math.RT — Pith

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math.RT 2026-07-03

New bounds on parts for Sylow character constituents in S_n

by Lorenzo Vanzi

Part bounds for the Sylow permutation characters of S_n

Partitions labeling the irreducible parts of the 2-Sylow permutation character are restricted in length.

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We study the Sylow permutation character of the symmetric group at the prime 2 and prove some new bounds on the number of parts of partitions corresponding to its constituents.

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math.QA 2026-07-03

Unitriangular R-matrices conjugate via T-series and Theta series

by Huafeng Zhang

Unitriangular R-matrices of quantum affine algebras and Yangians via Theta series

The formula applies to any finite-dimensional representation and extends to the Yangian case.

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The universal R-matrix of the quantum affine algebra associated to a finite-dimensional simple complex Lie algebra admits a Gauss decomposition into an uper unitriangular part, an abelian part, and a lower unitriangular part. In this paper, we provide a simple conjugation formula for the unitriangular R-matrices with one tensor factor evaluated at an arbitrary finite-dimensional representation of the quantum affine algebra. Our formula involves the T-series of Frenkel--Hernandez and the Theta series introduced in a previous work. We also extend our conjugation formula to the Yangian case, making use of associators for triple tensor product representations of shifted Yangians.

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math.RT 2026-07-03

Subregular W-algebra at critical level is orbifold of W-superalgebra limit

by Thomas Creutzig, Xuanzhong Dai +1 more

Feigin-Semikhatov duality at the critical level

The duality persists at the limit and supplies block-wise equivalences inside the category of weight modules.

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The Feigin-Semikhatov duality asserts that the Heisenberg cosets of the subregular $W$-algebra of $\mathfrak{sl}_n$ at level $k$ and the one of the principal $W$-superalgebra of $\mathfrak{sl}_{n|1}$ at level $\ell$ coincide when the levels satisfy the Feigin-Frenkel relation $(k+n)(\ell+n-1)=1$. A similar duality holds between the subregular $W$-algebra of $\mathfrak{so}_{2n+1}$ and the principal $W$-superalgebra of $\mathfrak{osp}_{2|2n}$. We study these dualities in the critical/large level limit. We describe the centerless subregular $W$-algebra at the critical level as an orbifold of the large level limit of the principal $W$-superalgebra times a lattice VOA. Our construction yields a functor between certain categories of the two involved vertex algebras. We show that in this set-up one in fact gets block-wise equivalences of categories. Studying the principal block of the large level limit of the principal $W$-superalgebra then gives us the structure of the principal blocks of the subregular $W$-algebras in the category of weight modules (which is much larger than the more common category of lower bounded modules).

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math.RT 2026-07-03

Induced modules classify all simple restricted modules over deformed Schrödinger-Virasoro

by Haibo Chen, Yongtao Liu +1 more

Simple restricted modules over the deformative Schr\"{o}dinger-Virasoro algebra

Any module meeting the injective conditions arises from a simple module over a positive-part quotient, including the original algebra and de

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This paper investigates simple restricted modules over the deformed Schr\"{o}dinger-Virasoro algebra $\mathcal{G}_{\lambda,\mu}$, which gives a complete classification of them for some $\lambda,\mu\in\mathbb{C}$. More precisely, we provide a systematic construction of these modules, including highest weight modules and Whittaker modules, by inducing simple modules from the positive part's quotient algebras. We prove that any simple restricted $\mathcal{G}_{\lambda,\mu}$-module satisfying certain injective conditions is isomorphic to such an induced module. As an application, we obtain some simple weak $V(c)$-modules over vertex algebras associated to $\mathcal{G}_{\lambda,\mu}$ for some $\lambda,\mu\in\mathbb{C}$. Note that our results include the Schr\"{o}dinger-Virasoro algebra and the deformed $\mathfrak{bms}_3$ algebra as special cases, thereby improving upon some of the previously reported results of [5,Theorem 3.4] and [6,Theorem 2]. This work effectively classifies and generalizes the representation theory of the deformed family.

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math.RT 2026-07-03

Alternative route produces Euler characters for gl(m,n)

by A.N. Sergeev

Euler characters for general linear Lie superalgebra

Results stated in new terms extend prior work on the general linear Lie superalgebra.

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M. Gorelik and Th. Heidersdors in the papers \cite{GH} investigated Euler characters for Lie superalgebra $\frak{gl}(m,n)$ and $\frak{osp}(m.2n)$. In the present paper we also investigate Euler characters for Lie superalgebra $\frak{gl}(m,n)$ but we use a different approach and our results are formulated in different terms.

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math.GR 2026-07-03

p-rational characters above principal Sylow character force normality

by Silvio Dolfi, Pham Huu Tiep +1 more

Degrees of p-rational characters and normality of Sylow p-subgroups

Extending Itô-Michler, the p'-degree condition on this restricted set suffices to prove the Sylow p-subgroup is normal, including the strong

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Several refinements of (the normality part of) the celebrated It\^o--Michler theorem were obtained during the last two decades, in which the condition of having $p'$-degree, for a fixed prime $p$, is imposed only on some subsets of complex irreducible characters of a finite group $G$. We prove further extensions of these results, where this condition is now imposed on the irreducible characters which lie above the principal character of a Sylow $p$-subgroup and are either $p$-rational, or strongly real when $p=2$.

