The equivariant cohomological Hikita conjecture for arbitrary quivers

Proof. Decompose N into G-isotypic summands. A commuting torus acts on each multiplicity space and is diagonalizable there. The nontrivial irreducible gauge representations that occur are the bifundamentals between distinct vertices, the dual standard representations from framing, and the traceless endomorphisms at vertices of rank at least two. All remaining summands are gauge-trivial. A weighted traceless endomorphism summand can be replaced by the full endomorphism representation by adding a scalar line of the same flavor weight. This gives the asserted description.

If E is gauge-trivial, the moment map for N\oplus E is the moment map for N, and semi-invariant generation shows that the stable and semistable loci are the corresponding loci for N, times T^*E. Their stabilizers are unchanged. The quotient gains a factor T^*E, which retracts F-equivariantly to zero. In the space of triples the condition on the E(\OO) coordinate imposes no lattice constraint, so this coordinate gives a common vector-space factor in every finite truncation. The normalized equivariant Thom isomorphism removes this factor, compatibly with convolution, both gradings, and the coefficient map. This proves the claim on both sides.□

Proof. We use the classical BFN localization and flavor construction [4, Proposition 3.22, Lemmas 5.9–5.13, Proposition 5.15, and Corollary 5.21], and its minuscule formula [4, Proposition 6.6][5, Appendix A, Proposition A.2], together with quiver minuscule generation [42, Proposition 3.1 and Remarks 3.3–3.4]. The form of the formula can be checked directly: the inclusion of the space of triples into the bundle of lattices gN(\OO) contributes the Euler class of

which is the numerator of (2.2). Localization on the closed minuscule orbit gives the tangent denominator. There is no rotation, so translating a modification by t^\eta changes the Laurent monomial but does not shift the coefficients. The localization theorem and the orbit filtration give injectivity before taking Weyl invariants. Equivalently, the image of the nonabelian algebra lies in the invariant part of the displayed ring.

For completeness, the minuscule coweights used in the generation theorem are subsets of slots at several vertices at once. Decomposing positive or negative integral coweights by successive level sets gives exactly these two families; no assertion that single-slot monopoles suffice is needed. Lemma 2.1 puts arbitrary commuting flavors in weighted quiver form. Flavor characters change the equivariant Euler factors but not the gauge-weight hyperplane arrangement used in the generation proof; the same chamber argument therefore retains those parameters, as discussed in [42, Section 1.6.1]. The embedding used here is the classical localization map, not a specialization of a representation in which loop rotation is inverted. Finally, put c_\lambda=\sum_{\beta\in\wt(N)} (-\langle\beta,\lambda\rangle)_+ and a_\lambda=\dim\Gr_G^\lambda. For homogeneous p, the BFN homological normalization gives

by [4, Lemmas 2.2 and 2.5–2.6]. These are exactly the degrees of the numerator, tangent denominator, and dressing in Equation (2.2). The bare matter-free torus class e^\lambda has degree zero. This proves the stated grading comparison. General dressings are decomposed into homogeneous summands. The additional shifts from a common finite section cutoff will be kept explicitly in Section 3. No nonnegative grading on A_F itself is assumed.□

Proof. Filter the affine Grassmannian by bounded unions of orbits and truncate all section lattices between sufficiently high positive and negative powers of t. Over an orbit, the truncated space of triples is a vector bundle of constant relative rank. The orbit is an affine-space bundle over a partial flag variety; its equivariant homology, with the relative shifts in the definition of A_F, is even. Its F'-equivariant homology is free over H_{F'}^*(\mathrm{pt}), and restricting the flavor action gives the displayed change of coefficients on this stratum. The exact sequences for adjoining strata have no odd terms, so these statements extend through each finite filtration. The free successive quotients also show that tensoring the sequences introduces no Tor term. Passing to the stabilized truncations and their exhaustive union proves the first assertion. Pullback, refined intersection, and proper pushforward defining convolution commute with this change of equivariance, so the assertion is an algebra isomorphism.

Flavor parameters have charge zero. Under the first isomorphism the ideal generated by all nonzero charge spaces is the extension of the same ideal before base change. Taking the quotient therefore gives the second assertion; no flatness of B_{F'} is required for this step.□

Proof. We identify a positive multiple of a rational character with the same stability condition. Fix a maximal torus of G and the finite set of its weights on N\oplus N^*. For a point x and g\in G, let C(gx) be the cone generated by the weights occurring in gx. The affine Hilbert–Mumford criterion says that x is semistable precisely when the character belongs to every such cone, and is stable with finite stabilizer precisely when it belongs to all their full interiors in the real weight space; see [27] and [22, Propositions 2.5 and 2.7]. Although g varies, only finitely many cones occur. A sufficiently small neighborhood of \theta remains inside each cone whose interior contains \theta, and outside each cone that does not contain \theta. A \theta-stable point therefore remains stable, and a \theta-unstable point remains unstable. Regularity accounts for all points of the closed subvariety \mu^{-1}(0). This proves the assertion about the entire loci, including when they are empty. The geometric quotient and its principal gauge bundle are unchanged, as is freeness. Choose a rational point in this neighborhood avoiding the finitely many proper hyperplanes specified in the statement, and then rescale by a positive integer. If G is trivial, its character space is a point and the list of bounded nonzero dimension vectors is empty, so the assertion is immediate.□

Proof. We first specify the correspondences and their degrees, then prove the identity by a faithful localization calculation.

Correspondences and shifts. For proper maps f:X\to L and g:Y\to L, write \omega_Y for the complex-linear dualizing complex of Y. Proper base change and adjunction give

For example, if q:X\times_LY\to Y is the projection, the left side is R\Hom_Y(q_*\C_{X\times_LY},\omega_Y), which gives the displayed identity by q_*\dashv q^!. Thus, when Y is smooth of complex dimension e, we have

The equivariant version follows on common Borel approximations; their additional dimensions cancel in the shift. These finite-type models have triangulable analytic topology and finite cohomological dimension. We use the proper-support base-change and adjunction formalism of [38, Theorems 5.10 and 7.7], with complex coefficients, and the duality and Borel–Moore identifications of [31, Example 2.7(3) and Corollary 2.11]. In particular, the source X need not be smooth. Composition across a smooth middle space is refined intersection along its diagonal followed by proper pushforward. These operations are the adjunction and base-change operations in the displayed identification.

Fix one word, temporarily suppressing its index w. Let B_j be the variety of chains of lattices with successive relative positions \lambda_1,\ldots,\lambda_j, starting at the identity. Set B_0=\mathrm{pt}. Each forgetful map B_j\to B_{j-1} is a bundle with fiber the closed minuscule orbit \Gr_G^{\lambda_j}, so the B_j are smooth and projective. Write

The intersection has constant rank by fixed relative position. It defines a vector bundle on B_j and a correspondence

over L, where v_j is the closed subbundle inclusion and u_j forgets the last lattice. Write c_j=\rk(E_{b_j}/I_j), using the same symbol I_j for the intersection bundle. The dressing p_j gives an equivariant characteristic class on B_j: use the two tautological blocks of the minuscule modification in the preceding lattice frame, together with the flavor classes. This is the twisted-orbit version of the stabilizer-invariant dressing in (2.2).

If \delta_j=\deg p_j, the class p_j\cap[I_j] defines a morphism \mathcal F_{B_j}\to\mathcal F_{B_{j-1}}[q_j], where

Indeed, \dim I_j=\dim B_{j-1}+a_j+r_j-c_j, and (3.6) uses the target dimension \dim S_{B_{j-1}}=\dim B_{j-1}+r_{j-1}. The endpoint map B_k\to Z induces a proper map S_{B_k}\to S_Z. Its pullback morphism on constant sheaves, followed by pushforward to L, gives \mathcal F_Z\to\mathcal F_{B_k} with no shift. Compose it with the k incidence morphisms to obtain \Psi_w. This already gives the stated first-step factorization.

