Papers

This paper studies simple Lie algebras in the Verlinde category, with canonical fundamental group action, using combinatorial methods. We give an exhaustive list of such Lie algebras which either have length at most 3, or appear as a subalgebra of the restriction of the Lie algebra of a simple algebraic group along a principal morphism. The results and exceptions answer several questions and conjectures regarding Verlinde categories of algebraic groups and their Lie algebras.

Let O be the ring of integers of a non-Archimedean local field with finite residue field, let p be its maximal ideal, and let O_2 = O/p^2. We show that the class equation of GL_n(O_2) depends on O only through the cardinality of its residue field: for two such rings O and O' with isomorphic finite residue fields, there is a canonical bijection between the conjugacy classes of GL_n(O_2) and of GL_n(O'_2) which preserves the size of every class. The argument constructs a section of the reduction map GL_n(O_2) -> GL_n(O_1) which is multiplicative on the centralizer of any element in its block Jordan canonical form, and uses it to transport the classification of conjugacy classes lying above a fixed class of GL_n(O_1) from one ring to the other. This note records the original argument for this result, developed in the author's 2010 doctoral thesis, which predates and is independent of two later proofs of closely related statements: the Ext-theoretic classification of similarity classes for n <= 4 by Prasad, Singla and Spallone, and the Hom-theoretic approach of Jambor and Plesken for general uniserial rings of length two. We record the centralizer-section argument here since it seems to be of independent interest and several colleagues have asked to see it in print.

We prove that all (possibly infinitely generated) semi-Gorenstein-projective modules over a virtually Gorenstein Artin algebra are Gorenstein projective. We also establish a new criterion for determining the projective (injective) dimensions of modules over virtually Gorenstein Artin algebras. This enables us to show the validities of the Auslander-Gorenstein Conjecture, the (Strong) Nakayama Conjecture and Tachikawa's First Conjecture for virtually Gorenstein Artin algebras. Finally, we prove that the Auslander-Reiten Conjecture holds for virtually Gorenstein Artin algebras of finite CM-type.

We introduce gemstones in a pretriangulated category, which are used to perturb pretriangulated structures. For a triangulated category $\mathcal T$ whose stable category of coherent functors admits a gemstone, we construct pretriangulated structures on $\mathcal T$ which fail the octahedral axiom (TR4). This confirms a long-standing conjecture on triangulated categories.

We categorify a lifting of the integral form of the Cartan subalgebra for quantum sl(2), via categories of equivariant coherent sheaves on projective spaces.

We give a negative answer to a question of Avramov, Buchweitz and Şega by constructing a commutative local finite-dimensional non-Gorenstein algebra $R$ with $\operatorname{Ext}_R^1(D(R),R)=0$; this question is related to the first Tachikawa conjecture. We also give a counterexample to a conjecture of Huneke, Şega and Vraciu on Tor vanishing over commutative finite-dimensional algebras. Finally, we construct a finite-dimensional commutative local self-injective algebra $R$ over $\mathbb{F}_2$ and an indecomposable non-projective $R$-module $M$ such that $\operatorname{Ext}_R^1(M,M)=\operatorname{Ext}_R^2(M,M)=0$, related to the second Tachikawa conjecture and answering a question of Dao.

Let $\mu_t$ be the full set of $t$-th roots of unity. Adjoining $r$ free reciprocal pairs gives a two-parameter family of alphabets; we settle three parts of it. At $r=1$, for every $t\ge2$ and every $\lambda$ with at most $t+2$ parts, $s_\lambda(\mu_t,z,z^{-1})$ is a signed product of exactly three factors over a fixed denominator, or zero, the arguments read off core and quotient. What it sees of $\lambda$ is a multiset of three integers and a sign, and exactly that: two partitions of any sizes share a nonzero value if and only if they agree on that datum. The proof is a Laplace expansion along the $t$ frozen rows with one cancellation lemma, and delivers the sign, of which Littlewood's is one factor. At $t=2$ and every $r$, $s_\lambda(1,-1,z_1^{\pm1},\dots,z_r^{\pm1})$ vanishes exactly when the beta set has constant parity or $\lambda$ is self-complementary of odd width; that direction is a corollary of complementation over an index family of Ayyer and Behrend, the converse an extremal argument in the degree filtration, modulo one rigidity theorem for Schur products. Equivalently: exactly those $V_\lambda$ restrict to $O(N,\mathbb{C})$ $\det$-stably. At odd $t$ and every $r$ it vanishes exactly when a residue class is absent, at no external cost. And for every $t$ and $r$, a reflection of the beta set's excess part with one increment hitting its centre forces vanishing. Three consequences of the first. A vanishing criterion: an empty residue class, or two distinguished classes concentric as intervals, the second only for even $t$. An extension of Ayyer-Kumari's independence criterion: on the reciprocal locus it acquires one further family, classified by core and quotient. And at $t=2$ a $(-1)$-enumeration of plane partitions in a box refined by a parameter that stays free. The factorization is isolated: it fails under each of four deformations of the alphabet, for one reason.