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math.RT 2026-07-03

Whittaker model vanishes iff Zelevinsky dual has segment of length n+1

by Taiwang Deng

A degenerate Whittaker criterion for mathrm GL_(2n)

Criterion for GL_{2n} gives explicit test in Langlands data and tests Prasad conjecture on L-functions

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Let $F$ be a non-Archimedean local field. Let $N$ be the unipotent radical of the standard parabolic subgroup of $\mathrm GL_{2n}(F)$ of type $(n,n)$ with fixed nondegenerate additive character $\psi$. For an irreducible admissible representation $\pi$ of $\mathrm GL_{2n}(F)$, a theorem due to Gomez--Gourevitch--Sahi on generalized Whittaker models gives a criterion for the vanishing of the twisted Jacquet module $\pi_{N,\psi}$ in terms of the wave-front set. We translate this orbit-theoretic answer into Langlands--Zelevinsky data: if $\pi=L(\mathfrak m)$, then $\pi_{N,\psi}=0$ if and only if the Zelevinsky dual $\mathfrak m^{\mathrm t}$ contains a segment of length at least $n+1$. We do this in response to a conjecture proposed by D.Prasad about the vanishing of $\pi_{N,\psi}$ in terms of the adjoint $L$-function $L(s,\pi\times\pi^\vee)$. We prove that, for every irreducible representation $\pi$, vanishing of $\pi_{N,\psi}$ implies the pole inequalities predicted by D.Prasad. However, we show that the converse implication is false by an explicit counterexample for $\mathrm GL_4(F)$. For the generalized Steinberg constituents $v_{P_\beta}^G$ of the principal series containing the trivial representation, we make an explicit calculation of when $\pi_{N,\psi}$ is zero. In particular, for $\mathrm GL_6(F)$, exactly three of the $32$ constituents of such a principal series violate the converse direction of the conjecture proposed by D.Prasad.

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math.AG 2026-07-02

Hecke operators establish chi-independence of BPS cohomology

by Ben Davison, Lucien Hennecart +3 more

Hecke operators on symplectic surfaces and chi-independence

Proves Toda conjecture for one-dimensional sheaves on quasi-projective symplectic surfaces and links BPS Lie algebra to tautological classes

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We prove Toda's chi-independence conjecture for the BPS cohomology of moduli spaces of one-dimensional sheaves on quasi-projective symplectic surfaces, relative to the Chow variety. We also identify the BPS Lie algebra associated with one-dimensional Mukai vectors with the subspace of tautological classes, giving an extension of Markman's tautological generation theorem from primitive to arbitrary Mukai vectors. The main structure input is a bialgebra structure on the cohomological Hall algebra of coherent sheaves on a quasi-projective symplectic variety S. The coproduct is obtained, by dimensional reduction, from a factorization coproduct for 3d cohomological Hall algebras, and gives rise to a global BPS Lie algebra attached to the stack of coherent sheaves on S. The link between this structure and the applications to chi-independence and tautological generation is provided by Hecke operators on BPS cohomology, which modify one-dimensional sheaves by zero-dimensional quotients. To make this construction work, we prove that there is an identification between the affinized BPS cohomology of the semistable locus and the primitive part of the coproduct on the entire moduli stack

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math.RT 2026-07-02

Types and beta-extensions built for all Bernstein blocks of quaternionic p-adic groups

by Daniel Skodlerack, Shuyang Ye

Semisimple types for quaternionic forms of p-adic classical groups and compatible beta-extensions

Construction uses transfer from the classical case to parametrize beta-extensions by chambers in the Bruhat-Tits building of the centralizer

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Let $G$ be a quaternionic form of a $p$-adic classical group ($p$ odd). We construct a Bushnell-Kutzko-Stevens type for every Bernstein block of the category of smooth complex representations of $G$. Further we construct a system of compatible $\beta$-extensions, i.e. a family of $\beta$-extensions parametrised by the points of a chamber of the Bruhat-Tits building of the centralizer $G_\beta$ which are related via transfer.

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math.RT 2026-07-02

One derivation governs every double transposed Poisson structure on an algebra

by Maxime Fairon, Nikita Safonkin

Double Transposed Poisson Algebras

It yields GL_N-equivariant transposed Poisson brackets on representation algebras and on their invariant rings via the trace map.

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We introduce double transposed Poisson algebras, a noncommutative analogue of the transposed Poisson algebras of Bai, Bai, Guo and Wu that is compatible with the Kontsevich--Rosenberg principle. We first consider a simplified version which we call id-adapted double transposed Poisson algebras and then explore the general definition. We prove that every such structure on a unital associative algebra $\mathbb{A}$ is governed by a single derivation $\mathbb{A}\to\mathbb{A}\otimes\operatorname{S}(\mathbb{A}/[\mathbb{A},\mathbb{A}])$. Furthermore, this induces a $\operatorname{GL}_N$-equivariant transposed Poisson structure on each representation algebra $\mathbb{A}_N=\Bbbk[\operatorname{Rep}_N(\mathbb{A})]$. We also define $H_0$-transposed Poisson structures, the transposed counterpart of Crawley-Boevey's $H_0$-Poisson structures, and use the trace map to obtain a transposed Poisson structure on the ring of $\operatorname{GL}_N$-invariants $\mathbb{A}_N^{\operatorname{GL}_N}$.

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math.RT 2026-07-02

Silting mutation extended to infinite dimensions

by Diego Alberto Barceló Nieves

Large silting mutation in extriangulated categories

Theory defined for n-cosilting complexes over any ring and infinite n-tilting modules over finite-global-dimension rings.

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Silting mutation in triangulated categories, both at the level of objects and of subcategories, was introduced in arXiv:1009.3370, and later generalized to extriangulated categories in arXiv:2303.08125. It simultaneously encompasses the mutation theories of cluster-tilting objects in cluster theory and of compact 2-term silting complexes and support $\tau$-tilting modules in $\tau$-tilting theory. In this article, we develop an infinite-dimensional analog of silting mutation in extriangulated categories with set-indexed (co)products, which we then apply to obtain a theory of mutation for $n$-cosilting complexes over an arbitrary ring, as well as for infinite-dimensional $n$-(co)tilting modules over a ring of finite global dimension. The former theory is also shown to reinterpret the cosilting mutation introduced in arXiv:2201.02147.

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math.RT 2026-07-02

Affine type A quiver Hecke algebras dualize to Iwahori-Hecke algebras

by Haruto Murata

Quantum imaginary Schur-Weyl duality

The parameter t is determined explicitly, deforming imaginary Schur-Weyl duality and enabling character computations with dual canonical bas

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We study quiver Hecke algebras of untwisted affine type $A$ with an arbitrary choice of parameters and establish a duality with the Iwahori-Hecke algebra of the symmetric group. The parameter $t$ of the Iwahori-Hecke algebra is explicitly determined by the parameters defining the quiver Hecke algebra. This duality provides a deformation of the imaginary Schur-Weyl duality introduced by Kleshchev and Muth. Furthermore, we prove that the characters of simple modules in the imaginary strata are computed in terms of the dual canonical basis and Kazhdan-Lusztig polynomials, and the characters of standard modules coincide with the PBW vectors of the corresponding quantum group under certain assumptions. In addition, we examine other untwisted affine types, where the quiver Hecke algebra is known to be independent of the choice of parameters and the imaginary Schur-Weyl duality with the symmetric group has been established. As in type $A$, we apply this duality to the computation of characters of simple and standard modules.

