Papers

Let $H$ be the generic Hecke algebra over $\mathbb{Z}[\mathbf{q}^{\pm 1}]$ associated to a split Kac–Moody group $G$, arising as the deformation of the group algebra of its Weyl group $W$. The cocenter $\overline{H} = H/[H, H]$ encodes the trace and character theory of $H$, playing a fundamental role in representation theory and harmonic analysis. Through deep combinatorial results on cyclic reductions in Coxeter groups, each conjugacy class $\mathcal{O}$ of $W$ determines a canonical element $T_{\mathcal{O}}$ in $\overline{H}$, and these elements are known to span the cocenter. However, establishing their linear independence has remained an open problem outside of finite and affine types. In this paper, we solve this problem: the canonical elements form a $\mathbb{Z}[\mathbf{q}^{\pm 1}]$-basis of the cocenter $\overline{H}$. Our approach differs from earlier representation-theoretic methods in finite and affine types. To construct explicit functionals that separate all conjugacy classes, we develop a new framework based on re-normalized orbital integrals. This framework synthesizes parabolic induction, traces of infinite-dimensional bimodules, and Kac–Moody harmonic analysis into an almost-dual basis for the cocenter. As a key local ingredient, we establish a generic duality theorem for finite groups of Lie type relating the cocenter to regular semisimple conjugacy classes, and determine precisely when this pairing is non-degenerate. Finally, we deduce the existence and uniqueness of generic class polynomials for $W$, and prove a uniform “dimension=degree” theorem for basic Deligne–Lusztig varieties of the split Kac–Moody group $G$.

In a 2023 survey paper, Gabriel Navarro posed the following problem: Given an irreducible character $\chi$ of some solvable group $G$ where $\chi$ either is 2-rational or has odd degree, does there exist some $g \in G$ such that $\mathbb{Q}(\chi(g)) = \mathbb{Q}(\chi)$? The answer was recently shown to be no in general by Ulrich Thiel. However, when $\chi$ is further assumed to be factorizable as a product of $p$-special characters, we are able to show that the answer is yes.

Let $(\mathcal{S},\mathcal{Z},\mathcal{V})$ be a premutation triple in an extriangulated category. Set $\mathcal{I}=\mathcal{S}\cap\mathcal{Z}=\mathcal{Z}\cap\mathcal{V}$. Iitsuka introduced premutation triples as a unified framework for various mutation constructions and showed that the quotient category $\mathcal{Z}/[\mathcal{I}]$ admits a canonical pretriangulated structure. In this paper, we prove that the compatibility condition $(\mathrm{MT4})$ is not only sufficient but also necessary for this structure to be triangulated. Therefore, $(\mathrm{MT4})$ precisely characterizes when Iitsuka's canonical pretriangulated structure becomes triangulated. Furthermore, we prove that, for concentric twin cotorsion pairs, the compatibility condition used by Nakaoka in the construction of triangulated quotients is equivalent to $(\mathrm{MT4})$. Finally, we construct an explicit reducible triple which is not a mutation triple. This gives a negative answer to a question posed by Iitsuka.

We develop an explicit operator and coefficient calculus for admissible (g,K)-modules of SU(2,2), where g = sl(4,C) and K = S(U(2) x U(2)). The K-types are indexed by triples (n,k,m). We introduce operators A_delta and B_delta associated with the noncompact roots, together with auxiliary operators Q and R. We establish their commutator relations, the central coefficient relations, and, in the unitary setting, the corresponding adjoint and norm-factor identities. This reduces part of the analysis of (g,K)-modules to the study of scalar coefficients arising from compositions of these operators. The coefficient formula applies to weights of multiplicity one, including all boundary weights. We derive necessary sign restrictions and explicit relations constraining the coefficient space. These coefficients also determine irreducibility. Several families of unitary (g,K)-modules are analyzed according to the minimal value of n+m. For N=0, the construction yields two-parameter families of unitary modules and describes reducibility when certain coefficients vanish. For N>0, the method produces both larger families of K-types and multiplicity-free ladder-type families. The resulting patterns are compatible with the known description of the unitary dual of SU(2,2) due to Knapp and Speh. The paper provides an explicit framework for recovering unitary (g,K)-modules from their K-type structure and suggests a possible approach to other real reductive groups.

