Papers

We study quiver Demazure algebras through residue covers and root-power quotients. The cover determined by the zero-root subsystem identifies the additive algebra with a finite fixed algebra of Sauter's generalized quiver Hecke construction. Her basis theorem then gives additive PBW in every finite Weyl type, for arbitrary admissible residue data and uniform quivers. A sequence-cover argument gives global multiplicative PBW in simply-laced type. Under a directed-path condition, root-power quotients are finite dimensional and the formal exponential identifies their additive and multiplicative forms. For arrow-free quivers, we reduce mixed-residue quotients to matrix algebras over finite skew group rings, obtain sharp nonvanishing criteria from coinvariant geometry, and determine their simple modules. In a mixed type-$D_4$ family we compute an explicit coefficient presentation, dimensions, Loewy lengths, and Cartan matrices. We also specify the direct and finite-cover comparisons with Liu's algebras.

We describe a Segre-Veronese construction of polynomial solutions to the interpolation conditions of the numerators of tree-level MHV gravity amplitudes given by Hodges' formula. The five- and six-point Hodges numerators arise from Segre determinants, while for every $n\geq7$ suitable Segre determinants lead to solutions linearly independent of the Hodges numerator. The existence of such independent elements for $n=7,8$ was recently established by Mai and Zhang.

We study the Iwahori Gaiotto conjecture for cotangent types and its application to the representation theory of the (mixed, small, and Lusztig) quantum supergroups. Using local and global models and SW-Zastava spaces, we construct factorization functors and compare the Iwahori Gaiotto category with the category $O$ of the mixed quantum supergroup. We then study the images of intersection cohomology sheaves and the metaplectic Hecke action. In type A, we describe the finite and affine monodromic categories, the corresponding Yokonuma–Hecke actions, and their role in the Kazhdan–Lusztig algorithm. The last section treats the pointwise purity.

We study the truncation $(-)^{\leq0}$ of dg categories. We first show that given a dg category $\mathscr{C}$ with shifts, for example a pretriangulated dg category, the canonical functor $\mathscr{C}^{\leq0}\to\mathscr{C}$ is a localization whose kernel is compactly generated by the $0$-th cohomology. Next we demonstrate that the derived category of the truncation serves as a triangulated analogue of the Auslander's category of coherent functors over abelian categories. We give a description of the Auslander-Reiten-Serre duality in terms of the inverse dualizing bimodule of the truncation. Building on truncations, we introduce the notion of connective resolutions of dg categories, defined as a localization functor from a connective dg category. This notion encompasses non-commutative resolutions of algebras in module categories, cluster tilting objects in triangulated categories, and also the truncation as the universal connective resolution. We give a sufficient condition for an object in a triangulated category to give a connective resolution of its enhancement in terms of finiteness of resolution dimension. Furthermore, we show that every proper dg module over a connective dg algebra is a direct summand of a dg module giving a connective resolution. Consequently, for every proper connective dg algebra, its bounded dg derived category has a connective resolution. Finally, we give some explicit description of the truncation for some categories including the derived and cluster categories of a Dynkin quiver, and the Yoneda category of a finite dimensional algebra.

We determine the first extension groups between simple restricted modules for the restricted simple Lie algebras of Cartan type over an algebraically closed field of characteristic $p\ge5$, which answers the restricted case of a problem of Benkart and Feldvoss. Extensions from parabolically induced modules are computed by finite restricted-cohomology complexes of the grading parabolic, and for the exceptional modules we obtain closed formulas in all four families. Neither the trivial module nor any nontrivial exceptional simple has self-extensions. In characteristic seven we determine the quiver of $S(3;1)^{(1)}$, which has 49 vertices and 566 arrows.