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math.GR 2026-07-02

Nilpotent groups have cyclic escape property

by Michael Björklund, Alexander Fish

Directional expansion in ergodic actions of countable groups

This forces directional expansivity in all their totally ergodic actions, while free groups of rank two or more lack the property.

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We study directional expansion for probability-measure-preserving actions of countable groups through a representation-theoretic group property, the cyclic escape property. An infinite countable group has the cyclic escape property if every totally ergodic unitary representation has arbitrarily small fixed-vector projections along infinite cyclic subgroups. This property implies directional expansivity for all totally ergodic actions. We prove that all infinite finitely generated nilpotent groups have the cyclic escape property, and conjecture the same for all infinite finitely generated polycyclic groups. We also prove the cyclic escape property for higher-rank simple lattices whose finite-dimensional unitary representations all have finite image; in particular, for $SL_n(\mathbb Z)$, $PSL_n(\mathbb Z)$, and $PGL_n(\mathbb Z)$, $n\geq 3$. By contrast, free groups of rank at least two do not have the cyclic escape property. The proofs exhibit two independent mechanisms: central spectral structure in nilpotent groups and stationary character rigidity in higher-rank lattices.

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math.RT 2026-07-02

Recurrence computes A_n subcategory counts via Fibonacci link

by Volodymyr Mazorchuk

On the number of extension closed additive subcategories for uniformly oriented A_n quivers

The number of extension-closed additive subcategories obeys a recurrence, connects to Fibonacci numbers, and admits exponential growth bound

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We provide a recurrence for computing the terms of the OEIS sequence A393920, introduced in \cite{KS}. We also describe a surprising connection between A393920 and the Fibonacci sequence A000045, obtain non-trivial lower and upper exponential bounds for its growth, and investigate connections with partial orders, Catalan numbers, and convex topologies on finite chains. For the representation-theoretic lattice underlying A393920, we describe its atoms, coatoms, join-irreducible and meet-irreducible elements.

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math.RT 2026-07-02

Strict polynomial functors regain universal property via category restriction

by Antoine Touzé (LPP)

The universal property of strict polynomial functors

Restricting the allowed tensor abelian categories restores freeness on one generator and shows Ext-algebras act on wider cohomology calculat

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In characteristic zero, the category of strict polynomial functors is well-known to be the tensor abelian category freely generated by one object. We show that this property fails in positive characteristic, but that it can be repaired by restricting the class of tensor abelian categories considered. The new universal property recovers several known constructions and shows that Ext-algebras of strict polynomial functors act on cohomological computations in many other contexts.

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math.RT 2026-07-02

Spetses inherit Harish-Chandra theories and all Ennola d-alities

by Gunter Malle

Harish-Chandra theories, Ennola d-ality and Rouquier blocks for spetses

Unipotent characters in the spets generalization satisfy dualities for every d and remain compatible with relative Hecke algebra blocks.

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It has been shown that the theory of unipotent characters of finite reductive groups admits a generalisation to objects whose Weyl group is a spetsial complex reflection group, called spetses. In this paper we prove several natural properties satisfied by the unipotent characters of spetses, in particular the validity of all Harish-Chandra theories as well as the existence of Ennola $d$-alities for all integers $d$, Alvis--Curtis duality, and compatibility with Rouquier blocks of relative Hecke algebras.

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math.RT 2026-07-02

B-stabilizers decide almost multiplicity-one for finite-field spherical varieties

by Fulin Chen, Fang Shi +1 more

Almost multiplicity-one property of spherical varieties over finite fields

The criterion on Borel stabilizers is the finite-field analogue of the strongly tempered condition.

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Let $H$ be a connected algebraic subgroup of a connected reductive group $G$ over a finite field $\mathbb F_q$ such that $G/H$ is a $G$-spherical variety, i.e., $G/H$ has an open dense $B$-orbit for each Borel subgroup $B$ of $G$. We formulate, for the pair $(G,H)$, an almost multiplicity-one property. Then we establish a criterion for this property in terms of the $B$-stabilizers on $G/H$. In particular, we will see that this property is analogous to the strongly tempered condition in characteristic $0$.

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math.RT 2026-07-02

Affine Hecke subalgebra carries canonical basis labeled by Weyl cosets

by Jonathan Gruber

Pseudo-centralizers in affine Hecke algebras

In types A_n, B_2 and G_2 the v-deformed Fomin-Stanley algebra is indexed by cosets of the finite Weyl group inside the affine Weyl group

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We introduce and study a subalgebra $\mathcal{B}$ of the affine Hecke algebra, which arises from a centralizer construction in the double affine Hecke algebra, and which may be regarded as a $v$-deformation of the affine Fomin-Stanley subalgebra introduced by Lam as a combinatorial model for the affine Grassmannian homology ring. In types $\mathsf{A}_n$ and $\mathsf{B}_2$ and $\mathsf{G}_2$, we show that $\mathcal{B}_\mathrm{aff}$ admits a canonical basis indexed by the cosets of the finite Weyl group in the affine Weyl group. We also discuss conjectural positivity properties of the canonical basis and explain how it can be used to study the center of the affine Hecke algebra.

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math.RT 2026-07-02

Reductive monoids classified over arbitrary base schemes

by Jingren Chi, Simon Jacques

Reductive monoids over general base

Classification theorem extends field results to general schemes and yields integral models plus orbit descriptions.