We prove a Plancherel formula for a family of affine $\Psi$-bundles. As an application, we construct Fourier transforms and asymptotic Schwartz spaces for the family. We then prove the corresponding Poisson summation formula under suitable assumptions. The choice of affine $\Psi$-bundles we consider is motivated by applications to triple product $L$-functions explored in another paper of the authors and Leslie.

This paper studies the asymptotic growth of tensor powers in affine Hecke categories, or equivalently of powers of Kazhdan–Lusztig basis elements in affine Hecke algebras. We prove general bounds in arbitrary affine type, determine precise asymptotics in affine type A1, and establish corresponding results for several natural families in affine type A2.

In 1991 Rickard asked whether every derived equivalence of algebras over a common field is standard. We construct a series of examples of non-standard derived equivalences, and thus answer the question negatively. Moreover, we give a necessary and sufficient condition for the question to be true.

Let $F$ be a non-Archimedean local field of characteristic zero and residue characteristic different from $2$, and let $(G,H)$ be an equal-rank split symmetric pair with $G$ split semisimple and adjoint and with an $F$-split maximal $\theta$-split torus. Motivated by the singular contribution near the nilpotent fiber on the $\theta$-fixed side of the infinitesimal local relative trace formula, we construct finite descent charts for regular families in $\mathfrak h$ approaching that fiber. On each chart, a Cartan lift defined over $F$ conjugates the sparse Jacobi family \[ Q(q)=\sum_{\alpha\in\Delta}(\gamma_\alpha n_{-\alpha}+q_\alpha n_\alpha) \] into $\mathfrak h$. The family $Q(q)$ is regular for every $q$ and takes the principal nilpotent value $Q(0)$. The restriction of the adjoint quotient to this family has generic rank equal to the number of odd exponents of $G$. For split symmetric pairs this rank equals $\operatorname{rank}H$; hence in equal rank the map is generically finite étale. Kummer theory classifies the twisted square-root covers on which the Cartan lifts are defined, and these covers exhaust the rational branches. On every branch, with $q_\alpha=z_\alpha^2$, the quotient Jacobian cancels the relative Weyl discriminant exactly: \[ \frac{ds}{|D_H^G|^{1/2}}=C\prod_{\alpha\in\Delta}d^\times z_\alpha. \] Thus the normalized $H^\circ$-quotient density is a multiplicative Haar measure on the parameter torus, uniformly across all rational twists. After a finite clopen refinement, the associated Iwasawa heights are piecewise affine in the valuations $\operatorname{val}(z_\alpha)$. For $(\operatorname{Sp}_{2n},\operatorname{GL}_n)$, the construction is explicit in Hurwitz coordinates, and the Cartan lift factors through commuting long-root $\operatorname{SL}_2$-subgroups.

Starting from the $(E_8,T_1)$ Nahm sum for the vacuum character of $T_{10}M_{\rm eff}(11,2)$, we construct a three-dimensional (3d) $\mathcal N=2$ $U(1)^8$ Chern–Simons matter theory $\mathcal T$ with level matrix $C_{E_8}$. Its $A$-twisted vacuum half-index reproduces the character exactly, while four distinguished Wilson loop half-indices agree with the remaining characters to the orders checked. Applying particle–vortex duality and gauging yields a second 3d $\mathcal N=2$ Chern–Simons matter theory $\mathcal T^\vee$ with level matrix $C_{E_8}^{-1}$. Exact superconformal-index matching supports the claim that the two theories flow to 3d $\mathcal N=4$ rank-zero mirror SCFTs. For the $(\mathcal D,D_c)$ boundary condition, the mirror map between the $A$-twisted Wilson loop half-indices of $\mathcal T$ and the $B$-twisted Wilson loop half-indices of $\mathcal T^\vee$ coincides componentwise with the Zagier transformation, realizing a Zagier duality between the complete $(E_8,T_1)$ and $(T_1,E_8)$ Nahm systems. Using the Bethe-equation formalism, we show that the corresponding $A/B$-twisted TQFT data agree up to orientation reversal and the $E_8$ framing phase $e^{-2\pi \mathrm{i}/3}$. The duality interface gives the torus bilinear $Z_{\mathcal I}(\tau)=\sum_i\chi_i(\tau)\chi_i^\vee(\tau)=\chi_{(E_8)_1}(\tau)$, where $\{\chi_i\}$ and $\{\chi_i^\vee\}$ are the corresponding five-component character bases. Thus Zagier duality is realized as part of a concrete 3d mirror and interface structure.