We consider the type AIII (quasi split) quantum symmetric pair subalgebras $U_q'$. We define a new Cartan subalgebra in one of the two subfamilies. Together with Letzter's Cartan subalgebra in the other, we construct a triangular decomposition of $U_q'$ via the Letzter map. Then we define Verma modules as induced modules and prove that finite dimensional simple $U_q'$-modules are quotients of our Verma modules. These treatments are uniform in both subfamilies. In the second part we define a diagrammatic category $Web^B$ that controls polynomial representations of $U_q'$. We prove a multiplicity-free decomposition of the quantum exterior powers $\bigwedge^k\mathbb{V}$ as a $U_q'$-module, using a explicit eigenspace decomposition of certain dot morphisms in $Web^B$. As applications, we deduce the multiplicity-freeness of the involved anti-spherical Hecke module. For $\bigwedge^k\mathbb{V}$, we explicitly determine the highest weight vectors and their weights, thereby determining the isomorphism classes of their irreducible summands. Finally, we determine the kernel of the $Web^B$-action on certain finite dimensional representation category of $U_q'$.

We construct an explicit rational \(\mathrm{SL}_2(\mathbb C)\)-action on the underlying vector space of the rank-two \(\beta\gamma\)-system. The construction is motivated by Creutzig's branching decomposition with respect to \(L_{-1}(\mathfrak{sl}_2)\) and a rank-one Heisenberg vertex algebra. We realize the raising and lowering operators as residues of lattice-type fields, prove that they preserve the original \(\beta\gamma\)-system despite being defined on localizations, and establish their commutation relations and integrability. The resulting action commutes with the affine \(L_{-1}(\mathfrak{sl}_2)\)-action and all nonzero Heisenberg modes, while the Heisenberg zero mode serves as its Cartan operator. Each affine multiplicity space is thereby identified with \(V_n\otimes\mathcal F_{\mathrm{osc}}\), where \(V_n\) is the irreducible \(\mathrm{SL}_2(\mathbb C)\)-module of highest weight \(n\) and \(\mathcal F_{\mathrm{osc}}\) is the Heisenberg oscillator Fock space. Thus the Heisenberg charge decomposition is realized as the weight decomposition of \(V_n\). The infinitesimal action is not by vertex algebra derivations and does not preserve the symmetric conformal grading.

We study the integer sequence that enumerates cokernel-closed, idempotent split, full, additive subcategories of the category of finite dimensional complex representations of a uniformly oriented $A_n$ quiver. Using a combinatorial model, we describe a non-obvious connection of this sequence to Catalan numbers and derive an implicit recurrence. We also describe some basic properties of the lattice underlying the sequences, in particular, we give an explicit description of the meet irreducible elements of that lattice.

Let $h_n$ denote the complete homogeneous symmetric function of degree $n$, $s_\nu$ the Schur function indexed by a partition $\nu$, and $\langle\cdot,\cdot\rangle$ the Hall inner product. The plethysm coefficient $\langle h_n[h_m], s_\nu\rangle$ is the multiplicity of the Weyl module $W_\nu(\mathbb{C}^N)$ in $\mathrm{Sym}^n(\mathrm{Sym}^m(\mathbb{C}^N))$ whenever $\nu$ has at most $N$ parts. For a partition $\lambda$, let $\lambda[mn]=(mn-|\lambda|,\lambda_1,\lambda_2,\dots)$. Bowman and Paget proved that $\langle h_n[h_m], s_{\lambda[mn]}\rangle$ is independent of $m$ and $n$ once $m,n\ge|\lambda|$. We prove that this coefficient is already constant for $m\ge\lambda_1$ and $n\ge\lfloor(|\lambda|+\lambda_1)/2\rfloor$, and that its stable value equals $\langle s_\lambda, H[H-1-h_1]\rangle$, where $H=\sum_{i\ge0}h_i$; the proof uses the skewing operators adjoint to multiplication by complete homogeneous symmetric functions. We give a short new proof of the Bowman-Paget theorem based on stability of vector-partition functions, which count the ways of writing a vector of non-negative integers as an unordered sum of non-zero such vectors (its parts). Using a Vandermonde-type determinantal identity, we deduce that the stable value is a signed sum of numbers of vector partitions all of whose parts have entries summing to at least 2. The same method gives a short proof of Brion's stability theorem. We show that the stable value equals the multiplicity of $W_\lambda(\mathbb{C}^k)$, where $k=\lfloor|\lambda|/2\rfloor$, in an explicit representation of $GL_k(\mathbb{C})$ independent of $m$ and $n$, obtained by modifying the polynomial induction of Narayanan, Paul, Prasad and Srivastava. Finally, we prove an explicit positive formula for the stable value, which counts the number of lattice points in a polytope, when $\lambda_1\le 4$.