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We develop a theory of affine algebraic monoids over general base schemes whose unit groups are split reductive groups. Our main result is a classification theorem for such objects, generalizing works of Vinberg and Rittatore over a field. As applications, we obtain combinatorial descriptions and normality properties of orbit closures, prove a Steinberg-type theorem on adjoint quotients of reductive monoids over general base schemes, and construct finite type integral models of the Vinberg monoids. A main tool in our construction is Lusztig's theory of modified quantum groups and their canonical bases.

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math.RT 2026-07-01

Fukaya category uniquely deforms NilHecke algebra under Z2 grading

by Jasper van de Kreeke

Fukaya categories of Coulomb branches as unique deformations

Removing the matter divisor from horizontal Hilbert schemes reduces their Fukaya category to the single Z^2-graded deformation of the NilHec

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The symplectic geometry of Coulomb branches is complicated and it is particularly difficult to determine their Fukaya categories. Relative Fukaya categories present an approach to circumvent these difficulties by first computing the Fukaya category of the complement of a divisor and then solving a deformation problem. In this paper, we apply this approach to the specific case of horizontal Hilbert schemes by removing their matter divisor and narrowing down the set of possible deformations through an additional $ \mathbb{Z}^2 $-grading. We utilize an existing description of the Fukaya category after removal of the matter divisor, in particular we use a specific generating Lagrangian and the identification between its endomorphism algebra and the NilHecke algebra. The core of this paper consists of solving the deformation problem, after which we recover the result of Aganagic et al.

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math.RT 2026-07-01

Central isogenies generalize Steinberg on centralizer components

by Sean Cotner

Central isogenies and conjugacy classes in reductive groups

The extension accounts for non-reduced centralizers of unipotents when the universal cover is not étale and yields multiplicity formulas for

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Steinberg described the group of components of the centralizer of a semisimple element of a connected semisimple algebraic group $G$ as a subgroup of the fundamental group of $G$. We show that this description can be generalized to explain the fact that centralizers of unipotent elements can fail to be reduced when the universal cover of $G$ is not \'etale. As applications, we compute generic multiplicities in the special fibers of moduli spaces of L-parameters and universal deformation rings, and we show there is no Springer isomorphism for $\mathrm{PGL}_p$ in characteristic $p$.

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math.RT 2026-07-01

Andrews-Gordon series generalize Rogers-Ramanujan to A2 odd modules

by Motoki Takigiku, Shunsuke Tsuchioka

A generalization of partition identities of G\"ollnitz-Gordon, Rogers-Ramanujan and Nandi

The q-series equal characters of level-2 standard modules and satisfy sum-product identities except for the 6n+3 family.

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We propose Andrews-Gordon type series for certain level 2 standard modules of type $A^{(2)}_{\textrm{odd}}$, and prove the corresponding sum-product identities except for $A^{(2)}_{6n+3}$. These identities generalize the identities of G\"ollnitz-Gordon (mod 8), Rogers-Ramanujan (mod 5) and (partially) Nandi (mod 14).

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math-ph 2026-07-01

Quantum Stokes matrices quantize Riemann-Hilbert-Birkhoff map

by Xiaomeng Xu

Quantum Stokes matrices and quantum Riemann-Hilbert-Birkhoff maps

Exchange relations turn them into an associative algebra homomorphism for systems with poles of order p+1.

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In this paper, we introduce quantum Stokes matrices for a noncommutative version of meromorphic linear systems of ordinary differential equations with a pole of order $p+1$. We prove that these quantum Stokes matrices satisfy natural quantum exchange relations. These relations allow us to interpret the quantum Stokes matrices as an associative algebra homomorphism, which may be viewed as a deformation quantization of the Riemann-Hilbert-Birkhoff map, regarded as a Poisson map, for meromorphic connections with a pole of order $p+1$.

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math.RA 2026-07-01

Super Jordan plane localizes to matrix algebra over Weyl algebra

by Tao Lu

Prime spectrum and representations of the super Jordan plane

The identification proves the ring is prime and classifies its ideals and modules.

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We study the ring-theoretic structure and representation theory of the super Jordan plane $\mathcal{J}$ over fields of characteristic different from $2$. We prove that $\mathcal{J}$ is prime and classify its prime, primitive, and maximal ideals. We determine its classical ring of quotients and classify the finite-dimensional simple modules, while relating infinite-dimensional simple modules to those of the first Weyl algebra. Our approach is based on showing that a localization of $\mathcal{J}$ is a matrix algebra over a localization of the first Weyl algebra.

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math.NT 2026-07-01

Local-global compatibility holds for torsion classes at p ≠ ℓ

by Bence Hevesi

Local-global compatibility at pneqell for torsion automorphic forms

Extends Varma's result to Betti cohomology torsion via Scholze determinants and Z_ℓ representations of p-adic groups.

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We prove local-global compatibility results at $p \neq \ell$ for the automorphic group determinants constructed by Scholze, generalising the result of Varma to torsion classes appearing in Betti cohomology. Our argument combines the construction of Scholze with the theory of representations of $p$-adic general linear groups with $\mathbf{Z}_{\ell}$-coefficients.

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math.RT 2026-07-01

Conditions equate Ginzburg-Rallis and trilinear models for GL_6

by Xinrui Wang

On the trilinear and Ginzburg-Rallis models

Sufficient conditions make the two models isomorphic for representations induced from parabolic type [2^3] over local fields of char 0.

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Let $k$ be a non-archimedean local field of characteristic zero. We give sufficient conditions under which the Ginzburg-Rallis models of the induced representations of $\mathrm{GL}_6(k)$ from a parabolic subgroup of type $[2^3]$ are isomorphic to the trilinear models of the inducing data. We also give nonvanishing criterion for these trilinear models and Ginzburg-Rallis models.

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math.RT 2026-07-01

Regular unipotents map to explicit SL_2 modules under GL_3 reps

by Dibyendu Biswas

Image of Regular Unipotent under a Representation of GL₃(mathbb{C})

Image under any irreducible polynomial representation decomposes as SL_2 module with Jordan form.

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We study the image of a regular unipotent element under any finite-dimensional irreducible polynomial representations of $\mathrm{GL}_3(\mathbb{C})$. This problem is equivalent to decomposing certain compositions of irreducible representations as $\mathrm{SL}_2(\mathbb{C})$-modules. We give an explicit decomposition of this finding, its Jordan decomposition.