At a modification of type \lambda_j, c_j=\sum_\beta(-\langle\beta,\lambda_j\rangle)_+, with multiplicities and with the pairing taken on the gauge part. The rational coefficient in Equation (2.2) has degree 2c_j-2a_j+\delta_j, counting linear forms with degree two. For the neutral word, the sum of these degrees is its BFN degree d, by Proposition 2.2. Moreover, Equation (3.2) gives r_k=r_0=r. Consequently

Thus \Psi_w and u\rho_Z belong to the same graded Hom space. No equality of intermediate ranks or identification of nonneutral degree conventions is used.

Faithfulness of the localization test. Put C_Z=S_{\mathrm{pt}}\times_LS_Z and let \iota:C_Z\hookrightarrow S_Z be the inclusion. The identity (3.6), with smooth target S_{\mathrm{pt}}, identifies the Hom space in (3.5) with H^{\mathsf K}_{2r-d}(C_Z). Under it, \rho_Z is the fundamental class of S_{\mathrm{pt}} over the identity lattice.

Let P=H^*_{T\times F}(\mathrm{pt}). The stratification of C_Z over the finitely many G(\OO)-orbits in Z has vector-bundle strata. Each orbit is an affine-space bundle over a partial flag variety and contracts to it under loop scaling t\mapsto at as a\to0. This contraction commutes with T, and F acts trivially on the orbit. Its ordinary equivariant cohomology is therefore even and free over P; smooth equivariant Poincaré duality gives the same assertion for Borel–Moore homology, with a dimension shift. The section bundle adds an even Thom shift. The closed-stratum exact sequences split into short exact sequences by parity, and their free quotients make each extension free over P. It follows that H_*^{T\times F}(C_Z) is free, in particular torsion-free. This is the finite version of the orbit-filtration argument in [4, Lemmas 2.5–2.6].

The T-fixed lattices in \Gr_G are precisely the diagonal lattices t^\eta: a fixed lattice in each V_i(\KK) splits along its distinct diagonal characters, and a lattice in a one-dimensional Laurent series space is t^m\OO. At t^\eta, every (T\times F)-weight-zero summand of N has gauge pairing zero with \eta. Its lattice is unchanged by intersection with N(\OO). Thus \iota identifies the fixed loci of C_Z and S_Z. Equivariant localization makes \iota_* an isomorphism after tensoring with \operatorname{Frac}P; torsion-freeness makes this localized pushforward a faithful test on H_*^{T\times F}(C_Z). Passing to Weyl invariants recovers \mathsf K-equivariance. This is also the fixed-fiber calculation of [4, Lemma 5.13].

After localization, the classes of the entire section fibers E_{t^\eta} form a basis for the homology of S_Z, indexed by its fixed lattices. These classes, rather than isolated points in a fiber, allow weight-zero summands of N without inverting a zero Euler class. We denote this basis formally by e^\eta.

The chain calculation. For each partial chain put C_j=S_{\mathrm{pt}}\times_LS_{B_j}, and let h_j:C_j\hookrightarrow S_{B_j} forget the initial section constraint. Let z_j be the Borel–Moore class of the composite \mathcal F_{B_j}\to\mathcal F_{\mathrm{pt}}[\sum_{h=1}^j q_h] formed by the first j incidence morphisms. Since S_{B_j} is smooth, write (h_j)_*z_j=\alpha_j\cap[S_{B_j}]. We claim that

with all factors pulled to B_k.

The initial class has \alpha_0=1. At the jth step, composition is refined intersection along the diagonal of the smooth middle space S_{B_{j-1}}. This intersection commutes with the proper pushforward that forgets the initial section constraint. We may therefore first replace z_{j-1} by (h_{j-1})_*z_{j-1} =\alpha_{j-1}\cap[S_{B_{j-1}}]. The remaining intersection pulls this cohomology class back to I_j, giving

Here the regular diagonal belongs to the smooth middle space; no smoothness of u_j is needed. The classes \alpha_{j-1} are pulled from the chain bases, and the subbundle inclusion v_j multiplies by \eu(E_{b_j}/I_j). This proves (3.8) by induction.

Under the same Hom–Borel–Moore identification, precomposition with the unit for the proper endpoint map S_{B_k}\to S_Z is proper pushforward along C_k\to C_Z. Forgetting the initial section constraint commutes with this pushforward. Thus the endpoint pushforward of \alpha_k\cap[S_{B_k}] is exactly the image of \Psi_w in the localization test.

Fixed chains are sequences of Weyl translates \lambda'_j of the chosen minuscule coweights; their successive lattices are t^{\eta_j} with \eta_j=\sum_{h\leq j}\lambda'_h. At such a chain the jth modification is translated by t^{\eta_{j-1}}. This translation commutes with the diagonal torus. There is no loop rotation, so it changes neither the relevant weights nor the dressing values. Hence its quotient Euler class is

and its relative tangent Euler class is \eu_T(T_{t^{\lambda'_j}}\Gr_G^{\lambda_j}). Localization on the smooth chain base therefore sends (3.8), after endpoint pushforward, to

This is exactly the product of the expressions (2.2) for the word. All dressings are transported with their coweights. The fiber ranks at the endpoint equal r, so no additional fiber factor enters endpoint pushforward.

The localized pushforward of u\rho_Z is u e^0. Equation (3.4) and the abelianization of Proposition 2.2 identify it with the sum of the displayed word expressions. The faithful test just proved implies the equality in H^{\mathsf K}_{2r-d}(C_Z) and hence (3.5).□

Proof. Choose a \nu^m-semi-invariant f, with m>0, that is nonzero at \sigma_\infty. Its evaluation on \sigma is a global section of a positive power of \mathcal L_\nu, nonzero at infinity. This proves nonnegativity of the degree.

If the degree is zero, this section is nowhere zero. Thus \sigma is everywhere semistable and determines a morphism

Every invariant polynomial evaluated on \sigma is a regular function on \mathbb P^1, hence constant. The image of \gamma therefore lies in one fiber of the projective morphism to the affine quotient (T^*N)/\!/G. A sufficiently divisible GIT polarization is relatively ample and pulls back to a positive power of \mathcal L_\nu. Its degree is zero, so \gamma is constant: an ample line bundle has positive degree on the image of a nonconstant morphism from \mathbb P^1. These are the usual character-linearized affine GIT constructions; see [27, Section 2].

The constant value is the stable quotient point represented by \sigma_\infty. Its fiber in the semistable locus is the single orbit G\sigma_\infty. Consequently \sigma is a section of the associated orbit bundle \mathcal P\times_G(G\sigma_\infty). Since G\sigma_\infty\simeq G equivariantly, this section trivializes \mathcal P. Freeness makes the trivialization unique once the coordinates of the section are prescribed to be \sigma_\infty; it agrees with the specified frame at infinity.

In the lattice presentation, compare this global trivialization with the fixed trivialization off zero. Their transition is a regular map \mathbb A^1_{t^{-1}}\to G equal to 1 at infinity. It is the loop h in (4.2), and the lattice of regular sections at zero is hG(\OO).□

Proof. Differentiating in y gives \operatorname{pr}_{L_+}n=s. Differentiating in the section variable n\in E_b says that p+y annihilates E_b in the residue pairing. The annihilator is the displayed truncated dual lattice. These conditions are zeros of algebraic bundle maps and hence define a closed locus.

This derivative calculation gives a support bound even when B is singular. Off \Sigma_B, a derivative in one of the vector-bundle fiber coordinates is nonzero. The function is affine linear in that coordinate; where its coefficient is invertible, replacing the coordinate by W_B gives a product chart over the unchanged base. The constant sheaf has zero vanishing cycles in this chart.

A bounded lattice in each general linear factor specifies a modification of the trivial vector bundle at zero, determined by finite jets. A point of \Sigma_B gives two global sections on \mathbb P^1 for the resulting principal bundle. In the off-zero frame they are the polynomials n and p+y in t^{-1}, with values s and p at infinity. The lattice conditions are exactly their extendability at zero: the discarded t^l tails already lie in the corresponding lattices. Their value at infinity is the stable pair (s,p). Lemma 4.2 now excludes negative character-line degree and, in degree zero, forces the entire topological class to be zero. This proves the last assertion.□

Proof. First, every point of \Sigma_Z lies over U. By Lemma 4.2, its principal bundle is trivial, with a based loop h as in (4.2). A vector in the kernel of E_b\to L_+ is a negative polynomial section of its associated N-bundle, extending at zero and vanishing at infinity. That associated bundle is trivial, so the vector is zero. The two spaces have dimension r, and the projection is an isomorphism.