We prove Hitchin's conjecture for $\mathfrak{sl}_n$ in every odd prime degree. More precisely, let $p=2k+1$ be prime and let $n\geq k+1$. Every nonzero primitive element $\alpha\in\Lambda^p(\mathfrak{sl}_n^*)^{\mathfrak{sl}_n}$ is nonzero on the top exterior power of the unique $p$-dimensional irreducible submodule of $\mathfrak{sl}_n$ for the adjoint action of a principal $\mathfrak{sl}_2$-subalgebra. The conjecture also holds for types $B_\ell$ and $C_\ell$, with $\ell\geq2$, in every prime degree $p=4r-1$ with $1\leq r\leq\ell$.

The Katz-Long-Moody construction associates to a representation of the semidirect product of a free group and a braid group, defined by the Artin action, and a nonzero parameter a new representation of the same group. On the pure braid group it corresponds to Haraoka's multiplicative middle convolution for KZ-type equations. For unitary input and a convolution parameter on the unit circle other than one, the construction equips the quotient representation with a canonical non-degenerate invariant Hermitian form. We give a closed formula for its signature in terms of the eigenangles of the input local monodromies, the eigenangles of their ordered product, and the convolution parameter. The formula accounts for the kernel of the form before passage to the quotient and for signature changes at resonant parameters. It determines precisely when the induced form is definite, answering the definiteness problem posed in the companion paper. Definiteness implies unitarizability of the output representation; the converse holds when that representation is irreducible. The proof uses elementary linear algebra: a determinant identity, explicit block-pivot formulas, and an inertia formula for sums of Cayley transforms of unitary matrices. As applications, we compare the rank-one construction explicitly with Haraoka's invariant form for Pochhammer systems, recover the Gauss case of the Beukers-Heckman interlacing criterion, and determine the definite parameter intervals for the Hecke and Temperley-Lieb specializations.

In this paper, we construct Gelfand–Tsetlin bases for representations of $\mathrm U_q(\mathfrak{gl}_n)$ at roots of unity. As an application, we obtain explicit bases for certain baby Verma modules of $\mathrm U_q(\mathfrak{gl}_3)$ and determine their composition series.

Let $H_n(q)$ be the generic Iwahori–Hecke algebra associated to the symmetric group $\mathfrak{S}_n$ of degree $n$ and let $\mathscr{Y}_{n,i}$ be the sum of standard basis of $H_n(q)$ with Coxeter length $i$ for $1\leq i\leq \binom{n}{2}$. We show that $\mathscr{Y}_{n,1}$ and $\mathscr{Y}_{n,2}$ are noncommutative whenever $n\geq 4$, which disproves Doikou's Conjecture on the commutativity of the shuffle elements of $H_n(q)$ in (Nuclear Physics B 1029 (2026): 117532). We also show that the matrix of the operator of the left multiplication by $\mathscr{Y}_{n,i}$ in $H_n(q)$ is $q$-symmetric for all $i$.

In this paper, we study finite dimensional representations of shifted quantum affine algebras of type A. We give an explicit description of the tensor product of simple evaluation modules of the quantum loop algebra and a one-dimensional representation of the shifted quantum affine algebra under the separation condition. As a consequence, we give the q-characters of some finite dimensional simple modules of the shifted quantum affine algebra.

We show that Jacobi-finite potentials need not exist on finite $2$-acyclic quivers. Our main tool is a matrix-valued Golod–Shafarevich–Vinberg inequality for quotients of completed path algebras by finitely many, possibly nonhomogeneous, topological relations. Applied to cyclic derivatives, it yields a potential-dependent obstruction to the finite-dimensionality of completed Jacobian algebras. We then construct a purely quiver-level criterion excluding every Jacobi-finite potential on a given quiver, and exhibit a family of quivers for which every potential has an infinite-dimensional Jacobian algebra.