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math.AC 2026-07-01

Rouquier dimension bounded below by Krull dimension

by Yuki Mifune

A lower bound for the Rouquier dimension of derived categories over commutative rings

Over commutative noetherian rings the dimension of the bounded derived category of finitely generated modules is at least the ring's Krull d

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We prove that the Rouquier dimension of the bounded derived category of finitely generated modules over a commutative noetherian ring is bounded below by the Krull dimension of the ring.

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math.RT 2026-07-01

Kottwitz-Viehmann varieties geometrize orbital integrals

by Jingren Chi

An overview of the geometry of Kottwitz-Viehmann varieties

Overview explains the geometry and supplies an explicit SL3 case for reductive groups over local fields.

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This is an update of an expository article on the geometrization of orbital integrals of spherical Hecke functions on reductive groups over non-archimedean local fields, appeared in Proceedings of ICCM 2019. Compared to the published version, we add a last section on an example in SL3 case.

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math.RT 2026-06-30

Embedding maps flag-line orbits into Bruhat order on S_{n+1}

by Mark Colarusso, Sam Evens

Orbits on a product of two flags and a line and the Bruhat order, II

The map into orbits on the next flag variety lets both closures and monoid actions be read from the symmetric group.

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Let $G=GL(n)$ be the $n\times n$ complex general linear group and let $\B_{n}$ be its flag variety. A Borel subgroup $B$ of $G$ acts on $\B_{n}\times \mathbb{P}^{n-1}$ diagonally with finitely many orbits. In this paper, we give an embedding of the $B$-orbits on $\B_{n}\times \mathbb{P}^{n-1}$ into the $B$-orbits on the flag variety $\B_{n+1}$ of $GL(n+1)$ and show that this correspondence respects closure relations and preserves monoid actions. As a consequence both closure relations and monoid actions on the set of all $B$-orbits on $\B_{n}\times\mathbb{P}^{n-1}$ can be understood via the Bruhat order on the symmetric group on $n+1$ letters by using our results in \cite{Shpairs}. This amplifies work of Magyar \cite{Magyar} by making the closure relation more transparent and allows us to compute the monoid action using Demazure products. If $S_i$ is the stabilizer in $B$ of the line through the ith standard basis vector, we give an embedding of the $S_i$-orbits on $\B_n$ into the $B$-orbits in a single $G$-orbit in $\B_{n+1},$ and this embedding plays an essential role in the above results. We extend results from our papers \cite{CE21I}, \cite{CE21II}, and \cite{Shpairs}, and in particular show that for $S_i$-orbits on $\B_n,$ the closure ordering is given by the Richardson-Springer standard order.

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math.RT 2026-06-30

2-local L_n representations extend to singular and virtual monoids

by Carmen Caprau, Mohamad N. Nasser

On the structure of the singular triplet monoid and its virtual extension

k-local and Phi-type methods succeed for every such representation on SLM_n and VSLM_n and coincide under conditions only in the singular ca

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In this article, we introduce two new algebraic structures associated with the triplet group on $n$ strands, $L_n$: the singular triplet monoid $SLM_n$ and its virtual extension $VSLM_n$, defined in analogy with the singular braid monoid and the virtual singular braid monoid. We begin by presenting these monoids in terms of generators and relations, and then derive several alternative presentations of $VSLM_n$. Second, we investigate the problem of extending representations of $L_n$ to these monoids. Two extension methods are developed: the $k$-local type extension, which applies to $k$-local representations, and the $\Phi$-type extension, which applies to representations satisfying suitable commutativity conditions. We show that every $2$-local representation of $L_n$ admits extensions to both $SLM_n$ and $VSLM_n$ via the two methods. As an application, we consider a specific representation $\mu : L_n \longrightarrow \mathrm{GL}_n(\mathbb{Z}[t^{\pm1}])$ introduced recently by Nasser et al. We explicitly determine all homogeneous $2$-local extensions of $\mu$ to $SLM_n$ and $VSLM_n$, and compute the corresponding $\Phi$-type extensions. Furthermore, we compare these two extension methods, showing that they coincide for $SLM_n$ under suitable parameter conditions, while they do not coincide for $VSLM_n$. These results provide a systematic framework for extending representations of $L_n$ to $SLM_n$ and $VSLM_n$.

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math.NT 2026-06-30

Kloosterman bounds imply regular Bessel distributions

by Li Cai, Jingsong Chai +1 more

Bessel Distributions and Kloosterman Sums

Germ expansions transfer nontrivial bounds from Levi subgroups to full regularity for generic representations on p-adic reductive groups.

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Let $G$ be a split reductive group over a $p$-adic field. We give germ expansions of Kloosterman integrals for $G$. As an application, we prove that Bessel distributions are regular for all generic representations on $G$ provided that Kloosterman sums for any Levi subgroups of $G$ have nontrivial bounds.

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math.RT 2026-06-30

Ideal n-cotorsion pairs match exactly in Frobenius categories and their stables

by Yixia Zhang, Panyue Zhou

Ideal n-cotorsion pairs in Frobenius extriangulated categories

The quotient by projective-injective objects preserves the n-cotorsion conditions, transferring approximation results between the two settin

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Motivated by the correspondence between ideal cotorsion pairs in Frobenius exact categories and those in their stable categories, we introduce the notion of an ideal $n$-cotorsion pair in an extriangulated category. We study the relationship between ideal $n$-cotorsion pairs in a Frobenius extriangulated category $\mathcal C$ and those in its stable category $\underline{\mathcal C}=\mathcal C/\omega$. Our main result shows that $(\mathcal I,\mathcal J)$ is an ideal $n$-cotorsion pair in $\mathcal C$ if and only if $(\mathcal I/\omega,\mathcal J/\omega)$ is an ideal $n$-cotorsion pair in $\underline{\mathcal C}$. This provides a bridge between higher ideal approximation theory in Frobenius extriangulated categories and its counterpart in their stable categories. Additionally, in Krull--Schmidt exact categories, we establish a bijective correspondence between complete cotorsion pairs and complete ideal cotorsion pairs, answering a question of Fu, Guil Asensio, Herzog and Torrecillas.