Over U, write E_b as the graph of a regular linear map A_b:L_+\to L_-. Thus n=a+A_ba for a\in L_+. Define the linear functional

Via P_-=L_+^*, Equation (4.3) becomes

Its fiber critical equations have the unique solution a=s, y=-\ell_{b,p}, depending regularly on (b,s,p). Consequently \Sigma_Z is the graph \Gamma\simeq U\times D. It is closed in all of M_Z because the equations defining \Sigma_Z are closed and every solution has already been shown to lie over U.

It remains to compute the critical value. Apply Lemma 4.2 to this solution. In the off-zero frame,

Invariance of the pairing gives \langle hp,hs\rangle=\langle p,s\rangle, a constant series. Hence

Write h=1+\xi t^{-1}+O(t^{-2}), where \xi\in\mathfrak g. The coefficient [t^{-1}](hp) is the infinitesimal dual action \xi\cdot p. Therefore

Thus the regular, equivariant translations a'=a-s, y'=y+\ell_{b,p} turn (4.10) into the fixed split form

over U\times D.

The vanishing cycles of this form on L_+\oplus L_+^* are one copy of \C[-2r] at the origin. The external-product theorem applies to the constant sheaf of U\times D even if U is singular [30, Theorem, pp. 1–2 of the arXiv preprint]. The splitting is global and equivariant. Its structure group acts as g\oplus g^{-T}, so the quadratic vanishing-cycle line has trivial equivariant monodromy. Choosing its identification with \C once gives compatible identifications over the whole base.

Lemma 4.3 says that the vanishing cycles are zero off the closed graph \Gamma. The product calculation therefore gives the first isomorphism in (4.8) on all of M_Z. In particular, there are no further cycles over Z\setminus U: every point there fails at least one of the fiber equations. The same computation at 1 has A_1=0, \ell_{1,p}=0, and the same 2r quadratic variables, giving the second isomorphism.

The map \rho_Z is restriction to the base lattice. To identify its test as a morphism, not just its source and target, put B=U\times D and consider the maps

These maps are \mathsf K-equivariant and satisfy \pi j=\id. Their ordinary cohomological pullbacks satisfy j^*\pi^*=\id. On the same base B, apply the natural external-product identification to the morphism \C_B\to Rj_*\C_D, with zero function on B and the fixed split quadratic on L_+\oplus L_+^*. Since j is closed, proper-direct-image compatibility identifies the target with the quadratic calculation over D. One common choice of generator of the quadratic line then identifies the resulting morphism with \C_B[-2r]\to Rj_*\C_D[-2r]. Taking ordinary equivariant cohomology gives exactly Equation (4.9). This uses naturality in a sheaf morphism and proper direct image, not commutation with an arbitrary nonsmooth pullback. The identity j^*\pi^*=\id supplies the R-linear splitting, without any properness requirement on \pi.□

Proof of Proposition 4.1. The assertion is immediate if D=\varnothing, so assume D is nonempty. Let u\in R be homogeneous and zero in B_F. Proposition 2.2 expresses u in A_F as a finite sum of positive-length words in nontrivial dressed positive and negative minuscule generators, with coefficients absorbed into the dressings. Taking the full topological-degree-zero part, every word has total topological degree zero.

The character-line degrees of its steps sum to zero. Choose a step with nonpositive character-line degree and, by commutativity, put it first. Its topological degree is nonzero. Apply Lemma 3.1, with a common neutral endpoint bound Z. The morphism u\rho_Z is a sum of compositions, each factoring through the first-step object f_{B,*}\C_{S_B}, with the prescribed shifts. By Lemma 4.3, the test of every such object is zero. It follows that

The degree calculation in Lemma 3.1 identifies the degree of this morphism with \deg u, including when intermediate lattice components are nonneutral. The test commutes with shifts, and Lemma 4.4 places the same quadratic shift on its source and target. Its R-linearity therefore gives the equality of graded maps

Apply this equality to \pi^*(1). Since j^*\pi^*(1)=1, we obtain u|_D=0 in H_{\mathsf K}^*(D).

The free stable-locus action identifies H_{\mathsf K}^*(D)=H_F^*(X), compatibly with the coefficient map. We have proved

Both rings are generated by their coefficient images. For B_F this follows from Proposition 2.2; for H_F^*(X) it follows from Kirwan surjectivity [32] and its equivariant extension recalled above. The displayed inclusion induces the graded R-algebra surjection (4.1).□

Proof. First, the direct sum on the right is an algebra. Indeed,

and every exponent is nonnegative. The multiset \mathcal B' is \mathcal W-stable, so taking invariants gives a subalgebra.

For a positive or negative subset coweight, the weights of the deleted loop at a vertex cancel the tangent Euler denominator in Equation (2.2), up to a nonzero constant sign. This is the Grassmannian tangent-weight calculation: for a positive subset I, the loop contributes \prod_{a\in I,\ b\notin I}(x_{ib}-x_{ia}), while the tangent weights are these roots, up to their common sign convention. For a negative subset the roots are reversed. This is the local Euler-factor calculation associated with minuscule flag localization; compare the pushforward identity in [42, Lemma 2.10]. Proposition 2.2 therefore places every generator of A_0 in the right-hand side of Equation (5.1). Conversely, it places every \mathsf S_\alpha(p) for a positive or negative subset coweight \alpha in A_0, since a constant sign can be removed.

We prove that all the orbit sums \mathsf S_\lambda(p) belong to A_0 by strong induction on

The coefficient and positive or negative subset-coweight cases have just been established. If \lambda has a positive coordinate, set

otherwise set

In either case define \alpha=f(\lambda) and \gamma=g(\lambda) coordinatewise. Outside the base cases both \alpha and \gamma have squared norm strictly smaller than that of \lambda. Both functions are nondecreasing on \Z, and their values have compatible signs relative to zero. Thus, for every pair of slots d,e and every slot d,

These statements apply across different colors as well. Consequently every exponent in Equation (5.2) is zero for this pair, and r_\alpha r_\gamma=r_\lambda.

Order the slots at each vertex by their \lambda-coordinates. Then \alpha and \gamma are ordered in the same way. The rearrangement inequality gives, for every relative matching w\in\mathcal W,

In the equality case, the resulting sum lies in the orbit of \lambda. To see the last assertion directly, sort the matched entries within each color. Swapping an inverted pair strictly increases the scalar product unless one of the two corresponding pairs is tied; swaps involving ties leave the multiset of sums unchanged. Hence every maximal matching has the same multiset of sums as the ordered matching. All other orbits occurring in a product \mathsf S_\alpha(p_\alpha)\mathsf S_\gamma(p_\gamma) therefore have strictly smaller squared norm.

We now verify that the leading coefficient can be an arbitrary \mathcal W_\lambda-invariant dressing. At the fixed representative \lambda, the decomposition into translates of \alpha and \gamma is unique. For example, in the positive case, a different assignment of the 1-entries of \alpha would assign 1 to some coordinate \lambda_d\leq0 and 0 to some coordinate \lambda_e>0 of the same color. The corresponding \gamma-entries would satisfy \gamma'_d=\lambda_d-1<\lambda_e=\gamma'_e, which is a strict inversion relative to \alpha'. This contradicts maximality. The negative case would similarly assign 0 to a negative coordinate \lambda_d and -1 to a nonnegative coordinate \lambda_e of the same color; then \gamma'_d=\lambda_d<\lambda_e+1=\gamma'_e is again a strict inversion relative to \alpha'. Thus the coefficient of r_\lambda in that product is precisely p_\alpha p_\gamma.