Let $F$ be a nonarchimedean local field of characteristic zero and let $n\geq2$. For $r=n,n+1$, let $\Pi_r$ be an irreducible tempered representation of ${\rm GL}_r(F)$ of conductor one and with trivial central character. We evaluate the Rankin–Selberg integral of opposite newforms in $\Pi_{n+1}\times \Pi_n$ explicitly and show that its central value is nonzero. As an application, this implies a case of Disegni–Zhang's conjecture on the nonvanishing of local relative characters.

In this paper, we develop a general theory of quasi-Whittaker supermodules over Lie superalgebras induced from an arbitrary ideal. We determine the quasi-Whittaker vectors in universal supermodules, establish an irreducibility criterion, and classify several families of irreducible supermodules. The odd part produces a new irreducibility phenomenon absent from the Lie algebra setting. As applications, we determine all irreducible quasi-Whittaker supermodules over the $N=1$ super Schrödinger algebra and the $N=1$ $\frac{3}{2}$-conformal Galilei superalgebra, and over the complete spectrum-generating superalgebra in a special case.

In this paper, we study the extensions of KLRW algebras to tensor products of Verma module representations of $\mathfrak{sl}_2$. Our motivation is to construct a theory of Khovanov homology for knot complements in $S^3$ (and which also categorifies the Gukov-Manolescu two-variable series for knot complements), which will be done in the second part of this work. We construct the categorification of R-matrices for Verma modules as functors given by derived tensor products with diagrammatic bimodules and explicitly compute their projective resolutions. We also prove these braiding functors induce an action of the braid group on the relevant categories. Then, we describe how to incorporate strands in finite-dimensional representations of $\mathfrak{sl}_2$, thereby establishing functors that serve as the Khovanov homology on a braid complement. In the case of the unknot, this gives knot homologies in $S^1\times D^2$, which we compare to Annular Khovanov Homology through several examples and show they are very closely related, conjecturing they are of the same dimension. We conclude with a proposal for the categorification of the cups and caps of Verma module colored strands, which we build upon in the next paper.

Let $A$ be an Artin algebra and let $T$ be a $\tau$-tilting right $A$-module. Put $C=A/{\rm Ann}_A(T)$. We prove that finite little finitistic dimension of $C^{\rm op}$, together with a finite range of self-extension vanishing for $T$, forces $T$ to be a classical $1$-tilting module. We also show that if the finitistic conjecture holds for all Artin algebras then the self-orthogonal $\tau$-tilting conjecture holds for all Artin algebras. Moreover, a counterexample to the self-orthogonal $\tau$-tilting conjecture will give a counterexample to the finitistic dimension conjecture. As an application, the self-orthogonal $\tau$-tilting conjecture holds for Artin algebras with radical cube zero.

For a partition $\lambda$, let $G_\lambda^{(\beta)}$ be the stable $\beta$-Grothendieck polynomial attached to $\lambda$. Each homogeneous component of the $\beta = 1$ specialization $G_\lambda^{(1)}$ is Schur-positive and hence positive in the fundamental basis of quasisymmetric functions. For $m\ge|\lambda|$, let $G_{\lambda,m}^{(1)}$ be the homogeneous degree $m$ component of $G_\lambda^{(1)}$. In this paper, we first give a direct proof of an expansion of $G_{\lambda,m}^{(1)}$ in the fundamental basis in terms of standard set-valued tableaux. We then use these tableaux as a basis to define a module of the $0$-Hecke algebra and show that the quasisymmetric characteristic of the resulting module is $G_{\lambda,m}^{(1)}$. We further show that this module decomposes as a direct sum of weak Bruhat interval modules.

