Papers

Motivated by equivariant neural networks, we study piecewise linear equivariant maps between finite-dimensional real representations of compact groups. We show that all genuinely non-linear piecewise linear behaviour is confined to the subspaces on which the identity component of the group acts trivially, while equivariance forces linearity on the corresponding orthogonal complements. As a consequence, we obtain a compact-group analogue of the finite-group existence criterion of Gibson–Tubbenhauer–Williamson for non-zero equivariant piecewise linear maps between irreducible representations, with the identity component giving rise to a rigidity phenomenon absent from the finite-group case.

Cluster varieties contain the union of cluster tori and the points not in this union are called $\textit{deep points}$, and the locus of these points is called the $\textit{deep locus}$. In arXiv:2402.16970, a description of this locus is conjectured for locally acyclic cluster algebras, in particular, stating that this should be the stabilizer locus of the cluster automorphism group. We resolve this conjecture for the case of cluster algebras arising from surfaces, introduced in arXiv:math/0608367, and the remaining finite type cases as categorized in arXiv:math/0208229. In particular, we show locally acyclic surface type cluster varieties have no deep points not contained in the stabilizer locus, and finite type cluster algebras also have no deep points not contained in the stabilizer locus, validating the mysterious points conjecture in these cases.

Let $G$ be a complex reductive group and $C_m(G)_1$ the identity component of the space of $m$-tuples of commuting elements in $G$. We prove that for every $m\geq 2$, the integral singular cohomology of $C_m(G)_1$ has torsion precisely at those primes which divide the order of the Weyl group of $G$. We deduce the analogous statement for compact Lie groups, settling a conjecture of Kishimoto and Takeda, and prove the corresponding result for compact Lie algebras. We prove these results using Smith theory for the conjugation action by semisimple elements of prime power order.

We give negative answers to two tilting-completion questions for finite-dimensional algebras. We construct two finite-dimensional basic connected quasi-hereditary $\mathbb C$-algebras. The first admits a faithful basic full-rank pretilting module with no tilting completion; the second admits an almost-tilting module with no tilting completion. The full-rank example also yields counterexamples to two conjectures: Enomoto's Self-orthogonal Wakamatsu-tilting Conjecture and the Self-orthogonal Faithful Conjecture of Chen, Li, Zhang, and Zhao. We further show that the Self-orthogonal Wakamatsu-tilting Conjecture holds for all finite-dimensional algebras if and only if the Self-orthogonal Faithful Conjecture holds for all finite-dimensional algebras. The construction ultimately stems from Krah's non-full exceptional collection of maximal length on a rational surface and Kalck's associated full-rank presilting example.

We introduce group-valued coloured interchange processes, a class of continuous-time random walks on wreath products generated by transpositions and arbitrary symmetric base-group transitions. We give a complete representation-theoretic characterisation of their spectral gaps by decomposing the associated quasi-regular representation and determining precisely which irreducible representations are required. For abelian base groups, we realise these representations on multislices and identify their Laplacians with discrete Schrödinger operators. We then classify the minimal families of irreducible representations that determine the spectral gap for every choice of transition rates.

We study $G$-equivariant Hörmander pseudodifferential operators on a noncompact symmetric space $G/K$. We define a notion of a complete symbol function, called the Harish-Chandra symbol function, on the spherical tempered dual of $G$, for operators that satisfy a rapid off-diagonal decay condition on their Schwartz kernels, and we characterize the class of such operators in terms of their Harish-Chandra symbol function.

Let $\text{F}$ be a non-archimedean local field of characteristic zero with residual characteristic $p$. In this paper we give an explicit description of the local Langlands correspondence for middle supercuspidal representations of $\text{GL}(2n,\text{F})$, under the tameness condition $p\nmid 2n$, in terms of the maximal simple types that define them. We achieve this by explicitly computing and comparing the gamma factors on the automorphic and Galois sides of the local Langlands correspondence. The computation on the automorphic side requires neither the tameness condition nor the characteristic zero condition.