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math.RT 2026-06-29

Quantum Langlands functor built via Whittaker coefficients in Betti setting

by Ekaterina Bogdanova

Quantum Betti geometric Langlands functor

The functor respects 2-Fourier-Mukai equivalence between gerbe 2-stacks for the center and the dual fundamental group.

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We construct the quantum geometric Langlands functor in the Betti setting via Whittaker coefficients. We show that the functor is compatible with the 2-Fourier-Mukai equivalence between sheaves of categories over 2-stacks $\operatorname{Ge}_{Z_G}$ and $\operatorname{Ge}_{\pi_1(\check{G})}$, which classify gerbes on $X$ with respect to the center $Z_G$ of $G$ and algebraic fundamental group $\pi_1(\check{G})$ of $\check{G}$.

0

math.QA 2026-06-29

Completed DAHA equals crossed product via formal star product

by Nikolay Grantcharov

Deformation theory of the Double Affine Hecke algebra of type (C_n^vee,C_n)

For every n the completed algebra of type (C_n^∨,C_n) is the universal deformation of the quantum torus crossed with the C_n Weyl group.

abstract click to expand

We study the double affine Hecke algebra (DAHA) of type $(C_n^\vee,C_n)$ from the perspective of deformation theory. First, we provide a zeros-and-residues realization of this algebra, extending the construction of Ginzburg, Kapranov, and Vasserot to the non-reduced affine root system setting. Specializing the parameters of the DAHA to the base point gives the crossed product of a quantum torus algebra with the finite Weyl group of type $C_n$. We then show that for all $n$, the completed DAHA is the formal universal deformation of this crossed product algebra, extending Oblomkov's result for $n=1$. Our proof explicitly identifies the completed DAHA with the undeformed crossed product algebra equipped with a formal star product.

0

math.SG 2026-06-29

3D left-invariant affine connections fully classified

by T. Aït Aissa, S. El Bourkadi +1 more

Three-Dimensional Real Affine Lie Groups

Decomposition into 2D plus 1D parts reduces the problem, quadratic solutions give all isomorphism classes and their properties.

abstract click to expand

We classify all left-invariant real affine connections in dimension three. Our approach reduces the three-dimensional problem to a two-dimensional one by decomposing each left-invariant affine connection into a two-dimensional part and an additional one-dimensional component. After characterizing all possible two-dimensional left-invariant affine connections, we return to the three-dimensional setting to obtain a simplified description of all three-dimensional left-invariant affine connections. We then explicitly solve the resulting simplified quadratic equations and perform a refined analysis up to isomorphism, leading to a complete classification. Furthermore, we determine several geometric and algebraic properties of these structures, including the Novikov, associative, radiant, and bi-symmetric conditions, as well as geodesic completeness.

0

math.RT 2026-06-29

Compatible Lie algebras break Lie

by Xabier García-Martínez, Manuel Ladra +2 more

Representations of compatible Lie algebras

Two-dimensional cases are fully classified and shown to have wild representation type with a Clebsch-Gordan rule for line representations.

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We study compatible Lie algebras from algebraic and representation-theoretic points of view, obtaining counterexamples to some fundamental theorems from classical Lie algebra theory, namely the theorems of Lie, Weyl and Levi. We also classify the two-dimensional compatible Lie algebras up to isomorphism and explore their representation theory, presenting families of indecomposable non-semisimple representations, showing that the solvable two-dimensional compatible Lie algebras have wild representation type, and classifying all irreducible finite-dimensional line representations. Finally, we prove a Clebsch-Gordan decomposition for tensor products of finite-dimensional irreducible line representations.

0

math-ph 2026-06-29

Wilson loop expectations factorize at large N on Z^d at strong coupling

by Thibaut Lemoine

The heat-kernel master field on mathbb{Z}^d at strong coupling

Infinite-volume limits exist, admit a full 1/N expansion with exponentially local coefficients, and the leading term obeys an area law.

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We solve large-$N$ Yang--Mills theory on $\mathbb{Z}^d$, for every $d\geq2$, at strong coupling, for structure group $\mathrm{U}(N)$ and for the heat-kernel action. More precisely, we prove that normalized Wilson loop expectations have infinite-volume large-$N$ limits, factorize at leading order, and admit an all-order $1/N$-expansion with exponentially local coefficients, whose leading order characterizes the master field. We also prove an area-law upper bound for the heat-kernel master field, with a stronger coefficientwise version. The proof is based on a rooted heat-kernel master loop equation. Unlike the Wilson-action equation or the two-dimensional Makeenko--Migdal equation, this equation does not close on Wilson loop observables alone; it closes on an extended space of loop observables coupled to compactly supported plaquette decorations. We prove a strong-coupling, order-truncated rooted trajectory expansion and then identify its leading term with the master field. The main inputs are the universal finite-$N$ duality formulas developed in the companion paper \cite{Lem26a} and large-$N$ heat-kernel estimates from \cite{LemMai25,LM2}.

0

math.AG 2026-06-29

Langlands equivalence extended to type A singular curves

by Yukinobu Toda

The Dolbeault geometric Langlands correspondence for type A groups beyond the elliptic locus

Dolbeault version holds for GL_r and SL_r/PGL_r on Hitchin base locus with at worst type A singularities.

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In this paper, we prove a Dolbeault geometric Langlands equivalence for $\GL_r$ and for the Langlands dual pair $\SL_r/\PGL_r$ over an open locus of the Hitchin base which strictly contains the elliptic locus. This open locus contains the points corresponding to spectral curves with at worst type $A$ singularities, without any restriction on the number of irreducible components. The Dolbeault geometric Langlands equivalence considered here is the one formulated in our previous work with Tudor P\u{a}durariu, which links categorical Donaldson--Thomas theory with the geometric Langlands correspondence. It relates coherent sheaves on moduli stacks of semistable Higgs bundles to the limit category associated with the full moduli stack of Higgs bundles. The use of limit categories is essential beyond the elliptic locus, where the full Higgs moduli stack is no longer quasi-compact and contains infinitely many Harder--Narasimhan strata. The key step is to prove the Whittaker normalization conjecture over the locus of spectral curves with type $A$ singularities, following and extending the strategy developed in the author's proof of the $\GL_2$ case over the reduced spectral curve locus. As a consequence, we also obtain the Dolbeault geometric Langlands conjecture for $\SL_2/\PGL_2$ over the reduced spectral curve locus.