Let $\mathfrak L$ be a Frobenius maximal parabolic subalgebra of $\mathfrak{sl}_n$. For any $F\in\mathfrak L^*$ for which the Kirillov form $B_F(x,y)=F([x,y])$ is non-degenerate, let $\widehat F$ denote the associated principal element. We prove that the multiplicities of the eigenvalues of $\operatorname{ad}_{\widehat F}$ on $\mathfrak L$ form a unimodal sequence symmetric about $\frac12$. We also prove that the multiplicities of the eigenvalues of $\operatorname{ad}_{\widehat F}$ on $\mathfrak{gl}_n$ form a unimodal sequence symmetric about $0$. The proof relates the ranked meander associated to $\mathfrak L$ to the order ideals of a related fence poset through Panyushev reduction. The known unimodality of the rank polynomial of the fence poset implies that of the meander, which in turn determines a good grading of $\mathfrak{gl}_n$ in the sense of Elashvili and Kac. We prove that this grading coincides with that induced by the principal element and that the pyramid associated to this grading may be filled in such a way that its good element $e$ lies in $\mathfrak L$. The two unimodality results then follow from the injectivity properties of $\operatorname{ad}_e$ coming from the good grading and the duality induced by the bilinear form $B_F$.

We study the restricted fan cut inside the dual fundamental Weyl chamber by the weights of a $27$-dimensional minuscule representation of $E_6$; the two such representations are dual and give the same arrangement. Only $11$ of the $27$ weights have kernels meeting its interior, and we prove that they determine the entire fan. It has exactly $14$ chambers and $18$ extreme rays, every chamber is a six-dimensional simplicial cone, and we determine all facets, rays, and incidence relations. The chamber count was previously obtained by Diaconescu and Entin; the simplicial structure, extreme rays, and incidence data are new. The geometry of the $27$ lines on a cubic surface then explains and organizes the resulting chamber architecture. We also enumerate all faces, compute both characteristic polynomials—the arrangement is not supersolvable—together with lattice indices and projective chamber volumes, and describe the oriented matroid. Our main result is representation-theoretic. A distinguished $14$-element visible subposet of the minuscule $\mathbf{56}$ of $E_7$, defined entirely inside $E_7$, has Hasse diagram equal to the chamber adjacency graph of the $E_6$ arrangement. Three canonical $7+7$ splittings of it, of types $A_7$, $D_7$, and $E_7$, are the visible traces of Levi-center $\mathfrak{u}(1)$-charge decompositions of the $\mathbf{56}$ and reproduce the three level-$8$ decompositions on the $E_6$ side. More strongly, the simple-root labels on its covers, transported by minimal-length coset representatives, recover chamber by chamber all six facets and, globally, the $11$ active weight hyperplanes and the boundary walls of the dual Weyl chamber. Thus the $E_7$ shadow records not merely the chamber graph but, once matched with the independent $E_6$ classification, the full local wall architecture of $I(E_6,\mathbf{27})$.

We characterize the standard modules of $\GL_{2m}(\C)$ that are distinguished by $\GL_m(\HH)$. Let $\delta_1,\dots, \delta_{2m}$ be characters of $\mathbb{C}^\times$. Assume that $S = \delta_1 \times \cdots \times \delta_{2m}$ is a standard module of $\GL_{2m}(\mathbb{C})$. For each $i$, define ${\delta_i^*} (z) = \delta_i(\overline{z})^{-1}$ for $ z \in \mathbb{C}^\times$. In particular, we conclude that a standard module for $\GL_{2m}(\mathbb{C})$ is distinguished by $\GL_{m}(\mathbb{H})$ if and only if there exists an involution $p\in S_{2m}$ without fixed points such that $\delta_{p(i)}=\delta_i^*$ for every $i$. We first verify the hypotheses of the multiplicity estimate theorem of Suzuki and Tamori in \cite{ST}. The orbit calculation of Matringe, Offen, and Yang in \cite{MOYglobal} then gives the necessary condition and a dimension bound. Then local intertwining periods prove sufficiency.

We establish local Ocneanu rigidity for separable algebra objects in Hom-finite monoidal categories C over an algebraically closed field k. A separability morphism contracts the first two Hochschild cohomology groups without requiring an abelian ambient category. Separable algebra structures and homomorphisms from separable sources have open algebraic-group orbits, and affine Bezout estimates give effective finiteness results. For Frobenius subalgebras, exchange relations replace the source and embedding data by a single self-dual idempotent. We prove that their separable inner-conjugacy classes are open in the exchange locus, yielding a bound of 2^(dim_k(End_C(X))) for any ambient Frobenius algebra X; connectedness gives the same bound on the actual number of subalgebras. As an application, we obtain an effective form of the Etingof-Walton finiteness theorem: every finite-dimensional semisimple Hopf algebra H over C has fewer than 2^(dim_C(H)) left coideal subalgebras. In the unitary setting, we bound E-compatible intermediates of C*-algebra and von Neumann algebra inclusions up to unitary conjugacy, under finite-index and finite-center hypotheses. For irreducible finite-index subfactors N < M, we improve the 9^[M:N] bound of Bakshi-Das-Liu-Ren to fewer than 2^[M:N] intermediate subfactors.