Let $A$ be a ring, let ${\bf T}$ be a finite-term big tilting complex over $A$, and choose a bounded projective representative $\cpx{P}$ of ${\bf T}$. Put $B=\End_{\D{A}}(\cpx{P})$. We prove that the kernel of the associated derived tensor functor ${\bf R}:\D{B}\to\D{A}$ is equivalent, through the derived inclusion, to $\D{\mathscr E}$, where $\mathscr E=\Ker({\bf R})\cap B\Modcat$ carries the inherited exact structure. Consequently, $\D{B}$ admits a recollement by $\D{\mathscr E}$ and $\D{A}$. The proof combines a dg double-centralizer argument with a functorial realisation of the kernel on unbounded complexes. The category $\mathscr E$ is definable and $d$-symmetric whenever $d$ is at least the projective amplitude of the underlying perfect right $B$-complex. We also characterise when the kernel is induced by a homological ring epimorphism: this holds precisely when $\mathscr E$ is an abelian subcategory, or equivalently when the standard $t$-structure restricts to the kernel. In amplitude at most one, this recovers the universal-localisation description for two-term big tilting complexes. Over semiperfect endomorphism rings, the amplitude is detected by reduction modulo the Jacobson radical. Finally, we construct non-compact examples of arbitrarily large minimal projective amplitude. One family yields algebraic Calkin quotients; another, over connected rings constructed from singular local hypersurfaces, has nonzero, non-abelian exact kernels and cannot be obtained from modules by derived equivalences between ordinary rings.

In this survey article, we give an introduction to the relatively young subject of $2$-representation theory, which studies categorifications of important players in classical representation theory. In particular, we provide a streamlined exposition of the main results leading to the classification of simple $2$-representations for Soergel bimodules associated to finite Weyl groups in characteristic $0$, obtained in [M. Mackaay, V. Mazorchuk, V. Miemietz, D. Tubbenhauer and X. Zhang, Simple transitive $2$-representations of Soergel bimodules for finite Coxeter types. Proc. London Math. Soc. 126 (2023)].

Cells for the canonical and $p$-canonical bases organise the representation theory and geometry of Hecke categories. We determine the relevant $p$-canonical basis elements and the resulting $p$-cell structure for the subregular and submaximal cells in all classical types.

Suppose that $p$ is a prime and $X$ is a finite group with a strongly $p$-embedded subgroup, for example a rank $1$ group of Lie type in characteristic $p$. Let $m$ denote the $p$-rank of $X$ and assume that $m \ge 2$. We say that a faithful $\FF_p X$-module is $k$-active if some element of order $p$ in $X$ acts with exactly $k$ non-trivial Jordan blocks. In this paper, we determine the non-trivial composition factors of the faithful $\FF_pX$-modules which are $k$-active for some $k\leq m$.

Suppose $F$ is a nonarchimedean local field whose residue field is a proper extension of $\mathbb{F}_p$ with $p > 3$. Generalizing results of Ghate–Le–Sheth, we show that any split, connected, reductive group $G$ over $F$ which is not a torus admits a smooth, irreducible, non-admissible mod $p$ representation. We use this to show that the functor of pro-$p$-Iwahori invariants does not induce an equivalence between the category of smooth mod $p$ $G$-representations generated by their pro-$p$-Iwahori invariant vectors and modules over the pro-$p$-Iwahori–Hecke algebra, contrary to what happens for $\textrm{GL}_2(\mathbb{Q}_p)$.

We investigate the action of the affine Yangian $Y_{t_1,t_2}(\widehat{\mathfrak{gl}}_1)$ on the singular cohomology of the moduli space of stable sheaves $\mathcal{M}$ on a smooth projective surface $S$. Through an intersection-theoretic analysis of nested moduli spaces, we obtain a proof of the representation $$Y_{t_1,t_2}(\widehat{\mathfrak{gl}}_1) \curvearrowright H_{\mathcal{M}}.$$ This intersection-theoretic approach extends classical constructions to the full Yangian structure and circumvents the $K$-theoretic machinery traditionally required to establish this action.

We consider the spherical Borel Eisenstein series induced from the trivial representation for a split semisimple linear algebraic group over a number field. We prove that its regularization at the special point corresponding to half the weighted marking of a distinguished coadjoint nilpotent orbit in the Langlands dual Lie algebra is nonzero and square-integrable. Our proof follows the philosophy of Kazhdan and Okounkov. We give a geometric interpretation of Langlands' square-integrability criterion in this setting and, using the equivariant integration formula, prove that the regularization satisfies this criterion. As an immediate consequence, we obtain a simple and uniform proof of Arthur's unitarity conjecture, without case-by-case analysis or machine computation.

