{/}(x-y)$, $D_2=(1-s_2)\mathrm{/}(y-z)$ in the three-variable calculation. For colors $(i,i,j)$ and $a=h_{ij}$, the two products are $D_2{M}_g{s}_1{s}_2$ and $M_g{s}_1{s}_2{D}_1$, where $g=(x-y{)}^a(x-z{)}^a$. Since $g$ is symmetric in $y,z$, $D_2{M}_g=M_g{D}_2$; also $D_2{s}_1{s}_2=s_1{s}_2{D}_1$, so the products agree. For $(j,i,i)$ put $b=h_{ji}$ and $g^{\mathrm{\prime }}=(y-z{)}^b(x-z{)}^b$. The products are $M_{g^{\mathrm{\prime }}}{s}_2{s}_1{D}_2$ and $D_1{M}_{g^{\mathrm{\prime }}}{s}_2{s}_1$; symmetry in $x,y$ and $D_1{s}_2{s}_1=s_2{s}_1{D}_2$ again give equality. Relabeling $i,j$ covers every exactly-adjacent repeated-color pattern. For three equal colors, direct expansion of either $D_2{D}_1{D}_2$ or $D_1{D}_2{D}_1$, using $s{M}_f=M_{sf}s$, gives ${\sum }_{w\in S_3}(-1{)}^{\mathrm{\ell }(w)}w\mathrm{/}((x-y)(x-z)(y-z))$ in the skew group algebra. Thus its braid difference is zero as well.

The outer-equal multiedge correction. It remains to consider colors $i,j,i$ with $i\mathrm{\ne }j$; put $a=h_{ij}$, $b=h_{ji}$, $N=a+b$, $x=x_r$, $y=x_{r+1}$, $z=x_{r+2}$. Expanding the two triple products in the skew group algebra using step 1.2 for the distinct-color crossings and step 1.3 for the middle equal-color crossing gives ${\tau }_2{\tau }_1{\tau }_2{e}_{\nu }=[(y-z{)}^a(z-y{)}^b-(y-z{)}^a(x-y{)}^b{w}_0]\mathrm{/}(x-z)$ and ${\tau }_1{\tau }_2{\tau }_1{e}_{\nu }=[(x-y{)}^b(y-x{)}^a-(x-y{)}^b(y-z{)}^a{w}_0]\mathrm{/}(x-z)$ with $w_0$ the longest permutation; the permutation terms agree and cancel, and the difference of the two remaining terms is $[Q_{ij}(x,y)-Q_{ij}(z,y)]\mathrm{/}(z-x)$, which is the stated correction with full multiplicities $a,b$. When $a=b=0$ the correction is zero, and when $i=j$ the nil-Hecke case of step 2.1 supplies the zero correction, so every relation of $S_{\mathrm{\Gamma }}(\beta )$ of braid type holds.

Homogeneity of the generators. For distinct colors, the operator ${\tau }_r$ raises the unshifted polynomial degree by $2h_{ij}$ and changes the summand by $d_{\nu }-d_{s_r\nu }=h_{ji}-h_{ij}$, so on the shifted summands it is homogeneous of degree $2h_{ij}+h_{ji}-h_{ij}=h_{ij}+h_{ji}=-a_{ij}$; for equal colors the divided difference has shifted degree $-2=-a_{ii}$, and the dots and idempotents have degrees $2$ and $0$. Each defining relation of [F1] is therefore homogeneous: the square relation has both sides of degree $-2a_{ij}$ on the $\nu$-summand, the mixed relations have both sides of degree $0$, and the braid correction has degree $2(h_{ij}+h_{ji}-1)$ on the outer-equal-color summand, matching the degree of the two triple products, which is $-a_{ij}-a_{ji}-2=2(h_{ij}+h_{ji})-2$. Hence the assignment is graded.

Transfer to the convolution algebra. The displayed operator identities of steps 1.2–3.1 hold on every shifted polynomial summand, and by step 1.1 the assignment $e_{\nu }\mapsto e_{\nu }$, $x_r\mapsto x_r$, ${\tau }_r\mapsto {\tau }_r$ respects all remaining relations; since by [F2] the action of $\mathfrak{A}$ on $P$ is faithful, the identities hold in $\mathfrak{A}$ itself, so the assignment is a well-defined graded $k$-algebra homomorphism $S_{\mathrm{\Gamma }}(\beta )\to \mathfrak{A}$. The Axiom of Choice entered only through [A1].

By [F2] the assignment on generators respects every defining relation of the presentation of [F1], hence induces a graded $k$-algebra homomorphism $\mathrm{\Phi }:S_{\mathrm{\Gamma }}(\beta )\to \mathfrak{A}$. The generators generate $S_{\mathrm{\Gamma }}(\beta )$ by definition of a free presentation, so any two graded algebra homomorphisms agreeing on $e_{\nu },x_r,{\tau }_r$ agree everywhere; this is the asserted uniqueness.

We establish the needed adjacent-transposition word reduction explicitly. Put $t_j=s_{m-1}\cdots s_j$ for $1\le j<m$, and $t_m=1$. Every permutation has a unique canonical expression $q{t}_j$, where $q$ fixes $m$ and $j$ is the position sent to $m$. Inductively choose the same expression for $q$ in $S_{m-1}$. Right multiplication by $s_i$ is absorbed using $t_j{s}_i=s_i{t}_j$ if $i<j-1$, $t_j{s}_{j-1}=t_{j-1}$, $t_j{s}_j=t_{j+1}$, and $t_j{s}_i=s_{i-1}{t}_j$ if $i>j$. The first formula commutes disjoint letters; the second appends one letter; the third cancels the final equal pair; the fourth follows by commuting $s_i$ left until a single braid move changes $s_i{s}_{i-1}{s}_i$ to $s_{i-1}{s}_i{s}_{i-1}$, then commuting $s_{i-1}$ past higher-index letters. For $t_m=1$ the generator lies in $S_{m-1}$ or equals $t_{m-1}$. Recursively absorbing the resulting left generator into $q$ converts any word into its canonical expression using only braid moves and adjacent-pair removals, never increasing length. The canonical length is the inversion number: $t_j$ contributes the $m-j$ inversions involving $m$, and $q$ contributes exactly the inversions on the remaining entries. Every word has at least that length because a swap changes inversion number by one. Therefore reduced words normalize using braid moves alone and any nonreduced word exposes a removable equal pair during normalization. Two reduced words for the same permutation are braid-connected through their common canonical word. The cases $m=0,1$ have only the empty expression.