30,590 papers in this slice of arXiv.
Emily Gullerud, Peter Webb
We introduce synthetic buildings for each finite group G and prime p. These are G-simplicial complexes, among which are the p
V. K. Dobrev
Recently we started building a bridge between two cases of Langlands duality. The latter is one of the most influential topics in mathematical research. It has many different appearances and influential subtopics. Yet there is a topic that until now seems unrelated to the Langlands program. That is the topic of invariant differential operators. That is strange since both items are deeply rooted in Harish-Chandra's representation theory of semisimple Lie groups. We started with the case of the group SL(2n). In the present paper we deal with the group SL(2n+1). The two mentioned groups are similar, but their representation theories is rather different.
Sanjay Amrutiya, Umesh Dubey
In this note, we construct moduli spaces of super-representations of quivers by extending King's Geometric Invariant Theory (GIT) framework to the super setting. Using the even part of super general linear groups and a parity-shifting operator, we construct moduli spaces that topologically parameterize super-representations of a fixed super-dimension. We also outline the construction of super quiver varieties for super-representations of framed quivers (without doubling); and realise super-Grassmannians and super-flag varieties as geometric quotients within this setting.
Semeon Arthamonov, Ievgen Makedonskyi, Daniil Sarafannikov
We construct three derived endofunctors on the derived category of graded integrable representations of the current algebra of a flat degeneration of D(2∣1,α), and prove that their classes in the Grothendieck group are the genus two Macdonald operators. Computing the graded Ext pairing explicitly, we deduce the self-adjointness of the genus two Macdonald operators and the orthogonality of genus 2 Macdonald polynomials with respect to the certain skew-bilinear form.
M. H. Shahzamanian
The determinant of a finite semigroup is closely related to the Frobenius property of its corresponding semigroup algebra, and it plays a significant role in coding theory applications. In this paper, we study determinants of twisted contracted semigroup algebras over finite fields. We extend Steinberg's determinant theory to this setting by establishing a Frobenius criterion for twisted contracted semigroup algebras over finite fields. We also develop a structural theory for two-generated twisted contracted monoid algebras. In particular, we identify classes for which the determinant of the associated matrix depends only on its zero--nonzero pattern, and is therefore independent of the values of the cocycle. Consequently, it suffices to verify the determinant condition for the trivial character, from which the corresponding result for all characters follows. Under suitable assumptions, these semigroups are completely determined by a small set of defining identities. Motivated by the character-theoretic construction of homogeneous weights on finite Frobenius rings, we construct homogeneous weights for Frobenius twisted contracted semigroup algebras and investigate the non-Frobenius case. In particular, we characterize the principal ideals on which the character-average weight satisfies the averaging property of homogeneous weights and thereby extend the classical notion of homogeneous weight to a prescribed family of principal ideals.
Erlend D. Børve, Maximilian Kaipel
We extend the notion of g-vector fan so that it is defined for a Hom-finite Krull-Schmidt 0-Auslander k-linear extriangulated category C with a projective silting object T. Moreover, we show that the g-vector fan admits an admissible partition, in the sense of the second-named author, which is induced by thick subcategories. One can thus define the picture category of C. We establish a bijection between thick subcategories of C generated by presilting objects containing all projective-injective objects and τ-perpendicular subcategories of the endomorphism k-algebra of T. This shows that our construction unifies all previous constructions of picture categories and τ-cluster morphism categories of finite-dimensional algebras. We introduce morphisms of partitioned fans to provide a common framework for the functorial relationships between picture categories of different algebras and categories.
Kevin Coulembier, Alexander Sherman
We study the relation between semisimple algebras, artinian simple algebras, and artinian absolutely flat algebras in ind-completions of tensor categories (rigid abelian monoidal categories). Our main results show that the three types of algebras coincide, if we restrict to commutative algebras in certain (all, if we accept the main conjectures in the field) symmetric tensor categories of moderate growth. We also provide examples to show that in general these notions tend to diverge, contrary to the classical case of algebras over a field. In one appendix we give the first, to the best of our knowledge, example of a tensor category with finite-dimensional morphism spaces but objects of infinite length. In a second appendix we establish a generalisation of Deligne's theorem that tannakian categories are neutral over algebraically closed fields.
Boming Jia
Let g be a complex simple Lie algebra of type G2, F4, or E8. We prove that the closure Omin(g) of its minimal nilpotent orbit is not isomorphic to (T∗X)aff for any smooth quasi-affine variety X. We do not assume that the isomorphism is equivariant or Poisson. We also prove the same statement in types A1 and A2.
Yeongseong Jo, Akash Yadav
We prove that the archimedean Asai L-factor attached to an irreducible generic representation of GLn(C) can be expressed as a finite sum of Flicker local zeta integrals. For an archimedean local field F, we also prove that the exterior-square L-factor attached to an irreducible generic representation of GLm(F) can be expressed as a finite sum of Jacquet--Shalika local zeta integrals.
Jérémie Pierard de Maujouy, Guillaume Neuttiens
Gibbs states are probability distributions defined on Hamiltonian G-manifolds. They are parametrized by an open set of the Lie algebra g and arise as natural equilibrium states with regard to the action of G on the system. In this paper, we focus on nilpotent orbits of SL(n,R). We show that for n>2, those orbits do not admit any Gibbs state.