Fix an integer $d\leq-2$. We characterize Hom-finite algebraic triangulated categories admitting a $(-d)$-simple-minded system as stable categories $\underline{\mathrm{CM}}(B)$ of proper $(-d)$-self-injective non-positive dg algebras; equivalently, each admits a $d$-stable locally finite strictly positive dg model whose cosingular dg quotient recovers the chosen enhancement. Under a $d$-Calabi–Yau hypothesis, we give a characterization theorem for acyclic negative cluster categories in terms of the finite graded extension algebra of a simple-minded system. At chain level, Hochschild and reduced cyclic localization identify right $(d+1)$-Calabi–Yau structures on the finite-dimensional part with normalized right $d$-Calabi–Yau structures on the cosingular quotient. Finally, when the Koszul dual is proper, the Brav–Dyckerhoff evaluation morphism is a quasi-isomorphism of mixed complexes, yielding left–right Calabi–Yau symmetry.

The Motzkin subproduct system (SPS) is constructed from the Jones Wenzl idempotents of the Motzkin algebras, which generalizes the Temperley Lieb SPS. The simplest Motzkin SPS, which is not a Temperley Lieb SPS, is constructed from a 3 dimensional Hilbert space. We explicitly describe the spectrum of the corresponding Cuntz Pimsner algebra, which remarkably admits only irreducible representations of dimensions 1 and 2, and can thus be viewed as a mildly quantum space. We moreover analyse representations of the Richard Thompson groups and of the Cuntz algebra that are associated to this spectrum.

We study Ringel's brick chain filtrations over finite-dimensional algebras: filtrations whose factors are filtered by copies of individual bricks, ordered so that morphisms from earlier bricks to later ones vanish. First, using submodule varieties, we prove that the largest submodule of a fixed module belonging to a torsion class takes only finitely many values as the class varies, answering Pavón's question. We deduce finiteness of brick chain filtrations and the bound $2^{d^2}$ for modules of dimension $d$. For $\tau$-tilting finite algebras, we bound their number by the multinomial coefficient determined by simple composition multiplicities. Finally, we construct families of bricks over the three-arrow Kronecker algebra whose filtration counts grow exponentially in the square of composition length. We prove this growth using a Littlewood–Richardson formula for submodule counts and the hook-length formula. In particular, the counts eventually exceed the factorial of composition length. These results answer Ringel's two questions.

We provide a categorification for the hybrid Kazhdan-Lusztig bases of the Hecke algebra by constructing and investigating a series of new stratified structures on the principal block of BGG category $\mathcal{O}$.

We study universal derived equivalences between diagram categories indexed by finite posets. Starting from a construction of Ladkani, we give an intrinsic criterion for determining when a finite poset admits a decomposition to which this construction can be applied. This leads to the notion of an admissible cut, formulated entirely in terms of the order structure of the poset. Our main result therefore provides a method for producing, from a given finite poset admitting such a cut, a new poset that is universally derived equivalent to it. The construction is reversible once the partition is retained, so that the original mixed order relations can be recovered from the transformed poset. As applications, we show that every finite poset of height at most one is universally derived equivalent to its opposite, give a criterion characterizing source-to-sink transformations at minimal elements, and recover the universal derived equivalence of all orientations of a finite tree through sequences of such local transformations. These results are also applied to persistence modules indexed by finite posets and to extensions obtained by adjoining further finite parameters.