Representations of the Kohno-Drinfeld Lie algebra associated with several classical multivariate distributions have played an important role in recent developments in the theory of quantum superintegrable systems. In this work, we analyze the corresponding Gaudin models for the multivariate Hahn, Dirichlet, multinomial, and negative multinomial distributions. More precisely, using tools from representation theory, we construct multivariate orthogonal polynomials with respect to these distributions as common eigenfunctions of Gaudin operators. The polynomials are parametrized by solutions to the Bethe ansatz equations or, equivalently, by the roots of Heine-Stieltjes polynomials.

We prove that every symmetric pair of a connected complex reductive group is regular in the sense of Aizenbud–Gourevitch. This settles the Aizenbud–Gourevitch regularity conjecture over the complex numbers. Generalized Harish–Chandra descent then makes the canonical central cover of every complex symmetric pair a Gelfand–Kazhdan pair. An anti-automorphism arising from a compatible Chevalley involution upgrades the resulting GP2 bound to GP1 on the cover, and finite central descent transfers GP1 to the original pair. In particular, van Dijk's conjecture on complex symmetric pairs follows. Rubio reduced the unresolved irreducible regularity problem to four families: the DIII family $(D_r,A_{r-1}+\mathbb{C})$, the balanced CII family $(C_{2r},C_r+C_r)$, some remaining Spin block pairs, and the EVII pair $(E_7,E_6+\mathbb{C})$. We treat these cases by four different mechanisms. For DIII we construct a sign-equivariant Schwartz distribution on the regular set and extend it across a common orbit boundary by the Chen–Sun theorem. For balanced CII we combine homogeneity, distinguished nilpotent orbits, and a stable-density theorem for the centralizer representation. For Spin blocks we prove pleasantness for unequal odd–odd blocks, use Przebinda's orthogonal-distribution theorem in odd smaller rank, and construct a finite orbit closure with automatic extension in even smaller rank. For EVII we compute the graded-$\mathfrak{sl}_2$ data for all twenty-two nilpotent orbits and use central-torus characters to eliminate the remaining resonances, including the two residual triple-centralizer cases. A finite-component assembly theorem then handles arbitrary connected central quotients and diagonal couplings among simple factors.

In 1996, Baranov and Kleshchev classified maximal ideals in the finitary symmetric group algebra over fields of characteristic not 2. The corresponding question in characteristic~2 has remained open since. In the current paper we solve the problem by establishing part of a conjectural connection between prime ideals in this group algebra and the recently defined higher Verlinde categories. To achieve this we investigate thick tensor ideals of tilting modules of the general linear group in characteristic 2.

Let $n$ be a natural number, let $q$ be a prime power, and let $U_n(q)$ denote the group of unitriangular $n\times n$ matrices over the finite field $\mathbb{F}_q$ with $q$ elements. Row closed and column closed subgroups $U$ of $U_n(q)$ are special pattern subgroups obtained by deleting entire rows or, respectively, entire columns (apart from the diagonal entries). The supercharacters of the André-Yan supercharacter theory of $U$ are afforded by the orbit modules arising from a monomial action of $U$ on the character group of the Lie algebra of $U$. We classify the corresponding orbits for row closed subgroups $U$, that is, we determine all orbits and identify which of the associated orbit modules are isomorphic and which afford orthogonal supercharacters. The classification for column closed subgroups follows from the mirror map, which reflects matrices across the antidiagonal.

We show that the subfactor planar algebra with principal graph $D_\infty$ is the Brauer planar algebra with bubble constant $\delta=2$. The Brauer algebra is similar to the Temperley-Lieb algebra, but with virtual crossings. At $\delta=2$, it relates to the category of representations of the orthogonal group $O(2)$. We work over any commutative ring with $1/2$. Its principal graph encodes information about the corresponding monoidal category.