Let $K/\Qp$ be a finite extension, $\bG$ a connected reductive group over $K$, and set $G = \bG(K)$, considered as a $p$-adic Lie group. We show first that the Bernstein center $\cC_G$ of the category of solid locally $\Qp$-analytic representations of $G$ is isomorphic to the center of the locally analytic distribution algebra $D^\la(G)$. We then consider the Emerton-Gee stack $\frX_{n,K}$ of rank-$n$ $(\vphi,\Gamma)$-modules over the Robba ring for $K$ and determine its connected components. The latter are in canonical bijection with the primitive idempotents of the ring $\cC_{\GL_n(K)}$. Moreover, under the assumption that the ring of global functions on $\frX_{n,K}$ has no non-zero locally nilpotent elements, we show that this ring is isomorphic to a Fréchet completion of $\cC_{\GL_n(K)}$.

We formulate a version of Schur–Weyl duality for the current Lie algebra $\mathfrak{gl}_V[x,y]=\mathfrak{gl}_V\otimes\mathbb{C}[x,y]$. Under this duality, the Garsia–Haiman modules of the $N!$-conjecture become cyclic and cocyclic $\mathfrak{gl}_V[x,y]$-modules, and we describe them as iterated fusion and cofusion products of tautological $\mathfrak{gl}_V$-modules. We show that Haiman's $N!$-theorem for a diagram is equivalent to the coincidence of the fusion and the cofusion filtrations on the tensor product of the local Weyl modules attached to its rows, and that the fusion and cofusion products of Garsia–Haiman modules are associative. Let $\lambda$ and $\mu$ be Young diagrams obtained by removing two different corners from the same diagram. We construct an iterated fusion product of $S^2V$ with local Weyl modules, which is a quotient of the common quotient of the Garsia–Haiman modules of $\lambda$ and $\mu$, fits into short exact sequences with each of them, and has Butler's intersection polynomial as its character. This gives a representation-theoretic proof of Butler's conjecture. The same construction gives lower bounds for the dimensions in the $N!/k$-conjecture.

Let $W$ be an irreducible complex reflection group with reflection representation $V$. A $W$-stable, faithful homogeneous system of parameters $\Theta \subseteq \mathrm{Sym}(V^*)$ of common positive degree $p$ is called a posh hsop; $\Theta$ carries a $W$-representation $U$ if $\Theta \simeq U$ as ungraded $W$-modules. We classify the posh pairs $(p,U)$ for which a posh hsop of degree $p$ carrying $U$ exists. Every such $U$ is a Galois twist of $V^*$. Extending work of Ito and Okada, we deduce that the quotient $S/(\Theta)$ is a permutation module for $W$ if and only if $U \simeq V^*$.

Building on the recently established de Sitter (dS)-covariant indecomposable Krein realization of the scalar discrete-series unitary irreducible representations $\Pi_{p,0}$, $p=1,2,\cdots$, of $\mathrm{SO}_0(1,4)$, naturally organized into a Gupta-Bleuler triplet, we uncover its underlying Araki-cohomological structure. We show that the Krein-Gupta-Bleuler construction provides a concrete realization of Araki's standard Gupta-Bleuler triplet, with its non-split indecomposable structure encoded by non-trivial first cohomology classes related by adjunction. We further show that the reducible intermediate sector naturally resolves into its two irreducible components, inducing a corresponding decomposition of the extension classes and yielding a four-layer refinement of Araki's standard triplet. The resulting Gupta-Bleuler quadruplet provides a finer cohomological resolution, distinguishing the physical representation from its complex-conjugate sector.