We use the full multiplication formula to study syzygy and suspension of cluster characters. In stably 2-Calabi–Yau Frobenius categories, syzygy and cosyzygy induce inverse automorphisms of localized character algebras; in Hom-finite 2-Calabi–Yau triangulated categories, the same uniqueness principle transports all cluster characters under suspension. The latter gives our Auslander–Reiten $F$-polynomial identity. No mutation reachability is required. We extend positroid twist identities to all objects, recover the Grassmannian and unipotent-cell formulas, identify partition-function algebras before boundary localization, and derive tropical covariance when the relevant tropical coordinate map is a homeomorphism.

We consider covariant algebras attached to certain hybrid qudit spaces. We describe these algebras in terms of generators and relations, and we show they are flat deformations of certain explicitly described semigroup algebras.

The periodicity conjecture asks whether a finite-dimensional algebra is periodic whenever all its simple modules are periodic. We construct a $36$-dimensional counterexample whose simple modules have period four. The algebra has a $2$-dimensional nonperiodic module, which shows that the algebra itself is not periodic.

We define a derived stack $\mathscr{Rep}_n(X)$ which generalizes the assignment $A\leadsto \operatorname{Rep}_n(A)$ to an algebra of its derived $GL_n$-representation variety of arXiv:1112.1449 from algebras $A$ to perfect stacks $X$ over characteristic 0 fields. In the case of a quasi-projective classical scheme $X$, we show that $\mathscr{Rep}_n(X)$ admits a subfunctor $\operatorname{Quot}^{n,\text{fr}}_{\mathscr{O}_X^n}(X)\subset \mathscr{Rep}_n(X)$, which is in fact represented by a derived scheme almost of finite type. We further construct a Fourier-Mukai integral transform between the derived categories of quasi-coherent sheaves on $X$ and $\operatorname{Quot}^{n,\text{fr}}_{\mathscr{O}_X^n}(X)$ which induces a trace map at the level of Hochschild homology generalizing the trace morphism constructed in arXiv:1112.1449 (for finitely presented commutative algebras). We show that the subfunctor $\operatorname{Quot}^{n,\text{fr}}_{\mathscr{O}_X^n}(X)$ is a derived enhancement of the framed locus of the Quot scheme of points and that this derived scheme is a $GL_n$-torsor over the stack of coherent length $n$ torsion sheaves. Hence, this stack is an analog of the derived character stack for quasi-projective schemes, and we show that for smooth, Calabi-Yau $X$, it inherits a shifted symplectic structure in the sense of arXiv:1111.3209 from the one constructed in arXiv:1812.11913 on the moduli stack of perfect complexes with proper support. This structure is shown to give a generalization of the classical symplectic structure on character varieties of surfaces of genus 1 constructed by Goldman \cite{gold}.

We give a complete characterization of the weak Lefschetz property (WLP) for monomial complete intersections over a field of positive characteristic. Richard Stanley observed that in characteristic zero WLP, and in fact the strong Lefschetz property (SLP), holds for all degree sequences, as a consequence of the hard Lefschetz theorem, and the same result was explained by Junzo Watanabe using the representation theory of $\mathfrak{sl}_2$. In positive characteristic, many partial results are known, most notably the classification for constant degree sequences due to Brenner–Kaid and Kustin–Vraciu. Our approach is based on a cohomological and representation-theoretic interpretation of WLP. Combined with an analysis of cohomology characters, this leads to a complete numerical criterion for WLP, expressed by simple inequalities involving the $p$-adic digits of the exponents. We also give a new proof of the known classification of SLP using Renaud's algorithm for multiplication in the Green–Han–Monsky ring.

We prove a modified invertibility property for the parabolic defects introduced in arXiv:2102.12283, arXiv:2505.14836. Namely, we show that the Borel defect cancels its dual, up to insertion of an invertible Weyl defect and up to restriction to an open-subcategory. The main result holds for an arbitrary reductive group $G$ – for illustration we give extended computations for $G=\mathrm{SL}_3$. Along the way, we introduce a defect skein theory with defects in both codimension one and two, which is compatible with gluing of defect 3-manifolds and defect surfaces. Our results provide a potential defect skein theoretic construction of certain standard charts which appears frequently in quantum cluster varieties associated to surfaces.