0

math.RT 2026-06-29

Every symmetric group block has a known column when p is odd

by David J. Hemmer, Pavel Turek

New columns in decomposition matrices of symmetric groups for every block

Columns for partitions with even arm lengths on p-divisible hooks are multiplicity free and given by the odd sequence.

abstract click to expand

The central unsolved problem in the modular representation theory of symmetric groups is to find the decomposition matrices, which describe how irreducible representations in characteristic zero decompose upon reduction modulo a prime characteristic $p$. In this paper we determine a large number of new columns in these decomposition matrices, namely those labeled by partitions whose $p$-divisible hooks have all even arm lengths. In particular in odd characteristic $p$, for every possible block of every possible symmetric group $S_n$, we determine at least one complete column. These columns are multiplicity free and are described by a recently introduced combinatorial statistic of partitions (depending on $p$), called the odd sequence. As an application, we determine the indecomposable summands of Foulkes modules $H^{(2^m)}$.

0

math-ph 2026-06-29

Shirokov method lists transitive realizations for Lie algebras to dim 4

by Severin Pošta

Shirokov realizations of low dimensional Lie algebras

Generic cases plus subalgebra-derived nongeneric cases are given for all real algebras of dimension at most four and checked against prior w

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We compute the transitive realizations for the low dimensional cases of real Lie algebras up to dimension four using Shirokov's method. First, the generic realizations are given, then, making use of the known list of subalgebras, nongeneric realizations are computed. The result is compared with the known classification of Popovych et al.

0

math.RT 2026-06-29

Finite-rank U(h)-free sl(2) modules yield identifiable coherent families

by Dimitar Grantcharov, Khoa Nguyen +1 more

On U(mathfrak{h})-free modules of finite rank over mathfrak{sl}(2)

The weighting functor maps these non-weight modules to families whose structure follows directly from freeness, with simplicity criteria for

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We study $\mathfrak{sl}(2)$-modules that are free of finite rank over $U(\mathfrak h)$, where $\mathfrak h$ is a fixed Cartan subalgebra of $\mathfrak{sl}(2)$. These modules form a natural class of non-weight modules. The coherent families obtained from this class via the weighting functor are identified. We also study a distinguished class of indecomposable $U(\mathfrak h)$-free modules defined in terms of Jordan blocks and give a recursive description of their socle filtrations. Finally, we apply the general results to exponential modules arising from the first Weyl algebra and obtain simplicity criteria for these modules.

0

math.RT 2026-06-29

Exterior cube gamma factors for GL(6) stabilize under ramified twists

by Taiwang Deng, Dongming She

Stability of the exterior cube γ-factors for GL(6)

Representations sharing a central character produce identical gamma factors once twisted by any sufficiently ramified character of F×.

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We prove the stability of the Langlands-Shahidi local $\gamma$-factor for the exterior cube representation of $\mathrm{GL}_6$. More precisely, if $\pi_1$ and $\pi_2$ are irreducible admissible generic representations of $\mathrm{GL}_6(F)$ with the same central character, then \[ \gamma(s,\pi_1\otimes\chi,\wedge^3,\psi)= \gamma(s,\pi_2\otimes\chi,\wedge^3,\psi) \] for every sufficiently ramified character $\chi$ of $F^\times$, where $\chi$ is regarded as a character of $\mathrm{GL}_6(F)$ through the determinant. The proof uses the realization of the exterior cube representation by the maximal parabolic subgroup of the simply connected group of type $E_6$. We give an explicit description of the relevant geometric quotient $U_M\backslash N'$, compute its invariant measure, and relate Shahidi's partial Bessel functions to partial Bessel integrals on the Levi subgroup. The desired stability then follows from an asymptotic expansion of these partial Bessel integrals and the vanishing of highly ramified Mellin transforms.

0

math.RT 2026-06-29

GL_2(F_q) character tables have ~q^4/2 entries not divisible by fixed primes

by Anwesh Ray, Mishty Ray

Average divisibility in character tables of GL₂(mathbb{F}_q)

The proportion tends to 1/2 overall and to 1 among nonzero entries, with arguments of nonzero values equidistributed on the circle.

abstract click to expand

Let $q$ range over odd prime powers and let $G_q=\mathrm{GL}_2(\mathbb{F}_q)$. Fix a prime number $\ell$. Motivated by work of Peluse and Soundararajan on Miller's conjecture for character tables of symmetric groups, we study the proportion of entries in the character table of $G_q$ which are not divisible by $\ell$, in the sense of divisibility in the ring of algebraic integers. We prove that $N_\ell(q)=\frac{q^4}{2}+O_\epsilon(q^{3+\epsilon})$ for every $\epsilon>0$, where $N_\ell(q)$ denotes the number of entries which are not divisible by $\ell$. We also show that the number of zero entries is $\frac{q^4}{2}+O_\epsilon(q^{3+\epsilon})$. Consequently, the proportion of all entries not divisible by $\ell$ tends to $1/2$, while the proportion of nonzero entries not divisible by $\ell$ tends to $1$. This differs significantly from the symmetric-group case, where almost every character-table entry is divisible by any fixed prime. We also prove an angular equidistribution result for the nonzero character values as $q\to\infty$. We show that the arguments become equidistributed in $[0,2\pi]$. This proves an analogue of Miller's question on the distribution of signs among the nonzero entries in character tables of symmetric groups.

0

math.QA 2026-06-29

Zonal spherical functions equal symmetric Koornwinder-Macdonald polynomials

by Philip Schlösser

Zonal Spherical Functions of Quantum Symmetric Pairs

The identification for quantum symmetric pairs now covers non-standard cases and non-reduced root systems, with a conjecture supplied for th

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We identify the zonal and character spherical functions for quantum symmetric pairs with the symmetric Koornwinder--Macdonald polynomials. To this end, the methods of Letzter's 2004 paper are translated to modern conventions and right coideal subalgebras, and extended to non-standard cases and cases with non-reduced restricted root systems. For the last elusive Satake type, FII, a conjecture is provided.