This paper introduces quasi-triangular Jordan D-bialgebras, which are constructed from solutions of the Jordan Yang-Baxter equation (JYBE) with invariant symmetric parts. We first develop a systematic operator approach to the JYBE by introducing the notion of extended relative Rota-Baxter operators on Jordan algebras. Such operators with their extensions are shown to induce new Jordan algebra structures. Moreover, the symmetrizer-antisymmetrizer decomposition of linear maps reduces the study of extended relative Rota-Baxter operators to both pairs of homomorphisms of Jordan algebras and relative Rota-Baxter operators, respectively. This operator framework subsequently yields a characterization of solutions of the JYBE whose symmetric parts are invariant. These operator forms are further investigated in the context of quadratic Jordan algebras and semi-direct product Jordan algebras, respectively, leading to explicit constructions of solutions of the JYBE. A factorizable Jordan D-bialgebra is then introduced as a special class of quasi-triangular ones, which naturally factorizes the underlying Jordan algebra. We prove that the Drinfeld classical double of any Jordan D-bialgebra admits a factorizable Jordan D-bialgebra structure. Finally, the operator perspective gives rise to the notion of quadratic Rota-Baxter Jordan algebras: those of zero weight yield triangular Jordan D-bialgebras, while those of nonzero weight are shown to be in one-to-one correspondence with factorizable Jordan D-bialgebras.

In this paper, we construct a class of non-weight modules over the BMS-Kac-Moody algebra by taking tensor products of finitely many irreducible modules $\Phi(\lambda,\a,\b,\r,h(t))$ with irreducible restricted modules. We obtain the necessary and sufficient conditions for these tensor product modules to be irreducible, and determine the corresponding conditions for two such modules to be isomorphic. Moreover, we compare these modules with other known non-weight modules, showing that these irreducible modules are new.

We give an explicit construction of monoidal structures on derived categories of right $A_\infty$-modules over an $A_\infty$-algebra $A$ equipped with a $B_\infty$-structure. Given such a $B_\infty$-algebra $A$, we construct an induction functor \[\iota\colon \mathcal{D}^{\rm{r}}_\infty(A) \longrightarrow \mathcal{D}^{\rm{bi}}_\infty(A)\] from right $A_\infty$-modules to $A_\infty$-bimodules and define \[M\boxtimes_A N=M\overset{\infty}{\otimes}_A\iota(N).\] We prove that $(\mathcal{D}^{\rm{r}}_\infty(A),\boxtimes_A,A)$ is a monoidal triangulated category: the unit and associativity constraints are induced by explicit quasi-isomorphisms of $A_\infty$-bimodules, including \[\iota(A)\simeq A \qquad\text{and}\qquad \iota(M)\overset{\infty}{\otimes}_A\iota(N)\simeq \iota(M\boxtimes_A N).\] We apply this construction to finite-dimensional Hopf algebras $H$, the Yoneda dg algebra $\mathcal{Y}(\Bbbk,\Bbbk)$ of the trivial $H$-module carries a natural brace $B_\infty$-structure, and hence its derived category carries the monoidal structure constructed above. We show that the Koszul duality functor \[\rm{Hom}_H(\mathcal{Y}(H,\Bbbk),-)\colon \mathcal{K}(\rm{Inj}\text{-}H)\longrightarrow \mathcal{D}(\mathcal{Y}(\Bbbk,\Bbbk))\] is triangulated lax monoidal, and that its restriction to the localizing subcategory generated by the injective resolution $\mathcal{Y}(H,\Bbbk)$ is a monoidal triangulated equivalence. If $H$ is local, this localizing subcategory is all of $\mathcal{K}(\rm{Inj}\text{-}H)$. In particular, this gives an alternative, purely algebraic proof of the monoidal equivalence conjectured by Krause and established by Benson–Krause through the classifying space $BG$. Finally, our examples recover the usual tensor product for graded commutative algebras and show that the resulting brace $B_\infty$ and monoidal structures can depend essentially on the chosen Hopf structure.