Write $\mu_t$ for the $t$-th roots of unity and $z^{\pm1}$ for $r$ free reciprocal pairs. We study $\Phi_{t,r}(\beta)=s_{\lambda}(\mu_t,z^{\pm1})$, $\beta=\lambda+\delta$, and the question the companion paper left open after $r=1$: when does it vanish? We factor the evaluation into classical branching followed by a torsion filter, and the shape depends on the parity of $t$: the point lies in the orthogonal group with determinant $(-1)^{t+1}$. For odd $t$ it sits in the identity component: an ordinary restriction $SO_{2R'+1}\downarrow SO_{2m'+1}\times SO_{2r}$, the filter an odd orthogonal character at a principal element of order $h+1$. For even $t$ in the other: a twining, a virtual expansion, and a torsion element regular but not principal; there we prove the filter, with its sign. One description covers both: the filter is nonzero exactly when the shifted point is regular semisimple. Both are minimal-level fusion projections: the even of type $C$, the odd's tensor sector of type $B$. Affine folding accounts for $0,\pm1$; what it does not survives as conjectures. The highest surviving weight is the dominant vertex of the numerator's Newton polytope minus the denominator's — the latter proved, the former conditional on a single-orbit property — and the class there is conjecturally primitive, $\pm$ the generator of the rank-one quotient. For odd $t$ and one $\Lambda$ that numerator is a signed transversal count in $\{0,\pm1\}$ by the equal-rank character formula, leaving one division. We invert it in closed form, along an arithmetic progression; the quotient is $\pm\epsilon_t\det M$ for an explicit $0/{\pm}1$ matrix — an interval matrix up to signs, hence totally unimodular, which settles (L1). The fibre count is a permanent, odd only when $1$, so at a dominant index a multi-hit fibre sums to zero. Two extremal statements remain. What is unproved is measured, in both parities.

We study subcategories of the module category of a finite-dimensional algebra that are closed under quotients or submodules. We propose the quotient–submodule equidistribution conjecture: over a representation-finite algebra, the number of quotient-closed subcategories of size $i$ is equal to that of submodule-closed subcategories of size $i$ for every $i$, where size is the number of indecomposable modules in the subcategory. We prove the following cases of the conjecture: (1) the five smallest and five largest values of $i$, hence all algebras with at most nine indecomposable modules; (2) Nakayama algebras; (3) algebras with radical square zero; and (4) representation-directed algebras. In the last case, we classify quotient-closed subcategories by a lower Bruhat interval in a Coxeter group constructed from the Auslander–Reiten quiver, extending the classification of Oppermann–Reiten–Thomas for Dynkin quivers. We also prove that quotient closure defines a finitary convex geometry and classify the functorially finite quotient-closed subcategories by Gen-minimal modules.

For any finite group, we pose the following question: given an irreducible real representation, is the associated representation sphere non-orientable if and only if the representation is nontrivial of real type? We prove that the answer to this question is yes if the group has a normal Sylow 2-subgroup, but exhibit a 2-nilpotent group of order 112 for which the answer is no via elementary arguments. We also link the question to the unit group of the Burnside ring, where we recover a basis discovered by Bouc for 2-groups, and pose a conjecture about detection of non-orientability from solvable subgroups. The question is motivated by issues arising from the construction of permutation twisted cohomology for finite groups.

We prove that every finite simple group of Lie type over a field of order at most 8 contains a conjugacy class whose square is the whole group. Together with the previously established cases, our results complete the proof of Thompson's conjecture. We explicitly construct the required class, address all low-rank identifications, and provide the precise computational certificates used in the exceptional character calculations.

Among finite simple groups, character values of alternating and sporadic groups have relatively low irrationality at any prime $p$, whereas those of simple groups of Lie type can have arbitrarily high $p$-irrationality. We provide concrete evidence supporting this phenomenon. In particular, we show that if $\chi:=R_{\mathbf{T}}^{\mathbf{G}}(\theta)$ is a Deligne–Lusztig character of a finite reductive group $\mathbf{G}^F$, with $\theta$ an irreducible character of a maximal torus $\mathbf{T}^F$, and if $\chi$ has degree prime to $p$, then the so-called $p$-rationality level of $\chi$ coincides precisely with that of $\theta$. We present further evidence suggesting that Lusztig induction preserves $p$-rationality for characters of $p'$-degree.

Let Phi be an irreducible reduced crystallographic root system of rank r, and let N be the number of positive coroots. A previous theorem (arXiv:2608.16363, Theorem 3.3) identifies the first pole below 2/h as q_2 = (r-1)/(N-1) and expresses its residue as a sum of periods attached to the simple walls of the dominant chamber. We evaluate that wall-period sum for all four classical families. The identity q_2 (N-1) = r-1 removes the radial variable from every wall integral. The resulting projective integrals reduce to mixed Dotsenko-Fateev chambers in type A_r, to those chambers together with a Selberg endpoint in types B_r and C_r, and to a single chamber family in type D_r. The chamber recurrences give explicit sine weights in types A, B, and C. In type D, a terminating basic-hypergeometric sum at a root of unity reduces to a finite cyclotomic product. We also show that the corresponding exceptional wall restrictions are not reflection arrangements, so this particular reduction does not extend directly to the exceptional types.

