arXane › Mathematics › Mathematics
Wednesday, 4 November 2026: 12 new papers, 7 cross-lists and 9 replacements in Representation Theory (math.RT), in arXiv's order, with full abstracts. The same papers as arXiv's own listing, set as a newspaper; every title links to arXiv.
Let $Q$ be a finite quiver and let $\Lambda$ be the complete path algebra of $Q$ with respect to the arrow ideal. We study the mutation of silting objects in the bounded homotopy category of projective $\Lambda$-modules and characterise in which cases the mutation theories of silting complexes and 2-term silting complexes of $\Lambda$ are determined by those of a finite dimensional quotient $\Lambda/I$. This characterisation is closely related to a characterisation of 2-term silting finite quotient algebras.
Let $\La_1$ and $\La_2$ be finite-dimensional $\ta_2$-finite complex algebras of global dimension at most two. We prove that, if they are cluster equivalent, then they admit $\bbZ$-gradings for which the bounded derived categories of graded modules are triangle equivalent. The equivalence is not required to commute with the internal grading shifts. The proof uses common multigraded dg models, relative gradings on two-term silting objects, and covering quotients. We also consider tilting lifts of cluster-tilting objects with endomorphism algebras of global dimension at most two. In acyclic cluster type, the existence of such a lift for every cluster-tilting object implies piecewise heredity. We construct a counterexample without this assumption using a surface algebra associated with a one-holed torus.
Let $K/\mathbb{Q}_p$ be a finite extension of degree $n$ with $p\geq3$, and let $D$ be the non-split quaternion algebra over $K$. We prove that every infinite-dimensional admissible smooth representation of $\mathrm{GL}_2(K)$ or $D^\times$ with a central character and satisfying a Casimir annihilation condition has Gelfand--Kirillov dimension $n$. As a corollary, every infinite-dimensional admissible Banach representation whose locally analytic vectors have an infinitesimal character has dimension $n$, strengthening Sorgdrager's bound to an equality. An appendix by Yongquan Hu proves that, when $K\ne\mathbb{Q}_p$ and the coefficient field is algebraically closed, the canonical diagram of an admissible supersingular representation satisfying the same Casimir annihilation condition is $(\pi,\pi,\mathrm{id})$, as in the case of local fields of characteristic $p$.
We investigate the structure of locally analytic representations of $p$-adic groups, and establish a sufficient condition for the irreducibility of locally analytic vectors in certain Banach representations of $D^\times$, where $D$ is the quaternion division algebra over $\mathbb{Q}_p$. As an application, we prove that $\mathrm{JL}(\rho)^{\mathrm{an}}/\mathrm{JL}(\rho)^{1\text{-}\mathrm{alg}}$ is a topologically irreducible locally analytic representation of $D^\times$ under certain genericity conditions on $\rho$. Here $\mathrm{JL}(\rho)$ is the $\rho(-1)$-multiplicity space in the image of $\Pi(\rho)$ under Scholze's functor, and $\Pi(\rho)$ is the representation associated to $\rho$ by the $p$-adic local Langlands correspondence.
We study the extension groups between one $\rm K$-type representations of the graded Hecke algebra $\mathbb H$. A one $\rm K$-type representation is a Hermitian representation of $\mbb H$ whose restriction to the Weyl group is irreducible. Using a Koszul-type resolution of $\mathbb H$-modules, we reduce the computation of $\operatorname{Ext}_{\mathbb H}^j(X,Y)$ to a combinatorial count, yielding explicit formulas for the classical types and a computer computation for the exceptional types.
We prove effective versions of several classical results in representation theory of reductive groups over non-Archimedean local fields of arbitrary characteristic. More specifically, we deal with the support of matrix coefficients of cuspidal representations and the recursions satisfied by matrix coefficients of representations of finite length. We provide effective generation results for irreducible representations. Finally, we study the effective smoothness of characters on the set of regular semi-simple elements.