In previous work with Harman and Snowden, we found explicit formulas for units of rank ideals of the matrix monoid algebra $\mathbb{K}\left[\operatorname{Mat}_n\left(\mathbb{F}_q\right)\right]$, where $\mathbb{K}$ is an algebraically closed field with characteristic not dividing $q$. In this work, we use these explicit formulas to establish a regularized Godement–Jacquet functional equation valid for irreducible representations of $\operatorname{GL}_n\left(\mathbb{F}_q\right)$ and of the algebra $\mathbb{K}\left[\operatorname{Mat}_n\left(\mathbb{F}_q\right)\right]$. Using this functional equation we give an explicit expression for primitive central idempotents corresponding to irreducible representations of $\mathbb{K}\left[\operatorname{Mat}_n\left(\mathbb{F}_q\right)\right]$, where the characteristic of $\mathbb{K}$ does not divide $\left|\operatorname{GL}_n\left(\mathbb{F}_q\right)\right|$. Interestingly, the coefficients in this formula are closely related to degenerate non-abelian Gauss sums.

Let $G$ be a simply connected simple complex algebraic group. It is proved by Oya, Qin, and Yakimov that the quantized coordinate ring $\mathcal{O}_q(G)$ is generated by generalized quantum minors, and therefore carries a quantized cluster algebra structure, for all $G$ but type $F_4$. In this article, we settle the $F_4$ case by an argument uniform across $G_2$, $F_4$, and $E_8$. The main idea is to bootstrap the existing proof in type $E_8$, which relies on Lusztig's canonical basis of the quantum adjoint representation, and replace it with the crystal combinatorics of the quasi-minuscule representation. As a consequence, we prove that $\mathcal{O}_q(F_4)$ also has a quantized cluster algebra structure.

A noetherian ring is called G-regular when all finitely generated Gorenstein projective modules are projective. In this paper, we study rings satisfying the stronger condition that all Gorenstein projective modules are projective, which we call strongly G-regular. We show that the notion of strongly G-regular rings is closely related to that of quasi-dominant rings introduced by Takahashi and to the covariant/contravariant finiteness of a certain thick subcategory. We also answer a series of questions due to Chen in the negative, showing that the Gorenstein projective analogue of the Auslander-Ringel-Tachikawa theorem fails even for commutative local artin algebras which are weakly Gorenstein in the sense of Ringel and Zhang.

We show that isoclasses of iterated tilted algebras in type $A_n$ are in bijection with non-crossing spanning trees up to rotation on a convex n+1-gon. This is done by constructing a relationship between iterated tilted algebras up to isomorphism and exceptional sets up to isomorphic Hom-Ext quiver.

Let $G$ be a finite group acting on a Grothendieck category $\mathcal{A}$ with enough projectives, such that $|G|$ is invertible in $\mathcal{A}$. We prove a general lifting theorem for abelian model structures from $\mathcal{A}$ to its equivariant category $\mathcal{A}^G$, and establish a triangle equivalence up to retracts between the corresponding homotopy categories. We also construct a commutative square whose horizontal functors are triangle equivalences and whose vertical comparison functors are triangle equivalences up to retracts. This square relates derived functors on the lifted equivariant model categories to the equivariantizations of the derived functors on the original homotopy categories. In the module category setting, we illustrate the above results using the PGF Hovey triples, and apply them to homotopy squares induced by a Frobenius bimodule and by a stable equivalence of adjoint type.

For any symmetrizable generalized intersection matrix (GIM) $C$, we construct an acyclic valued quiver $(Q,\mathbf{d})$ endowed with an involution $\theta$. Let $\mathcal{D}$ be the bounded derived category of finite-dimensional representations of $(Q,\mathbf{d})$, and let $\Sigma$ stand for the suspension functor of $\mathcal{D}$. We show that the orbit category $\mathcal{D}/(\theta\circ\Sigma)$ carries a canonical triangulated structure and is $2$-periodic. Applying Peng–Xiao's construction to this orbit category, we prove that the GIM algebra $\operatorname{gim}(C)$ is isomorphic to the integral Ringel–Hall Lie algebra associated with $\mathcal{D}/(\theta\circ\Sigma)$. As a further application of the above machinery, we investigate elliptic Lie algebras of types $D_4^{(1,1)}$, $E_6^{(1,1)}$, $E_7^{(1,1)}$ and $E_8^{(1,1)}$. For each elliptic Dynkin diagram, we define a finite-dimensional algebra $A$ by taking an appropriate quotient of the acyclic quiver $Q$ attached to the GIM matrix $C$. From the resulting $2$-periodic triangulated categories, we build the corresponding Ringel–Hall Lie algebras, and establish a surjective Lie algebra homomorphism from each elliptic Lie algebra to its integral Ringel–Hall counterpart. This map is conjectured to be injective, and its injectivity on real root spaces is confirmed.