A perfect isometry $I$ (introduced by Broué) between two blocks $b$ and $c$ is a frequent phenomenon in the block theory of finite groups. It maps an irreducible character $\psi$ of $c$ to $\pm$ an irreducible character of $b$. Broué proved that the ratio of the codegrees of $\psi$ and $I(\psi)$ is a rational number with $p$-value zero and that its class in $\mathbb{F}_p$ is independent of $\psi$. This element is called the Broué invariant of $I$ by Boltje. The goal of this paper is to show that if $I$ comes from a Morita equivalence with an endopermutation source $V$, then, up to a sign, the Broué invariant of $I$ is determined by local data of $b$ and $c$. Therefore, up to a sign, it is independent of the endopermutation-source Morita equivalence. Moreover, we show that the sign factor is given by the reduction of the rank of $V$ modulo $p$. As a corollary, we obtain that the Isaacs–Navarro refinement of the Alperin–McKay conjecture holds for inertial blocks.

The classical Murnaghan-Nakayama rule is a recursive formula for computing the values of complex irreducible characters of the symmetric group $S_n$. Alun Morris proved a recursive formula for evaluating the values of irreducible spin characters of $\widetilde{S}_n$, where $\widetilde{S}_n$ is one of the Schur covers of $S_n$ defined by $\widetilde{S}_n:=\langle t_1,t_2,\cdots,t_{n-1},z\ |\ z^2=1,\ t_i^2=z,\ (t_it_{i+1})^3=z, \ t_it_j=zt_jt_i\ \text{if}\ |i-j|>1\rangle$. A recursive formula for evaluating the values of complex irreducible characters of the wreath product $G\wr S_n$, where $G$ is a finite group, was proved by J. Stembridge. In this article, we state and prove a recursive formula to compute the values of the irreducible spin characters of the wreath product $G\wr \widetilde{S}_n$. For the convenience of implementing these recursive formulas, we extend the notion of 0-1 boundary sequence of a Young diagram to shifted Young diagrams.

We establish that the skew-symmetric bicharacter on the group of $\ell$-weights of a quantum affine algebra is invariant under Chari's braid group action. Furthermore, we define an extension of this bicharacter to the group generated by prefundamental $\ell$-weights and show that this invariance continues to hold under the extended braid group action of Frenkel-Hernandez. As immediate applications, we obtain braid group actions by automorphisms on the quantum tori associated respectively with the category of finite-dimensional representations of the quantum affine algebra and with the category $\mathcal{O}^{\mathfrak{b}}$ of its Borel subalgebra. In addition, we derive explicit formulas for quantum $QQ$-systems in all simply-laced types. Finally, we show that our extended bicharacter yields a quantization matrix compatible with the Hernandez-Leclerc cluster algebra categorified by $\mathcal{O}^{\mathfrak{b}}$ for all finite Dynkin types, extending Bittmann's construction in the simply-laced setting.

We establish a criterion for the irreducibility of Verma modules over symmetrizable Kac–Moody superalgebras. Our main tool is the Weyl-Kac denominator identity.

We classify reductive monoids over arbitrary base schemes by descent from the split classification in arXiv:2607.00322. For a reductive group scheme $G$, we construct its sheaf of based root data, abstract Cartan torus, abstract Weyl group, and a scheme of Weyl cones. The last of these represents the functor of isomorphism classes of rigidified reductive monoids with unit group $G$. As applications, we deduce a criterion for the existence of monoids that are not groups, prove an extension theorem over normal integral bases, and construct the Vinberg monoids without a quasi-split hypothesis.

We prove the automorphic commutator relation conjectured by Liu and Wang arXiv:2504.00275, which is the geometric input for the automorphic Clifford relations in their framework relating higher period integrals to higher derivatives of $L$-functions over function fields. The relation was previously established only under additional cohomological vanishing assumptions and for curves of genus different from one; we remove these assumptions. The main new ingredient is a "universal-local" stage between the local and the semi-local settings: we introduce the classifying stack of formal multidisks with two ordered marked sections, and we construct universal-local analogues of the Hecke stacks, unit objects, Hecke actions and special cohomological correspondences. Fusion and purity on this stack yield a universal-local commutator identity, which is proved using only the local assumptions on the Plancherel algebra that enter the conjecture. Specialization to a curve and the six-functor formalism then give the conjecture for every smooth projective curve.