Sam K. Miller
Tornehave and Yalçin proved that the tom Dieck homomorphism, which sends a virtual real representation to a unit of the Burnside ring, is surjective for any p-group S. We prove that this homomorphism, and the sign homomorphism it factors through, remain surjective when restricted to fusion-stable subgroups associated to a saturated fusion system on S. As a corollary, we close the main question posed by Mazza--Miller in arXiv:2508.07404 by showing that given a field k of positive characteristic, the Lefschetz homomorphism from the Picard group of the bounded homotopy category of p-permutation modules to the unit group of its Grothendieck ring is surjective for all finite groups if and only if k=F2.
Roger Casals
The main result of this article shows that the quiver mutation class of a malleable real Morsification uniquely determines the integral monodromy module of a plane curve singularity. In particular, we show that the quiver mutation class of a malleable divide determines the complex topological type of an irreducible plane curve singularity. This establishes the algebraic-to-topological implication of a conjecture of S.~Fomin, P.~Pylyavskyy, E.~Shustin and D.~Thurston in the malleable irreducible case. The result is proven by developing representation-theoretic techniques based on an equivariant Euler pairing in the derived category of continuous finite-dimensional dg modules over a differential bigraded Ginzburg algebra. A key step uses these techniques to show that the quiver mutation class of a plabic fence uniquely recovers the torsion part of the Alexander module of the associated smooth link.
Charles Eaton, Florian Eisele, Radha Kessar +2
We determine the Morita equivalence classes of 2-blocks with quaternion defect groups of arbitrary 2-power order, thereby completing the proof of Donovan's conjecture for blocks of tame representation type.
Sihai Jin
We prove the Shan--Yan--Zhao affine left-cell conjecture for the simple affine vertex algebras L−1(Dℓ), ℓ≥5. The distinguished vacuum block has exactly ℓ+1 simple objects, indexed by the subregular affine left cell containing s0. Two finite-dimensional ingredients enter the proof. A primitive-ideal inclusion shows that the predicted non-vacuum modules descend to the relevant quotient, while a separate exhaustion argument rules out further highest weights. It follows that the Grothendieck group is the q=1 specialization of the corresponding dual affine left-cell module. The same identification yields uniform character formulas in terms of subregular inverse Kazhdan--Lusztig coefficients.
Shuoyang Xu, Haibo Chen
In this paper, we classify all modules over the Takiff Block type Lie algebra B(q) for q∈C∗ that are free of rank one over U(h), where h=CL0,0⊕CW0,0, and give explicit formulas for their actions. The irreducibility and isomorphism classes of these modules are also determined. We then consider their tensor products with irreducible restricted modules and obtain necessary and sufficient conditions for irreducibility, together with criteria for isomorphism. Finally, restricting these modules to several natural subalgebras of B(−1) yields families of non-weight modules.
Robert Boltje, Serge Bouc, Deniz Yılmaz
Let p be a prime number. A DΔ-pair is a pair consisting of a finite p-group and a p′-automorphism of the group. In this paper, we introduce diagonal DΔ-pair bisets and a category whose objects are DΔ-pairs and whose morphism groups are Grothendieck groups of diagonal DΔ-pair bisets. Our main result shows that, over suitable coefficient rings, the category of diagonal p-permutation functors is equivalent to the category of linear functors on a natural quotient of this new category. In this way, diagonal p-permutation functors can be studied through a category built only from finite p-groups and their automorphisms of p′-order.
Youqi Yin, Shiping Liu
This paper introduces a novel classification approach for representation-finite artin algebras in terms of the maximal length of their indecomposable modules of finite length, which we call the representation bound. To this end, we develop methods to compute almost split sequences and establish lower bounds for the lengths of the Auslander-Reiten translates of certain modules over artin algebras. We apply these techniques and results to explicitly classify artin algebras of representation bound n for each positive integer n≤4.
Joseph Chuang, Kai Meng Tan
We obtain necessary and sufficient conditions for a block of an Ariki-Koike algebra to have the property that all its associated multipartitions have no addable node with a given residue. This leads to a classification of Scopes equivalence classes for Ariki-Koike algebras in terms of their pyramid numbers and Scopes vectors, generalising that for level 1 and for core blocks. Our proof is independent of the results of Richards for level 1.
Baoyu Zhang
For a finite group H, let ν(H) denote the maximum order of a nilpotent subgroup of H. We prove that every finite solvable transitive permutation group P on a finite set Ω has a subset Δ⊆Ω such that ∣P:PΔ∣≥ν(PΔ). We also prove that if a finite solvable group H acts faithfully and completely reducibly on a finite module V, then some x∈V satisfies ∣H:Hx∣≥ν(Hx). Consequently, we settle Gluck's well-known conjecture, open since 1985: every finite solvable group G satisfies ∣G:F(G)∣≤b(G)2, where F(G) is the largest normal nilpotent subgroup of G and b(G) is the largest degree of an irreducible complex character of G.
Stephen D. Miller
One of the challenges of the unitary dual problem is the daunting number of possible representations to consider. This article describes an approach to narrowing the search space for minimal principal series representations, in terms of the ``fundamental parallelepiped'' (or ``FPP''): the set of linear combinations of fundamental weights with coefficients in the interval [0,1]. An earlier conjecture of the author, which was rooted in work of Barbasch and then subsequently vastly generalized by Vogan (and recently proven by Davis and Mason-Brown), asserts that the FPP houses all dominant infinitesimal characters for which minimal principal series have a unitarizable quotient. We present a technique to prove this ``FPP inequality'' in specific examples, demonstrated here for the split real form E8(8), by introducing a limiting theory of intertwining operators as ν approaches ∞ in directions of fundamental weights. This limiting theory is inspired by Wilfried Schmid's work on variation of Hodge structure. As an application, we show that the unitary set for a particular minimal principal series consists of the closure of a single open alcove.