Deep level Deligne-Lusztig representations, as natural extensions of the classical Deligne-Lusztig representations, have recently seen various applications in the Langlands program. In this note, we address their distinction problem for symmetric pairs over finite fields, yielding a generalization of a classical result by Lusztig.

We develop a uniform framework for “big” categorification of representation categories of towers of finite classical groups and wreath product groups. We construct actions of symmetric products of Heisenberg categories, quantum in the finite classical group case and degenerate in the wreath product case. These actions lead to categorical actions of suitable symmetric products of Kac–Moody 2-categories, and hence to actions of large Lie algebras on Grothendieck groups. In characteristic zero, the resulting actions control the full graded centers of the group algebras through diagrammatic central elements, and the associated colored weight functions separate all irreducible ordinary characters. We also obtain modular block descriptions for wreath product groups through categorification.

We develop the representation theory of imaginary Verma modules for twisted quantum affine algebras and construct the corresponding Kashiwara algebras. The twisted case presents substantial new difficulties compared with the untwisted setting: the PBW root vectors have nontrivial orbit structure, the imaginary root spaces occur with multiplicities, and roots of unity enter essentially into the defining commutation relations. Our first main result is an explicit PBW-type basis for twisted quantum imaginary Verma modules associated with the natural imaginary partition of the affine root system. This provides a precise compatibility between the twisted quantum and classical theories that is not immediate from the standard PBW theory. We then determine the structure and irreducibility of the twisted quantum imaginary Verma modules. We prove that the Heisenberg submodule generated by the imaginary root vectors is irreducible precisely at nonzero central charge, and establish the corresponding irreducibility criterion for the reduced twisted imaginary Verma modules at zero central charge. The second main part of the paper introduces the Kashiwara algebra in the twisted setting. The required current formula for the generators differs essentially from the untwisted formula and is needed to construct the Omega operators. We derive the resulting Omega-operator commutation relations, including the root-of-unity factors specific to the twisted cases. These relations lead to a new presentation of the Kashiwara algebra associated with the reduced twisted imaginary Verma modules. We prove that the negative current algebra is a simple module over this Kashiwara algebra and construct a symmetric non-degenerate bilinear form characterized by the Omega operators.

For every finite-rank crystallographic affine Coxeter system $(W,S)$ and Coxeter element $c$, we construct a minimal linear graded free resolution of the trivial module over $k[M([1,c]_T)]$ supported on a rectified exceptional cluster complex. Hence the affine dual braid monoid algebra is Koszul over every field $k$. The exceptional complex is introduced to recover the principal-fibre topology missing from the direct Reading–Stella labelling. Half-orbit rectification replaces the transjective root labels by ordinary exceptional modules, so that a face $F$ determines an exceptional wide subcategory and the intrinsic weight \[ \omega(F)=\operatorname{cox}(\operatorname{wide}\langle F\rangle). \] The resulting principal fibres are induced subcomplexes and split canonically as joins of subcomplexes attached to connected Dynkin and affine blocks; these subcomplexes are contractible. Affine non-lattice divisibility creates the genuinely nonprincipal case. The McCammond–Sulway completion shows that whenever no greatest interval right divisor exists, all maximal interval right divisors share a common nontrivial complete finite Coxeter component. In the associated exceptional-wide decompositions, this common Coxeter component is the Coxeter element of a Dynkin block, and the subcomplex attached to that block occurs as a common contractible join factor. Thus every nonidentity fibre is contractible, and the weighted-face complex is exact, minimal and linear. In particular, $\operatorname{Tor}^{A_c}_q(k,k)$ is indexed by $q$-vertex exceptional cluster faces in internal degree $q$, and $\operatorname{pd}_{A_c}k=|S|$.

We show that the $ \mathrm{K}_1 $-functor modeled on locally isotropic reductive groups is hypoabelian if the base ring has finite Bass–Serre dimension and, for the Tits index $ \mathsf{E}_{8, 2}^{78} $, contains a field. For classical or globally isotropic reductive groups schemes $ \mathrm{K}_1 $ is actually solvable. This implies that the elementary subgroup (or its derived subgroup) is the maximal perfect subgroup of the reductive group.