We formulate and solve a $G$-equivariant analogue of the Seiberg-Witten equations on any irreducible Hermitian symmetric space $G/K$. We use a canonical $K$-invariant form to define a twisted Hodge operator generalising self-duality to higher dimensions. The resulting equivariant Seiberg-Witten equations arise, for a compact $G/K$, as conditions for the absolute minima of an energy functional. Their linear part has a representation theoretic interpretation: the same Weyl group data control the highest-weight realisation of the positive Dirac kernel in the compact case and its discrete-series realisation on the non-compact dual. This space of harmonic spinors collapses almost entirely by the quadratic conditions of the monopole equation. In the compact case, the equivariant moduli set, when non-empty, is a single gauge class, identifying with the $G$-invariant part of Bradlow's trichotomy for vortices, while in the non-compact case, every irreducible solution sits outside the discrete series.

We give a simpler proof of the theorem of J.~Tao and R.~Travkin saying that (the neutral block of) the affine Hecke category is the monoidal colimit of finite Hecke categories. The argument is axiomatized to apply to various sheaf-theoretic and group-theoretic contexts. We also prove an extension of the Tao-Travkin equivalence for the whole affine Hecke category as a Morita equivalence.

Let $F$ be a non-Archimedean local field of characteristic zero, let $D/F$ be the quaternion division algebra, and put $G_n = GL_n(D)$ and $H_n = Sp_n(D)$. We prove Prasad's revised conjecture on tempered representations with symplectic period. If $\rho$ is a unitary $H_r$-distinguished supercuspidal representation and $\delta_m(\rho)$ denotes the generalized Steinberg representation associated to $\rho$, set $B_0(\rho) = \rho, B_m(\rho) = \delta_{m+1}(\rho) \times \delta_m(\rho), m \geq 1.$ We prove that an irreducible tempered representation $\pi$ of $G_n$ is $H_n$-distinguished if and only if $\pi$ is isomorphic to the product over $i$ of $B_{m_i}(\rho_i),$ where each $\rho_i$ is a unitary distinguished supercuspidal representation.

Given a morphism of Abe-Bott-Samelson bimodules, we construct an algorithm for computing its double-leaves coefficients. The key ingredient is a modification of Bruhat upper-triangularity which allows us to use forward substitution to solve for the double-leaf coefficients. We provide a complete implementation of the algorithm in the Sage computer algebra system. We use this to compute the Zamolodchikov 3-color relation in type $H_3$ for balanced realizations. We further simplify the 3-color relation using thick Soergel calculus and express it as a difference of two thick Elias–Williamson diagrams.

We construct a presentation of the Khovanov arc algebras of type $D$ by a quiver with quadratic relations. As an application, we use this presentation and the Koszul dual to compute some Hochschild cohomology groups of $\mathbb{D}_4$, and show that $\mathbb{D}_4$ is intrinsically formal, that is, it admits no non-trivial $A_\infty$-deformations.

The exceptional group $F_4(\mathbb{C})$ has two conjugacy classes of involutions: an involution $t$ with centralizer $\mathrm{Spin}(9,\mathbb{C})$, and the principal involution, with centralizer $(\mathrm{SL}(2,\mathbb{C})\times\mathrm{Sp}(6,\mathbb{C}))/\{\pm1\}$. We determine the values of all irreducible characters of $F_4(\mathbb{C})$ at $t$. For the irreducible representation with highest weight $a\omega_1+b\omega_2+c\omega_3+d\omega_4$ (in Bourbaki's labelling), we show that the character vanishes at $t$ if and only if $c$ and $d$ are both odd. In all other cases its value is $\pm 2^{-7}$ times the dimension of an explicit irreducible representation of $\mathrm{Spin}(9,\mathbb{C})$. The proof is a limiting argument in the Weyl character formula, combined with a decomposition over the three cosets of $W(B_4)$ in $W(F_4)$ and the triality action of $W(F_4)=W(D_4)\rtimes S_3$ on the weight lattice modulo the $D_4$-lattice. At the principal involution, Nadimpalli, Pattanayak and Prasad showed that the character value is $\pm 1/2$ times the dimension of an irreducible representation of $\mathrm{SL}(2,\mathbb{C})\times\mathrm{Spin}(7,\mathbb{C})$. Comparing the two results, we find that the characters of $F_4(\mathbb{C})$ vanish at the two involutions for exactly the same highest weights. Together these results determine all irreducible characters of $F_4(\mathbb{C})$ at all involutions. We also explain how our formula at $t$ relates to the generalized Weyl character formula of Gross, Kostant, Ramond and Sternberg.