We describe an algorithm and a computer program computing the relative Tate-Shafarevich kernel $Sha^S(K,G)$ of a reductive group $G$ over a global field $K$ with respect to a finite set $S$ of places of $K$ containing no archimedean places. We assume that we know the algebraic fundamental group $\Lambda=\pi_1^{alg}(G)$, the Galois group $\Theta=Gal(L/K)$ acting on $\Lambda$, where $L$ is a finite Galois extension of $K$ having no real places, the decomposition subgroups $\Theta_v\subseteq \Theta$ for all places $v$ in $S$, and all non-cyclic decomposition subgroups $\Theta_v$ for $v$ outside $S$. We also compute $Sha^S(K,G)$ in some cases where $S$ contains archimedean places.
We study derived Ringel duality for modular category $\mathcal O$ on downward-closed, quasi-bounded supports. We reconstruct global tilting objects and their morphisms from compatible images in finite highest-weight Serre subquotients, and realize the complete Ringel algebra as an inverse limit of finite-dimensional algebras. The duality identifies the bounded homotopy category of finitely generated projectives over the associated locally unital algebra with the bounded derived category of the Ringel dual category. We characterize tiltings recovered, together with their outgoing morphisms, by a single finite subquotient, and give a sharp criterion for equivalence under Frobenius restriction. For $SL_2$, we compute tilting endomorphism algebras and their quotient maps, and obtain sharp recovery bounds for finite tilting families.
We prove Douglas' conjecture for $m$-Kronecker quivers for each $m \geq 3$ on the normalized logarithms of Euler characteristics of moduli spaces. Also, we state a log-concavity conjecture on Euler characteristics near slope one for the 3-Kronecker quiver.
For each pair of integers $m\ge 1$ and $n\ge 2$, we construct a finite inverse semigroup $S_{m,n}\subseteq M_m(\mathbb{C})$ and explicitly determine its marginal inverse subsemigroup $M_{x^n}(S_{m,n})$ associated with the word $x^n$. We prove that the three conditions $\sim_{x^n}=\equiv_{x^n}$ on $S_{m,n}$, $M_{x^n}(S_{m,n})\trianglelefteq S_{m,n}$, and $m\le n$ are equivalent. As a consequence, we answer Problem 11 of Ara\'ujo, Ara\'ujo and Kinyon in [1]. Moreover, for a word $\omega$ in any inverse semigroup $S$, we prove that $\sim_\omega=\equiv_\omega$ if and only if $M_\omega(S)\trianglelefteq S$ and $\sim_\omega$ is invariant under inversion.
Let $p$ be a prime. We classify homomorphisms $H:V_1\to V_2$ between abelian $p$-groups that are direct sums of cyclic groups, under the assumption $p^2V_2=0$. We prove that every such homomorphism is isomorphic to a direct sum of finite indecomposable representations and that this decomposition is unique up to isomorphism and reordering of the summands.
Almost 30 years ago Littelmann conjectured that there is a unitriangular base change between the dual canonical basis and the path vector basis for all symmetrizable Kac-Moody Lie algebras. In this paper we give a positive answer to the conjecture for the case of classical Lie algebras and type $G_2$, when the path model is given by standard Young tableaux.
We describe Hochschild cohomology of Temperley--Lieb algebras by explicit minimal-resolution coordinates and multilinear operations on Kauffman diagrams. For each doubled-line basic algebra, a finite four-term bimodule complex is verified by a signed-shuffle comparison and an augmented contracting homotopy. We compute cocycles and coboundaries and recover the known cup product and Gerstenhaber bracket in explicit representatives. Over $\mathbb C$, at nonzero root-of-unity loop parameters, we construct a tower-compatible derivation and the marked-circle two-cocycle. Their classes are nonzero on every nonsemisimple block; their cup products therefore represent all positive classes over the centre. Relative full twists give a compact expression for the derivation. Worked planar diagrams distinguish the cup product from insertion and strict cochain equalities from equalities of classes. A finite-lattice section interprets the marked-circle operation as loop-weight response, derives its planar Potts normalization, and exhibits a six-vertex realization. It also shows why passing to a deformable spin representation may annihilate the class with different coefficients. No thermodynamic conclusion is claimed, and the even-strand zero-parameter family remains separate.