Within each color, the level blocks of \alpha and \gamma are intervals in the chosen ordering. Their combined cuts are exactly the cuts of the level blocks of \lambda: two distinct values cannot have both the same f-value and the same g-value, since f(t)+g(t)=t. For every k\geq1, the power sum on each \lambda-block is therefore a linear combination of power sums on \alpha-blocks and \gamma-blocks. Explicitly, obtain prefix power sums at all cuts from whichever block partition contains the cut, and subtract successive prefixes. Power sums generate each symmetric-polynomial ring over \C, so

As the rings commute, every element of the ring on the right is a finite sum of products p_\alpha p_\gamma from the two indicated subrings. Such products of dressed orbit sums realize any prescribed leading dressing. The other terms have polynomial dressings by Equation (5.2); their dressings are invariant under the stabilizers of their coweights because the full product is \mathcal W-invariant. The induction hypothesis removes all those lower terms. This proves the reverse inclusion, without inverting any polynomial, and completes the proof.□

Proof. Choose, for each color, an identification of its l_i positive slots with l_i of its v_i gauge slots. On the remaining v_i-l_i slots put s(d)=0; call these the zero slots and write their set as \Omega_m^0. Define a coweight by

The word “positive” refers to the part size, so at a rank-three vertex the sizes (2,1,0) give the coweight (1,0,-1). The coweight \lambda_m has topological degree zero, since the sum of its coordinates at vertex i is \sum_s s m_{i,s}-v_i=0. Its stabilizer is

The last factors permute the zero slots.

By the degree convention in Proposition 2.2, the half cohomological degree of r_{\lambda_m} is the polynomial degree of its Laurent coefficient. For a remaining arrow i\to j, that degree is

because \sum_a s(ia)=v_i. The framing contribution at i is w_i\sum_{d\in\Omega_m^+,\ i(d)=i}(s(d)-1)=w_i(v_i-l_i). Summing gives

This also proves integrality and nonnegativity. The formula includes remaining loops: their ordered pairs of gauge slots give precisely the diagonal term with c_{ii}=2n_{ii}.

For -1\leq L\leq\sum_i v_i, define \mathcal F_{-1}=0 and, for L\geq0, let \mathcal F_L\subseteq B_0 be the sum of the images of

Here and below an orbit sum denotes its image in the quotient when it is viewed in B_0. This is a finite filtration by graded vector subspaces. The array m_{i,1}=v_i has \lambda_m=0, so the last stage contains the image of all coefficients. Those coefficients generate B_0, by Proposition 2.2; hence \mathcal F_{\sum_i v_i}=B_0. Exhaustion is asserted in the quotient, and requires no assertion about spanning the entire topological-degree-zero part of A_0 before quotienting. For later use denote the symbol map of each orbit type by

It adds the half-degree shift \delta_m.

We first eliminate dressings in the zero slots. For each color i and k\geq1, define an ambient Laurent-ring derivation by

Additivity of the displayed multiplier in \lambda proves the Leibniz rule. This derivation commutes with \mathcal W, preserves the algebra in Equation (5.1), and preserves every topological homogeneous summand. It therefore preserves the ideal defining B_0. Its induced derivation on B_0 is zero, since it kills the coefficient generators. Applying it to an arbitrarily dressed orbit sum gives

for every p\in P^{\mathcal W_{\lambda_m}}. The invariant dressing ring is the tensor product of the symmetric-polynomial rings on the positive type blocks and the zero blocks. Power sums generate each zero-block ring over \C. Equation (5.6), including its arbitrary multiplier p, replaces each such power sum by the indicated positive-slot polynomial. Thus positive-slot dressings alone span the image of this orbit type. These replacements preserve degree.

Fix now a nonempty S\subseteq\Omega_m^+, and split the array of sizes on all gauge slots as s=a+b, by putting b_d=s(d) on S and b_d=0 elsewhere. Set

The topological degree of \beta is a nonzero vector with nonnegative entries, so every product

vanishes in B_0. The dressings here may be arbitrary polynomials invariant under their respective coweight stabilizers, by Proposition 5.1.

To analyze the terms of this product, fix a and match it with a colorwise permutation b' of b. The resulting coweight is t-1, where t=a+b' has nonnegative entries and \sum_{i(d)=i}t_d=v_i. It therefore determines another array in \mathcal M(\mathbf v). Writing A=\operatorname{supp}(a) and B'=\operatorname{supp}(b'), its length is

Every overlap term belongs to \mathcal F_{|l|-1}. For a disjoint matching, the positive sizes of t are the multiset union of those of a and b at each color. Hence its array is exactly m; no other orbit type occurs at length |l|.

For the chosen disjoint decomposition, the excess coefficient in r_\alpha r_\beta/r_{\lambda_m} is q_S up to a nonzero constant sign. Here is the calculation for each weight. For an arrow from slot d to slot e, its excess exponent is

This is zero unless one slot belongs to \operatorname{supp}(a) and the other to S. In that case it is \min(s(d),s(e)). For the framing weight -x_d, the excess is

which equals one on S and zero elsewhere. These give exactly the exponents in Equation (5.3). With the weight conventions above, the sign is (-1)^{\epsilon_S}, where

This integer depends only on the numbers of selected and unselected copies of each type. In particular the sign is constant on the averaging orbit considered next.

It remains to check the full span of the leading dressing relations. Let

Then H_S\subseteq H. Its blocks are the selected and unselected copies of each positive type, together with the zero block at every color. All disjoint decompositions of the fixed representative \lambda_m into translates of \alpha and \beta form exactly the set H/H_S. The coefficient of its leading orbit sum in the product above is (-1)^{\epsilon_S}\operatorname{Tr}_{H_S}^H(q_Sp_ap_b). Since the sign is nonzero and all other terms have lower length, we obtain

Choose p_a to depend only on the positive-size blocks of a and p_b only on the positive-size blocks of b. Such choices are allowed by their coweight stabilizers. Their products span the H_S-invariant polynomial ring on the positive slots, since that ring is the tensor product of the symmetric-polynomial rings on the selected and unselected copies of every type. Moreover q_S is H_S-invariant. For an arbitrary f\in P_m, Reynolds averaging first over H_S and then over H consequently shows that

is a linear combination of the relations in Equation (5.7). All normalizing factors are nonzero constants in \C. Averaging the generators q_S with arbitrary multipliers spans J_m\cap P_m^{\mathcal W_m}, because J_m is stable under \mathcal W_m and averaging is exact in characteristic zero. Therefore all invariant cut-ideal relations annihilate the symbol of this orbit type.

Combining the zero-slot elimination with these cut relations gives, for every L, a surjection of graded vector spaces

Here [\delta_m] means that polynomial half degree d is placed in half cohomological degree d+\delta_m. Exactness of invariants identifies the source with the invariant dressing quotient just obtained. Summing its Hilbert-series bound over the finite exhaustive filtration proves Equation (5.4).□

Proof. The assertion for the empty array is immediate. Suppose that x is a common zero of the cut polynomials and that one of its coordinates has a nonzero value a. Take

Every framing factor in q_S(x) is nonzero, and every difference across the cut is nonzero because S contains all slots with value a. Thus q_S(x)\ne0, a contradiction. The common zero set of J_m is therefore contained in the origin. By the Nullstellensatz, either J_m=P_m or a power of each variable belongs to J_m. Both alternatives give a finite-dimensional quotient. Taking invariants under the finite group \mathcal W_m preserves finite dimensionality and the nonnegative grading.□

Proof. We realize the cut products as Euler classes of incoming coordinates and stratify by unreachable slots to obtain both the coefficient-ring presentation and the recurrence. Make a directed graph whose vertices are the slots of m. For each unordered pair of distinct slots d,e, put b_{de} arrows from d to e and b_{de} arrows from e to d, with each ordered arrow counted once. Add a fixed framing vertex and w_{i(d)} arrows from it to d. Assign a one-dimensional vector space to every slot and to the framing vertex. Let E_m be the space of representations of this graph, and let

act by change of basis at the slots, leaving the framing line fixed. The class x_d\in H_{T_m}^2(\mathrm{pt}) is the first Chern class of the standard character at slot d.