We introduce a new algorithm for computing Kostka-Foulkes polynomials. It arises from a combinatorial procedure that computes the transition coefficients between successive pre-canonical bases for spherical Hecke algebras, introduced by Libedinsky, Patimo, and the first author, in order to interpolate between the standard and canonical bases of the spherical Hecke algebras. Using this algorithm, we prove that the coefficients of the polynomials in the transition matrix from any pre-canonical basis to the next one has nonnegative coefficients. This establishes the positivity conjecture for pre-canonical bases and, moreover, yields a strictly stronger result that uncovers additional positivity phenomena beyond the original scope of the conjecture. As a by-product, this method yields a combinatorial algorithm for computing Lascoux's atomic decomposition of canonical basis elements. Finally, our results show that, after applying the Satake isomorphism and dualizing with respect to the Hall inner product, pre-canonical bases correspond to a new family of Schur-positive symmetric functions.

We develop the cellular structures for the endomorphism algebras of cyclotomic $q$-webs, which form a new family of quantum algebras sitting in between cyclotomic Hecke algebras and cyclotomic $q$-Schur algebras. We show that the Grothendieck group of a module category of the cyclotomic $q$-webs for $q$ generic or a root of unity is isomorphic to an integrable highest weight module over quantum affine $\mathfrak{gl}_p$; moreover, the isomorphism maps the classes of projective indecomposable modules to the canonical basis. This substantially generalizes the classic works of Lascoux-Leclerc-Thibon, Ariki, and Varagnolo-Vasserot.

We construct an explicit tilting equivalence between the incidence algebra of every rational Dyck staircase and a canonical idempotent corner of a higher Auslander algebra of type~$A$. In the coprime case, this corner identifies with the algebra $B_0$ introduced by Xing. The resulting Dyck-corner equivalence supplies the missing link in the previously known chain of equivalences and thereby proves the Chapoton-Ladkani-Rognerud conjecture for coprime positive integers. The Dyck-corner equivalence itself requires no coprimality hypothesis and is compatible with replicated algebras. Our main tool is a linear-categorical extension of the interval-tilting mechanism of Chapoton-Ladkani-Rognerud. The relative theorem applies to finite $\kk$-linear categories under finite-global-dimension assumptions on the total category and its fibers. In contrast with the incidence-category setting, it allows arbitrary finite-dimensional $\Hom$ spaces and zero composites of nonzero morphisms, and it does not require the diagonal endomorphism algebras to be semisimple. The tilting object is constructed from exact right Kan extensions of fiberwise representables. We compute its opposite indexed endomorphism category, including all forced-zero compositions, and hence its opposite endomorphism algebra. Iterating this construction one coordinate at a time yields an explicit derived equivalence between the incidence algebra of every finite coordinate staircase and an idempotent corner of a higher Auslander algebra of type~$A$. We further realize the resulting staircase derived categories as triangulated subcategories generated by product Lagrangians in partially wrapped Fukaya categories of stopped-disk symmetric products and, in the coprime Dyck case, as Fukaya-Seidel categories of symmetric Brieskorn–Pham singularities.

We give the first positive combinatorial rule for the multiplicities of irreducible $GL$-representations in the coordinate rings of type $A$ quiver orbit closures, valid for every orientation of the quiver. Previously, for the special case of varieties of complexes, work of De Concini–Strickland in the early 1980s gave an implicit description of these multiplicities. The combinatorial objects in our rule carry a crystal structure whose highest-weight elements compute these multiplicities.

We prove that two finite-dimensional linear representations of the braid group are isomorphic. One representation comes from the monodromy of the boundary Casimir connection, and the other comes from the $\iota$quantum Weyl group. Both representations are defined for any split symmetric pair $\mathfrak{k}\subset \mathfrak{g}$ and for any integrable representation of $\mathfrak{k}$. Our proof uses $(O_m,\mathfrak{so}_{2n})$ spin Howe duality, so it is only for the pair $\mathfrak{so}_m\subset \mathfrak{sl}_m$.