In this article we establish new obstructions to the categorification of near-group fusion rings by combining the values of irreducible characters of the Grothendieck ring with the Galois symmetry of the induced unit object $I(1)$ in the Drinfeld center. As a result, we show that a near-group ring of type $G+m$, where $G$ is an abelian group and both $|G|$ and $m$ are odd, can only categorify if $m=|G|$.
Let $R$ be a commutative noetherian ring. We ask whether complexes with finitely generated cohomology and a fixed support can be generated by relatively compact objects, using coproducts which exist in the half-bounded derived category. We prove that $\mathbf D^+_{\mathrm{fg},W}(R)=\operatorname{Loc}_{\mathrm{ex}}(\mathbf D^b_{\mathrm{fg},W}(R))$ whenever $W\subseteq\operatorname{Spec} R$ is specialization closed and has finite dimension. The same equality holds for every specialization-closed subset of $\operatorname{Spec} B$ when $B$ is a finite-type algebra over a noetherian domain $A$ of finite character and either $A$ or $B$ is regular. These equalities classify the thick Hom-submodules closed under existing coproducts. We also prove the corresponding bounded-above statement from the support results of Matsui--Takahashi. For an arbitrary noetherian ring, the bounded-below generation conjecture reduces to torsion complexes over quotient domains, with their support unchanged. Finally, over a regular noetherian domain of infinite dimension, we give a bounded-below object which is generated by bounded objects with the same support but has no bounded-below representative by finitely generated modules. The comparison functor from the bounded-below derived category of finite modules is therefore neither full nor essentially surjective.
Three constructions of vertex operator algebras realizing the Haagerup center were announced independently and almost simultaneously. The relations between these constructions are studied in this paper. The generators are expressed in a common lattice basis for $E_6\oplus A_2$. It is proved that the constructions of Riesen and Huang give isomorphic vertex operator algebras, and that this algebra and the algebra constructed by Jiao have isomorphic fixed point subalgebras under automorphisms of order two. The isomorphisms are given explicitly on generators. The common fixed point subalgebra contains the parafermion vertex operator algebras $K(\mathfrak{sl}_2,3)$ and $K(\mathfrak{sl}_2,13)$. A general rationality criterion for unitary vertex operator algebras is given in terms of positive linear combinations of conformal vectors and is applied to these constructions. The two vertex operator algebras are simple current extensions of the common fixed point subalgebra and have equivalent module categories as ribbon categories. The equivalence follows from decomposition with respect to a nondegenerate pointed subcategory with four simple objects in the module category of the common fixed point subalgebra. It is also proved that the conformal nets associated with these unitary vertex operator algebras are completely rational.
Hecke operators can be used to relate characters of rational conformal field theories with different central charges, but it has remained a mystery how to interpret these relations physically. We offer a simple explanation in terms of self-duality under asymmetric orbifolds. Self-dual orbifolds also give more general identities beyond the familiar Harvey-Wu construction, which we illustrate through examples.
In this paper we classify the algebraic K-theory of sets equipped with an action by a finite abelian monoid. The main tool developed for this purpose is an Orbit-Stabilizer Theorem, which gives a decomposition of monoid sets analogous to the decomposition given by the Orbit-Stabilizer Theorem for G-sets.
We propose a new refinement of the (blockwise) Alperin weight conjecture taking dimensions of simple modules (or degrees of irreducible Brauer characters) into consideration. This is similar to Giannelli's refinement of the (Alperin--)McKay conjecture. We prove this new refinement for blocks with a cyclic or Klein four defect group, for inertial blocks (which include the Brou\'{e}--Puig nilpotent blocks), and for $p$-blocks of $p$-solvable groups.