A slot is reachable if there is a directed path from the framing vertex to it on which every arrow coordinate is nonzero. Let U_m be the open subset where all slots are reachable. We will show that the natural coefficient map gives

and that this cohomology is concentrated in even degrees. Reachability is an open condition: it is a union of conditions requiring the coordinates on finitely many paths to be nonzero.

We use induction on the number n of slots. For n=0, the representation space and its reachable open are both a point, and both assertions hold. For 0\leq r\leq n, let U_r\subset E_m be the open subset with at most r unreachable slots. Thus U_n=E_m and U_0=U_m. For a subset S of r\geq1 slots, let Z_S\subset U_r consist of representations whose unreachable set is exactly S.

Let L_S\subset E_m be the linear subspace on which all framing arrows into S and all arrows from the complement of S into S vanish. These equations force every slot of S to be unreachable. Conversely, within U_r, unreachability of all r slots in S forces every complementary slot to be reachable, so these incoming coordinates must vanish. It follows that

In particular, Z_S is smooth and closed in U_r, and U_r\setminus U_{r-1} is the disjoint union of the Z_S with |S|=r.

Let k record the types of the slots in S, and put m'=m-k. Restricting to the complement identifies Z_S with a vector bundle over U_{m'}. Its remaining coordinates are the internal arrows on S and the arrows from S to its complement, all of which are unrestricted. Contracting these vector-bundle fibers is equivariant. The factor of T_m acting at the slots of S acts trivially on the base, so the inductive hypothesis gives

The normal bundle is the representation formed by the incoming coordinates set to zero. Its complex rank and equivariant Euler class are

Only arrows entering S vanish; this accounts for the exponent b(k,m'), even though the graph has arrows in both directions.

Multiplication by e_S on Equation (6.3) is injective. Indeed, view it as a polynomial in the variables indexed by S, over the possibly nonreduced coefficient ring P_{m'}/J_{m'}. Its term of highest total degree in those variables is the single monomial

with coefficient one. Multiplication by this monomial is injective over any coefficient ring, and all other terms have lower degree in the new variables. A highest-degree argument proves the claim, including the case e_S=1.

Write i_S:Z_S\hookrightarrow U_r. The self-intersection identity says that i_S^*i_{S,*} is multiplication by e_S. Restriction to a different, disjoint stratum kills the Gysin image. Therefore the direct sum of the Gysin maps is injective. The equivariant Thom exact sequence, applied successively from U_n down to U_0, gives short exact sequences

It also proves at each step that the cohomology is even and that restriction from P_m=H_{T_m}^*(E_m) is surjective. These statements start with U_n=E_m; evenness of the strata follows from the induction on n.

We also identify the kernel at each step. By Equation (6.3), every class on Z_S is the restriction of a polynomial in the slot variables. The incoming linear equations cut out Z_S regularly inside U_r, so its fundamental class i_{S,*}(1) is the restriction of the Euler product q_S. The projection formula now shows that the image of i_{S,*} is the ideal generated by that class. Removing all strata therefore gives exactly the ideal generated by the nonempty cuts, proving Equation (6.2).

Finally, choose arrow labels compatibly with permutations of equal types. Then \mathcal W_m acts on the entire construction, permuting the slot torus and its variables. Complex orientations give equivariant Thom classes without permutation signs. Taking \mathcal W_m-invariants is exact over \C. Subsets S with a fixed count vector k form one orbit, whose stabilizer is \mathcal W_k\times\mathcal W_{m'}. Their total invariant cohomology contribution in Equation (6.6) is consequently

There is one such contribution for each count vector k; taking invariants of the sum over its subset orbit supplies no binomial multiplicity. The all-reachable open supplies the term k=0. Summing all strata and using \Hilb(P_m^{\mathcal W_m};q)=1/(q)_m proves Equation (6.1).□

Proof. The quadratic form satisfies

This identity includes the diagonal entries of the type matrix: slots in the two separate blocks are distinct even when their types agree. Multiply Equation (6.1) by z^{\|m\|}q^{-D_m-w(m)} and sum over m. Its summand factors as

For each monomial in the z_i, only finitely many arrays occur. The resulting convolution identity may therefore be divided by the first factor’s series, whose constant coefficient is one. This is Equation (6.7).□

Proof. We first specify the moment-level comparison. Give N a Hermitian metric invariant under G_c=\prod_iU(V_i), and equip T^*N with its flat hyperkähler structure. The hyperkähler moment map has one real and one complex component. Under the trace identification of the central real dual with \mathbb R^{Q_0}, let \eta=(1,\ldots,1) be the level corresponding to the determinant character. The GIT–moment-map correspondence identifies X_{\det}(v,w) with the compact-group quotient at real level \eta and complex level zero. Rotation of the three moment-map components carries this quotient to the one at real level zero and complex level c\eta, for a common nonzero normalization constant c. At real level zero, affine Kempf–Ness identifies the latter quotient with

These are identifications of underlying topological spaces; they do not assert that rotation is holomorphic in the original complex structure. The comparison uses the direct rotation on the flat representation [20, Section 3(D)] and the topological affine Kempf–Ness theorem recalled in Section 2. The quiver GIT description is explained in [35]; the flat-representation argument just given includes loops. Multiplying the cotangent coordinate by c^{-1} is an algebraic isomorphism from this affine quotient to

because the cotangent moment map is linear in that coordinate. Thus M_\eta(v,w) and X_{\det}(v,w) have the same ordinary cohomology, including simultaneous emptiness.

Hausel’s formula for M_\eta(v,w) explicitly allows loops and multiple arrows [16, Equation (9)]. We record its grading and all factors needed here. For partitions \lambda^i and \lambda^j, write

Let m(\lambda)_{i,s} be the number of parts of size s in \lambda^i, and set

The nonempty smooth variety has complex dimension 2\delta(v,w). With q=t^2, where t is the ordinary cohomological variable in Hausel’s formula, that formula reads

Here \lambda ranges over all multipartitions, including the empty one. In particular, the left side uses ordinary cohomology. Its normalization can also be checked by Poincaré duality: if u is the arithmetic variable, a smooth variety of complex dimension 2\delta satisfies P_c(u)=u^{2\delta}P(u^{-1}). The normalized count u^{-\delta}P_c(u) therefore becomes q^{-\delta}P(q) after setting u=q^{-1}, as in Equation (6.9).

Now write m=m(\lambda). Removing one loop at each vertex cancels the vertex self-pairing in E_Q, and therefore

Consequently the right side of Equation (6.9) is exactly the ratio in Equation (6.7). Taking its z^v coefficient and multiplying by q^{C(v)+w\cdot v} proves Equation (6.8).

The right side is a finite polynomial before this identification: Lemma 6.1 makes each h_m finite, there are only finitely many multipartitions of v, and the shifts are nonnegative integers. Explicitly,

The diagonal entries c_{ii} are even and the off-diagonal terms pair, which also verifies integrality.□

Proof. Proposition 4.1 gives the displayed graded R_0-algebra surjection. Proposition 5.2 and Proposition 6.4 bound the dimension of each graded piece of its source from above by that of its target. Surjectivity gives the reverse inequality. Its kernel therefore vanishes in every degree. Since the bijection is the original coefficient-compatible algebra map, this is an isomorphism over R_0 and an equality of the two coefficient ideals, with the quotient rings retaining nilpotents. The same argument includes the zero ring in the empty case.□

Proof. By Lemma 2.4, each regular character has a real open ball in which every rational parameter gives the same stable and semistable loci, geometric quotient, and principal gauge bundle. We may therefore perturb within this ball without changing the quotient.