We construct relative modular categories from the weighted representation theory of the unrolled quantum group of the orthosymplectic Lie superalgebra $\mathfrak{osp}(2 \vert 2n)$.

We prove that a generic spherical parameter for the graded affine Hecke algebra with equal parameters is unitary if and only if its normalized intertwining forms are positive on the reflection representation and on the irreducible constituents of its second symmetric power. The proof combines a signature formula for the reflection representation, Yu's uniform two-wall operation, a simply-laced Jantzen recursion, and a root-poset analysis. In types B_n and C_n, we prove more sharply that the reflection representation V and the traceless-diagonal constituent of Sym^2(V) suffice. We also give a shorter second proof when arbitrary Weyl-group types are allowed. The consequence of the first proof (via Barbasch-Vogan petite K-types) is that the generic spherical unitary dual of a Chevalley group over the real numbers or a nonarchimedean local field, and for simply-laced groups and complex symplectic groups also over the complex numbers, is independent of the field. Moreover, the answer has a simple uniform description as the Weyl conjugates of a disjoint union of a power of 2 alcoves in the affine Weyl group arrangement in the fundamental Weyl chamber, where the exponent is the matching number of the Dynkin diagram of the subsystem of short coroots.

We characterize four groups of Clifford circuits for Calderbank–Shor–Steane (CSS) codes that are relevant to fault-tolerant logical operations. First, we show that every code-preserving Clifford circuit is a product of Z-diagonal circuits, composed of S and CZ gates, and their X-basis analogues. Second, we define the two-fold transversal group, generated by depth-one two-local code-preserving circuits, and show that each of its elements can be expressed as a product of layers consisting of either Z-diagonal, X-diagonal, or CNOT gates. As a corollary, every transversal gate is a product of three transversal diagonal circuits; for connected non-self-dual codes, two such circuits suffice. We further show that every code-preserving automorphism circuit, consisting of single-qubit Clifford gates and permutations, has a normal form comprising a Hadamard layer, a permutation, and two diagonal circuits. We also define a two-fold automorphism group, in which a depth-one two-local circuit may be code-preserving up to a permutation, and show that its logical image can be larger than that of the two-fold transversal group. For 231 CSS codes, we provide explicit generators and determine the logical image of the two-fold transversal group. We tabulate 173 codes whose full logical Clifford group is generated by two-fold-transversal circuits, including codes of distances 3, 4, 5, 6, 8, 10, 12, and 13, with respective best rates 2/5, 3/4, 34/77, 17/28, 14/25, 1/27, 3/56, and 1/33. We construct families of CSS codes from bipartite grids, cut-complements, and quadrics, and census the self-dual codes invariant under rank-3 permutation groups. Many of these codes realize the full logical Clifford group in this way.

For coprime $1<a<b$, let $M_n^{a,b}(\mathbb{F}_q)$ be the set of commuting pairs of nilpotent $n\times n$ matrices over $\mathbb{F}_q$ with $X^a=Y^b$. Huang, Jiang, and Oblomkov assembled their orders as an Eulerian $q$-series $Z_{a,b}(q)$. They conjectured that it is an explicit product $P_{a,b}(q)$ involving Jacobi's theta function and Dedekind's eta-function, implying the threefold equality $$\underbrace{\prod_{n\ge1}(1-q^n)\cdot\Biggl(\sum_{n=0}^{\infty}\frac{|M_n^{a,b}(\mathbb{F}_q)|}{|\mathrm{GL}_n(\mathbb{F}_q)|}\Biggr)\Biggr|_{q\mapsto q^{-1}}}_{\text{point count}}\;=\;\underbrace{Z_{a,b}(q)}_{q\text{-series}}\;=\;\underbrace{P_{a,b}(q)}_{\text{theta quotient}}$$ If true, the point count on $X^a=Y^b$ is essentially a modular function on $\Gamma(a+b)$. The conjecture is layered in $a$, with an identity for each $b$. The $a=2$ layer is classical, including identities of Rogers–Ramanujan and Andrews–Gordon. For $a\geq3,$ nothing was known. We prove the $a=3$ layer in full: a new infinite family of Rogers–Ramanujan identities, and a geometric origin for Warnaar's products. AxiomProver verified these new identities in Lean assuming existing literature.