We study the commuting scheme of the exceptional Lie algebra $\mathfrak{g}_2$ via computational methods. Over a field of characteristic zero, we show that the commuting scheme of $\mathfrak{g}_2$ is reduced, Cohen-Macaulay and normal. Our computations give an explicit homogeneous system of parameters, determine the Hilbert series and degree, identify the first syzygies of the commuting ideal and show that it is not Gorenstein. We extend some of the above results to positive characteristic.

We classify nontrivial simple super-Virasoro modules finitely generated over \(\C[L_0]\) using simple Laurent Weyl algebra modules finite free over \(\C[D]\). The classified modules are finite Cartan-free with equal parity ranks. We also prove finite length without simplicity and construct explicit nonsplit self-extensions.

We investigate the explicit geometric and combinatorial structure of Mirković-Vilonen (MV) cycles in twisted affine Grassmannians associated with special parahoric group schemes. Using the twist map on configuration spaces of decorated flags, we obtain an explicit parametrization of twisted MV cycles by Lusztig data. We show that the changes between reduced words are governed by the usual Lusztig transformations for the reductive group arising from ramified geometric Satake correspondence. As a consequence, the moment polytopes of twisted MV cycles coincide with the usual MV polytopes for this group. Our construction applies to all special parahoric subgroups in the tamely ramified setting.

Let $(\mathfrak g, \mathfrak k)$ be the complexication of a Hermitian symmetric pair and let $\gamma_1 <\cdots<\gamma_r$ be the Harish-Chandra strongly orthogonal roots, with $\gamma_1$ the lowest non-compact root, and let $\zeta$ be the normalized central character of $\mathfrak k$. We consider the unitary highest weight representations $L(\lambda(x))$ of $\mathfrak g $ of highest weights of the form $\lambda(x)=-k\gamma_1 -x\zeta$, $x\in\mathbb{R}$. In particular, we are interested in the case $x=a$, the last reduction point in the analytic continuation in $x$. We prove that $L(-k\gamma_1-a \zeta)$, $k\ge 1$, is $K$-multiplicity free and we describe all its K-types.

The main aim of this article is to obtain explicit formulas for the root multiplicities of a class of finite-rank symmetrizable Borcherds-Kac-Moody algebras g(A) whose real part is of type A. The Dynkin diagram of each such algebra is obtained from the disjoint union of a Dynkin diagram of type A and a finite simple graph G_im, whose vertices correspond to imaginary simple roots, by choosing one vertex in each and joining the chosen vertices by a single edge of multiplicity c>0. More generally, we consider the same construction with the type A diagram replaced by the Dynkin diagram of an arbitrary finite-rank symmetrizable Kac-Moody algebra g_re. The resulting symmetrizable Borcherds-Kac-Moody algebra g(A) has real part g_re and imaginary part represented by G_im. Our starting point is a graph-theoretic description of the quotient of the denominator products of g(A) and g_re. This quotient is a multivariate independence polynomial of G_im, with the variable corresponding to the attachment vertex weighted by normalized highest-weight characters of g_re. We prove that the coefficients of powers of this quotient factor into a generalized chromatic polynomial and a tensor-power weight multiplicity. Using this, we obtain explicit formulas for all root multiplicities with nonzero imaginary component in terms of generalized chromatic polynomials of G_im and weight multiplicities in tensor products of highest-weight modules of g_re. When G_im is chordal, these formulas specialize to finite divisor sums involving products of binomial coefficients; paths and complete graphs provide particular examples. In particular, we obtain explicit formulas for the root multiplicities of g(A) when its real part g_re is of type A.