Here is a comparison of such neighborhoods for different regular characters. Write G_c=\prod_i U(V_i) for a compact form and \mathfrak c=(\mathfrak g^*)^G for the space of complex central moment parameters. Equip T^*N with its standard flat hyperkähler structure. The character space identifies with a real form of \mathfrak c. For a rational parameter, clear its denominators and then dilate the flat representation by the inverse square root of that common denominator; the quadratic moment maps thereby return to level \theta. Kempf–Ness identifies X_\theta with the quotient by G_c at real moment level \theta and complex moment level zero. Quaternionic rotation commutes with G_c and rotates the three moment maps. Thus it identifies this space with the quotient at real level zero and complex level c\theta, where c\ne0 is one fixed scalar determined by the normalization of the moment maps. Applying affine Kempf–Ness at this latter level gives a homeomorphism

This uses the flat hyperkähler quotient construction [20, Section 3(D)] and the topological affine Kempf–Ness identification recalled in Section 2; the quiver-variety description appears in [35]. The rotation here acts directly on the flat representation and is valid with loops. It requires no properness of a moment-map family. One can normalize c=1; alternatively a linear dilation of T^*N identifies any two common nonzero scalar multiples of the complex level.

Consider the affine family

Its fiber at \zeta is \mu^{-1}(\zeta)/\!/G. To check this without a flatness assumption, let A be the coordinate ring inside the brackets. The ideal specifying \zeta is generated by invariant parameter functions. Exactness of invariants and the Reynolds operator give

The morphism \pi is of finite type. Constructibility of compactly supported direct image and its fiber-stalk description [31, Theorem 2.5(b)] [38, Theorem 5.10] therefore imply that R\pi_!\C_{\mathcal Z} has constructible cohomology sheaves with stalks H_c^*(\pi^{-1}(\zeta);\C). On a common dense Zariski open subset \Omega\subset\mathfrak c, their ranks, hence all compactly supported Betti numbers of the fibers, are constant. We may take \Omega connected. Every real open ball in the character space is Zariski dense after multiplication by c, and its rational points are dense in the usual topology. Each regular character can thus be perturbed, without changing its quotient, so that c\theta\in\Omega.

By Equation (7.1), all regular quotients have the same compactly supported Betti numbers. A nonempty such quotient is smooth of complex dimension 2(\dim N-\dim G): freeness makes the moment map a submersion along its stable locus. Poincaré duality now gives equality of ordinary Betti numbers. If one regular quotient is empty, all compactly supported Betti numbers vanish for every regular quotient; a nonempty smooth quotient would have nonzero top compactly supported cohomology, a contradiction. This also covers nonfaithful gauge actions and unframed components: the generic central fiber is allowed to be empty. If G is trivial, the parameter space is a point and the assertion is immediate.□

Proof. Set S=\C[\mathfrak f] and let S_+ be its positive-degree ideal. Kirwan surjectivity implies H^{\mathrm{odd}}(X)=0. The Borel spectral sequence for X\to EF\times_F X\to BF has second page H^*(BF)\otimes H^*(X), concentrated in even total degrees. It degenerates, and its edge map to H^*(X) is surjective. Lifting a finite homogeneous basis of H^*(X) and applying Leray–Hirsch shows that H_F^*(X) is a free S-module with

This remains valid for X=\varnothing, with zero modules.

Let J be the kernel of the surjection B_F\twoheadrightarrow H_F^*(X) from Proposition 4.1. Freeness of the target makes \operatorname{Tor}_1^S(H_F^*(X),S/S_+) vanish. Tensoring its defining exact sequence with S/S_+ consequently gives

Lemma 2.3 identifies the middle term and map with B_0\xrightarrow{\sim}H^*(X); hence J=S_+J. The module J is graded and bounded below, since B_F is a quotient of the nonnegatively graded ring R. If J had a nonzero homogeneous element of least degree, the identity J=S_+J would express it using elements of smaller degree, which is impossible. Thus J=0. This argument uses positivity only for the coefficient quotient B_F, and does not assume that the Coulomb algebra itself is nonnegatively graded.□

Proof. Discard zero-rank gauge factors. Add one loop at each remaining vertex and write

Let H=\C^\times act trivially on N and with weight one on every added loop. Put H^*_H(\mathrm{pt})=\C[z], with \deg z=2, and use a superscript + for the enlarged theory. At determinant stability, Corollary 6.6 and Lemma 7.2 give

We compare this with the original theory endowed with the trivial H-action. Let \mathfrak m_0\subset R_0 be the maximal ideal of the graded origin and set

The comparison over \Lambda is ungraded in cohomological degree but retains topological charge. The equality of homogeneous ideals in R_0 obtained below will restore the cohomological grading.

The Coulomb comparison. Use the common rational Laurent realization of Proposition 2.2, and put P=\C[x_{ia}]. For a positive or negative minuscule coweight \lambda, the added numerator in Equation (2.2) is

It is invariant under the stabilizer \mathcal W_\lambda and has value one at (x,z)=(0,1). Its inverse belongs to P[z]^{\mathcal W_\lambda}\otimes_{R_0[z]}\Lambda: indeed,

and E_\lambda^{-1}=(\Delta_\lambda/E_\lambda)/\Delta_\lambda. If M(\lambda,p) denotes an original dressed generator, the enlarged generator with dressing p is M(\lambda,pE_\lambda). Conversely, dressing by p/E_\lambda recovers M(\lambda,p) after localization. Generation therefore identifies the two localized Coulomb algebras, with their coefficient maps and their full topological gradings. The elements inverted in Equation (7.3) have topological degree zero. Thus localization commutes with quotienting by all nonzero topological degrees, and gives

The cohomological comparison. Let D^+\subset T^*N^+ be the stable zero moment locus. An H-fixed point of the quotient may be fixed only after a compensating gauge transformation. We therefore localize the actual H-fixed locus of this prequotient, retaining G\times H equivariance, and first restrict to T\times H. Brion’s subgroup localization theorem [6, Lemma 4] says that, for a torus space embedded equivariantly in a finite-dimensional representation, restriction to the fixed locus of a subtorus becomes an isomorphism after inverting finitely many characters nontrivial on that subtorus. Apply it to the compact forms of T\times H and H and to the invariant locally closed subset D^+\subset T^*N^+. Every character nontrivial on H is nonzero at the central Lie algebra element (0,1). The restriction

is consequently an isomorphism there. Taking \mathcal W-invariants recovers G\times H equivariance. This is compatible with localization over \Lambda: the point is \mathcal W-fixed, and the product of the Weyl translates of any denominator is an invariant denominator still nonzero there. This is also the central-parameter case of the prequotient localization in [13, Proposition 2.3].

Both the added loops and their cotangent partners have nonzero H-weights, so (D^+)^H sets all these coordinates to zero. The enlarged moment map then restricts to the original moment map. Stability restricts to the original stability as well: under a G-equivariant closed inclusion, the Hilbert–Mumford test of a point and of every one-parameter subgroup is unchanged. Equivalently, semi-invariants lift by reductivity, and orbit closures remain in the closed subvariety. Hence

with H acting trivially on the right. Freeness of the stable actions now gives a coefficient-compatible isomorphism

The localization theorem used here does not require compactness of D^+ or X^+.

Descent. Let I_B and I_X be the kernels of the coefficient maps from R_0 to B_0 and H^*(X), respectively. Equations (7.2), (7.4), and (7.5) identify the localized quotients over the same ring \Lambda. Therefore I_B\Lambda=I_X\Lambda. Setting z=1 uses the identity

and yields (I_B)_{\mathfrak m_0}=(I_X)_{\mathfrak m_0}. Both ideals are homogeneous. If h\in I_B is homogeneous, there is s\in R_0 with s(0)\ne0 and sh\in I_X. The component of degree \deg h of this relation is s(0)h\in I_X, whence h\in I_X. Interchanging the ideals proves I_B=I_X globally. This is equality of ideals and gives the asserted graded isomorphism of quotient rings, with all nilpotents retained. The proof applies unchanged to empty stable loci; if there are no nonzero gauge ranks, the assertion is immediate.□

Proof. At determinant stability, Proposition 7.3 identifies B_0 with ordinary cohomology, so B_0 is finite-dimensional. For any other regular stability, Lemma 7.1 gives the same total Betti number. The graded coefficient-compatible surjection B_0\twoheadrightarrow H^*(X) of Proposition 4.1 is therefore an isomorphism. Lemma 7.2 proves the assertion for every commuting flavor torus, with the generality supplied by Lemma 2.1. If a quotient is empty, the same argument gives B_0=B_F=0. Uniqueness follows from surjectivity of the coefficient map R\to B_F.□

Proof. We specify the operator algebra and the sign convention. Let D_{ia}w_{jb}=(w_{jb}+\hbar\delta_{ij}\delta_{ab})D_{ia}, with all the D_{ia} commuting. Allow coefficients in

This ring is preserved by every shift D_{ia}^{\pm1}. For subsets A_i\subseteq\{1,\ldots,d_i\}, write w_A=(w_{ia})_{a\in A_i,i\in\Gamma_0}. Define

These are the fundamental formulas [23, Equations (78)–(79)]. Let \mathcal C_S be the S-subalgebra generated by these operators and the polynomials symmetric in w separately at each vertex.

We construct the maps first over K, then prove that their polynomial halves land in \mathcal C_S. Use Tsymbaliuk’s maps on the full trigonometric shuffle algebra and its opposite [41, Theorem 5.7, Equations (93) and (96)], with no framing, and make the formal substitutions

Here \varepsilon is an auxiliary formal variable, not \hbar. The multiplicative shift in the source satisfies D^{\mathrm{trig}}_{ia}W_{ia}=q^{-1}W_{ia}D^{\mathrm{trig}}_{ia}. Thus its inverse becomes our positive shift D_{ia}, while it becomes D_{ia}^{-1} in the negative map. The leading coefficient of the trigonometric shuffle kernel is

It is precisely the kernel for \star in Equation (8.1): the incoming-arrow signs have been removed from the kernel of [23, Equation (31)]. This distinction is necessary. For the quiver consisting of one arrow i\to j, the untwisted product gives e_{(1,0),1}*e_{(0,1),1} =-e_{(1,1),z_i-z_j+u_a}, whereas the corresponding fundamental operator product has a plus sign.

Here are the order calculations that justify taking leading coefficients, including for loops and repeated gauge slots. Put

Lift a polynomial E of dimension \mathbf n to \varepsilon^{\kappa(\mathbf n)} E((X_{ia}-1)/\varepsilon). The cross-kernel order is \kappa(\mathbf n+\mathbf m)-\kappa(\mathbf n)-\kappa(\mathbf m); thus the leading source product is \star. In an operator summand, let m_{ia} be the length of the string assigned to gauge slot (i,a), so \sum_a m_{ia}=n_i, and set P_2=\sum_{i,a}\ell_i m_{ia}^2. The three prefactors in Tsymbaliuk’s formulas have orders

Together with the lifted input, their total order is L^\pm\cdot\mathbf n, independently of the composition. Multiply the operator image of the lift by \varepsilon^{-L^\pm\cdot\mathbf n} and take its \varepsilon^0 coefficient to define the normalized leading map. The remaining denominators have nonzero leading coefficients over K(w). There are only finitely many compositions. Consequently an input with zero normalized leading coefficient also has zero normalized leading output. The construction is independent of the lift. Since L^\pm\cdot\mathbf n is additive, the normalized leading maps preserve multiplication. Multiplying them in dimension \mathbf n by \hbar^{-|\mathbf n|} preserves that property.

The use of formal parameters in this argument requires no analytic specialization theorem. For real \hbar<u_a<0 and small positive \varepsilon, the substitutions satisfy Tsymbaliuk’s parameter inequalities. The monomial factors involving logarithms of torus parameters are expanded near W=1; they have leading coefficient one and rational Taylor coefficients in \hbar,u_a,w_{ia}. Each coefficient of the multiplication identities is therefore a rational identity valid on this nonempty real open set, hence valid over K(w). This justifies the formal leading maps, denoted \Psi_+ and \Psi_-.

For the distinguished elements the maps are especially simple:

Indeed, the same-color factor in e_{\mathbf n,g} vanishes on every additive string of length at least two. Only selected subsets of distinct slots remain. Its diagonal factors contribute \hbar^{|\mathbf n|}, which cancels the normalization \hbar^{-|\mathbf n|}; its off-diagonal factors cancel the same-color denominators in the operator formula. The factors for arrows between two selected slots cancel as well. In the positive formula the single-entry factors contribute (-1)^{\sum_i n_i(L_i^+-\ell_i)}, and the internal-arrow cancellations contribute (-1)^{\kappa(\mathbf n)-\sum_i\ell_i n_i}, giving the stated sign. In the opposite formula the selected variables are w_A-\hbar; the incoming-arrow factors cancel without these signs. This proves both formulas, and proves that the image is zero when \mathbf n\not\leq\mathbf d.

By [23, Theorem 2.4], the distinguished elements generate \mathcal S_S^+ over S. The twist preserves their word spans, and the opposite statement follows by reversing words. Equation (8.2) therefore proves that both polynomial halves land in \mathcal C_S. Although we constructed the maps over K, their restrictions are thus defined over S; no inverse of a parameter is needed in the identities in \mathcal C_S.

Finally specialize \hbar=0, w_{ia}=x_{ia}, and u_a=-f_a. Every allowed denominator becomes x_{ia}-x_{ib}, with a\ne b, so this defines a homomorphism to \operatorname{Frac}(P_F)[X_*(T)], sending each D_{ia} to the monomial for its standard slot coweight. For a selected-variable dressing g, also denoting its specialization by g, direct comparison with Proposition 2.2 gives

The first sign comes from the outgoing matter factors; the second comes from reversing the minuscule tangent weights. The coefficient roots themselves are unchanged. Thus every generator of \mathcal C_S specializes to an element of A_{\Gamma,\mathbf d,F}, and so does every image of the polynomial halves. Only this homomorphism is needed; no injectivity or flatness after specialization is asserted.□

Proof. Polynomial Hall-algebra input. Use the independent polynomial ring S defined above. This is the equivariant coefficient ring H^*(BT_{\mathrm{univ}};\Qbb) for T_{\mathrm{univ}}=(\C^\times)^{1+|\Gamma_1|}, with independent rotation and arrow characters. It lies in the geometric parameter setting of [23, Equation (30) and Remark 2.6]; no extension to arbitrary field-valued exponential parameters is needed. On the tripled quiver use weights -u_a, u_a-\hbar, and \hbar on an original arrow, its reverse, and an added loop, respectively. These choices satisfy all the genericity hypotheses of [23, §2.2]; in particular there is no nonzero nonnegative integral relation among u_a,\hbar-u_a, and u_a\notin\Z\hbar for every original loop.

We use the following polynomial-form results of that source: the loop-nilpotent Hall algebra identifies with its shuffle half [23, Proposition 3.7]; its BPS descendants generate over S [23, Theorem 3.10, Equation (109), and Corollary 3.15]; and the BPS comparison and Kac formula show that the nonzero dimension-vector support of these generators is \Delta_\Gamma [23, Proposition 3.14 and §3.10]. Here the Kac identity retains the full cohomological grading: the coefficient of t^{d/2} is (-1)^d\dim H^d. A zero Kac polynomial therefore forces every graded piece of the corresponding BPS space to vanish. Proposition 3.14 of that source transfers this vanishing to the loop-nilpotent BPS space. The concrete twist in Equation (8.1) preserves these homogeneous word spans. Here polynomial form refers to equivariant parameters over \Qbb, not integral cohomology.

Apply the generation statement to e_{\mathbf b,g} or f_{\mathbf b,g}, and retain dimension-vector degree \mathbf b. All nonconstant Hall degrees are nonnegative, so the result is a finite S-linear combination of products of BPS descendants with degrees in \Delta_\Gamma summing to \mathbf b. Applying Lemma 8.1 gives the corresponding finite identity in \mathcal C_S. Its coefficients lie in S; no parameter from S is inverted in this identity. Every factor has degree at most \mathbf b\leq\mathbf d; no assertion about vanishing of entire Hall components outside this bound is needed.

Specialization. Now extend scalars to \C and apply the specialization homomorphism of Lemma 8.1. Each finite identity becomes an identity in A_{\Gamma,\mathbf d,F}. The signs in the images of the distinguished elements are nonzero constants and can be absorbed in its coefficients. This operation is performed on the abstract polynomial-form algebra and its map, not on the ambient polynomial representation of the shuffle algebra. An element whose ambient representative is divisible by \hbar need not belong to \hbar times the polynomial shuffle half, and is therefore not discarded before applying the map.

General dressings. It remains to allow an arbitrary stabilizer-invariant dressing p. At each vertex, the polynomial algebra symmetric separately in the selected and complementary variables is generated by the polynomials symmetric in all variables together with those symmetric in the selected variables. Indeed, each complementary power sum is the total power sum minus the selected power sum. Thus p is a finite sum of coefficient polynomials times selected-variable dressings g. The already proved identities for g give Equation (8.3).□

Proof. First, \Gamma has an indecomposable complex representation x of dimension \boldsymbol\gamma. For completeness, the required passage from Kac support to characteristic zero can be made by constructibility. A representation is decomposable over an algebraic closure exactly when its endomorphism algebra contains an idempotent other than 0 and 1. The equations for such idempotents, together with these two open exclusions, define a constructible incidence condition over \Z. Chevalley’s theorem therefore gives a constructible locus of geometrically indecomposable representations. A nonzero Kac polynomial gives points of this locus in infinitely many residue characteristics. Its generic fiber cannot be empty, since a constructible subset of \Spec\Z that omits the generic point is finite. Base change of the nonempty characteristic-zero locus to \C gives x.

The following trace-annihilator argument is the lifting criterion of Crawley–Boevey [7, Lemma 3.1 and Theorem 3.3]; we recall it before passing to the real moment level. Every endomorphism e of x has the form c\id+n, where n is nilpotent. Indeed, its generalized eigenspaces are subrepresentations, so indecomposability permits only one eigenvalue. Consequently

The cotangent moment map with x fixed is the transpose of the infinitesimal orbit map:

Its image is the annihilator of \End_\Gamma(x)=\ker(\xi\mapsto\xi x). Hence the complex moment fiber at \tau_{\boldsymbol\theta} is nonempty, as is the fiber at any common scalar multiple of that value.

This fiber is a closed invariant affine subset, since the moment value is central. The closure of any orbit in it contains an orbit closed in the ambient affine representation. By the zero-character affine Kempf–Ness theorem, that orbit meets \mu_{\mathbb R}^{-1}(0); see [27, Theorem 6.1 and Corollary 6.2]. Thus there is a point with hyperkähler moment value (0,\tau_{\boldsymbol\theta},0), using the real and imaginary parts of the complex moment map and adjusting the common scalar to the chosen normalization.

The quaternionic rotation on the flat cotangent representation commutes with the compact gauge group and rotates its three moment maps. A single rotation sends this value to (\tau_{\boldsymbol\theta},0,0) [20, §3(D)]. This proves the asserted nonemptiness. The simultaneous scalar acts trivially on all arrow spaces and their duals, proving the last statement.□

Proof. Lemma 2.1 allows us to suppose that F scales the individual arrows and framing rows. Adding or removing the gauge-trivial cotangent summands in that lemma preserves both regularity and all the fixed quotients under consideration. Write

Adjoin one vertex \infty of dimension one and w_i arrows i\to\infty, keeping the original arrows. Denote this unframed quiver by \widehat Q and its dimension vector by \mathbf d=(\mathbf v,1). Its representation is the original N, after a choice of basis in each framing space.

There is a group isomorphism

The second factor acts trivially on N. The BFN space of triples and its convolution therefore split, giving

Here \xi is the coefficient parameter of the extra scalar and z is its fundamental Laurent monopole, of cohomological degree zero. This z is unrelated to the degree-two added-loop mass in Section 7. In the original vertex coordinates, z has topological charge \mathbf d: it translates every gauge coweight entry, including the entry at \infty, by one.

Extend the character by

Then \widehat\nu\cdot\mathbf d=0, and under the displayed group isomorphism \widehat\nu is \nu on G and trivial on the extra scalar. In particular, both \xi and z^{\pm1} have \widehat\nu-weight zero. Taking the cocharacter fixed quotient in Equation (8.4) consequently gives

where the subscript denotes the quotient by nonzero cocharacter weights. This extension is faithfully flat.

Let M_{\mathbf b',p}^{+} be a nontrivial positive minuscule generator of A_F, and put \mathbf b=(\mathbf b',0). If \widehat x_{ia} denotes an enlarged gauge Chern root, the group isomorphism above identifies the original root with x_{ia}=\widehat x_{ia}-\xi. Let p^{\uparrow} be the polynomial obtained from p by these substitutions, leaving every flavor parameter unchanged. The original generator is then the positive minuscule class M_{\mathbf b,p^{\uparrow}}^{+}. Central translation also expresses it using the complementary negative coweight:

The two coweights have the same Weyl stabilizer, so the same polynomial p^{\uparrow} is an admissible dressing on both sides. It need not depend only on the variables selected by the complementary negative coweight; Lemma 8.2 allows all stabilizer-invariant dressings. The equality follows directly from Equation (2.2): all matter weights pair trivially with the central coweight, and central translation identifies the minuscule orbits and their equivariant tangent bundles. Equivalently it follows from convolution with the invertible central class. Thus it holds with compatible fundamental-class conventions; rescaling minuscule generators changes only a nonzero constant.

Suppose that this generator has nonzero image in the cocharacter fixed quotient. Apply Lemma 8.2 in the enlarged quiver to M_{\mathbf b,p^{\uparrow}}^{+}. Any word containing a factor of nonzero \widehat\nu-weight vanishes in that quotient. A nonzero image therefore requires a decomposition

Since z is an invertible element of weight zero, Equation (8.5) implies that M_{\mathbf d-\mathbf b,p^{\uparrow}}^{-} also survives. Applying the negative half of the same lemma yields

Both sums are nonempty: \mathbf b\ne0, while its \infty-entry is zero and that of \mathbf d is one.

Together these decompositions express \mathbf d as a sum of at least two nonzero dimension vectors \boldsymbol\gamma_1,\ldots,\boldsymbol\gamma_k in \Delta_{\widehat Q}, each orthogonal to \widehat\nu. For each summand choose the point furnished by Lemma 8.3, with real central parameter \widehat\nu. Use orthogonal decompositions of the vertex spaces and take the direct sum of these points. Moment maps are block diagonal on direct sums, so the resulting point has

Its stabilizer contains k independent block scalar circles.

The enlarged reduction is exactly the original framed reduction: the extra diagonal scalar is ineffective, its moment equation is redundant, and \widehat\nu is trivial on it. After dividing by this scalar, the stabilizer still contains a torus of dimension k-1>0. By the affine Kempf–Ness correspondence at the character level, the resulting point is \nu-semistable in the original zero complex moment fiber [27, Theorem 6.1 and Corollary 6.2]. This contradicts the assumed freeness of every such point. The argument also applies when the stable locus is empty: it would then produce a semistable point where none exists.

Every nontrivial positive minuscule generator therefore vanishes. For a negative generator use the identity M_{\mathbf b,p^{\uparrow}}^{-} =z^{-1}M_{\mathbf d-\mathbf b,p^{\uparrow}}^{+} and interchange the two halves in the argument; it vanishes as well. By Proposition 2.2, the ideal they generate is the ideal generated by the nonzero full topological-charge spaces. It is therefore contained in the ideal generated by the nonzero \nu-weight spaces. For m\ne0, the \nu-weight space (A_F)_m is a sum of nonzero topological-charge spaces. This gives the opposite inclusion. The ideals coincide, proving the proposition. If all gauge ranks are zero there are no such generators and the same conclusion is immediate.□

Proof of Theorem 1.1. Proposition 8.4 and Theorem 7.4 give graded R-algebra isomorphisms

Thus \ker\phi_C=\ker\phi_H. The inverse of \alpha\circ\beta is the isomorphism in Theorem 1.1; surjectivity of \phi_H makes it the unique isomorphism compatible with the coefficient maps. All quotient rings are scheme-theoretic and retain their nilpotents.□