0

math.RT 2026-06-29

Algebras exist with arbitrarily high brick chain complexity

by Claus Michael Ringel

The brick chain complexity of an artin algebra

The shortest brick chain filtrations for indecomposables grow without bound over all Artin algebras.

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We consider the category of finitely generated modules over an artin algebra $A$. It is known that any module $M$ has a brick chain filtration. We say that M has brick chain complexity at most $t$ provided $M$ has a brick chain filtration of length at most $t$. The brick chain complexity of A is by definition the supremum of the brick chain complexity of the indecomposable $A$-modules. The aim of this note is to calculate the brick chain complexity for some algebras. We will exhibit algebras with arbitrarily large brick chain complexity.

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0

math.NT 2026-06-29

Automorphic forms split into cusp forms plus Eisenstein coefficients

by Devadatta G. Hegde

On Franke's theorem in the simplest case

Direct proof for level one on the half-plane uses only growth conditions and Green's identity, bypassing Langlands spectrum construction.

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For level one spherical automorphic forms on the upper half-plane, we prove directly that every automorphic form is a sum of a cusp form and a linear combination of Laurent coefficients of the standard Eisenstein series. This is the simplest instance of Franke's general theorem, which asserts that automorphic forms on a reductive group are spanned by Laurent coefficients of Eisenstein series induced from cuspidal automorphic forms on Levi subgroups. Unlike Franke's general argument, ours does not invoke Langlands' construction of the discrete automorphic spectrum from cuspidal Eisenstein series. It rests instead on basic analytic properties of automorphic forms and Green's identity.

0

math.RT 2026-06-29

All p(3) irreducibles classified in char 3

by Ye Ren

The representations of the Lie superalgebra p(3) in characteristic 3

The paper supplies the full list of simple modules and their character formulae for this rank-2 Lie superalgebra over algebraically closed f

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Let $g$ be the Lie superalgebra $p(3)$ of rank 2 over an algebraically closed field $K$ of characteristic $p=3$. We classify all irreducible modules of $g$, and give the character formulae for irreducible modules.

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math.RT 2026-06-29

Semisimple rings make infinite species representations hereditary

by Raphael Bennett-Tennenhaus, Job Daisie Rock

Representations of infinite species

Purity then yields conditions for decomposition into indecomposables with local endomorphism rings and produces examples in homology and fie

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We consider species, consisting of a possibly infinite set of rings, and bimodules between them. Simson realised the category of representations as a functor category, which we prove is hereditary when each of the rings is semisimple. We use purity to provide sufficient conditions, in order for a representation to decompose into indecomposables with local endomorphism rings. For any bifunctor valued in bimodules, we functorially construct species equipped with commutativity conditions. This generates examples coming from a range of topics, such as subobject lattices in abelian length categories, the field choice problem in persistent homology, and topological field theories with defects.

0

math.CO 2026-06-26

C-vector projections fill bands except on Ã_n source-sink diagonals

by Sarah B. Brodsky

Coordinate projections of c-vectors of cluster algebras from the annulus

The Auslander-Reiten defect blocks the source-sink pair in annulus type while all other coefficient-one bands fill completely.

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For an acyclic cluster algebra, the $c$-vectors are, up to sign, the real Schur roots of the associated root system. We study the two-coordinate projections $(c_v, c_w)$ of this configuration: when the difference $c_v - c_w$ is bounded the image lies in a band of lattice lines, and we ask when the projection fills that band. A band-existence dichotomy, valid in every acyclic type, shows the difference is bounded if and only if the null root satisfies $\delta_v = \delta_w$. For affine type $\widetilde{A}_n$ (the annulus), in the source-sink orientation, we resolve the filling question completely: every coordinate projection fills its band except along the source-sink diagonal, which carries only the finite regular part. The obstruction is the Auslander--Reiten defect, which a projection sees on its diagonal exactly when the defect is a coordinate difference; the only such pair is the source-sink pair of $\widetilde{A}_n$, so the pattern depends on the chosen seed. More generally, every banded pair of null-root coefficient one fills, except these diagonals. Off the diagonal a banded pair in $\widetilde{E}_7$ fails to fill, so non-filling is not confined to type $\widetilde{A}_n$; a computation classifies the pairs of coefficient at least two over a range of affine types, where this $\widetilde{E}_7$ pair is the only further failure, and the general classification remains open.

0

math.CO 2026-06-26

Scalar product counts matrix extensions in adjoint orbits

by Samrith Ram

Enumerating matrices with prescribed entries in an adjoint orbit

Hall product of skew modified Hall-Littlewood and q-Whittaker functions enumerates endomorphisms with prescribed columns and invariants over

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We study intersections of conjugacy classes of square matrices over a finite field with affine coordinate subspaces, or equivalently matrices in a fixed adjoint orbit with prescribed entries. Our main result treats the case of prescribed columns: for a partially defined linear map we give a Hall scalar product formula for the number of extensions to an endomorphism with prescribed similarity invariants. This formula is expressed in terms of skew modified Hall--Littlewood functions and $q$-Whittaker functions. As applications, we count monic matrix polynomials over $\mathbb{F}_q$ with prescribed Smith normal form and with prescribed determinant, and recover the Gerstenhaber--Reiner formula for the number of square matrices with a fixed characteristic polynomial. We also note that known point-count formulas for Hessenberg varieties imply related formulas for Hessenberg supports involving chromatic quasisymmetric functions, motivating polynomiality questions for more general supports and prescribed affine slices.

0

math.AG 2026-06-26

Motivic Hall algebra recovers Auslander-Reiten sequences

by Aayush Verma

Hall Geometry and Auslander-Reiten Quiver

The algebra and its moduli-stack correspondence determine the sequences and the quiver they form.

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We show how the geometric information in the motivic Hall algebra and the correspondence of the moduli stack recovers the Auslander-Reiten sequences and the Auslander-Reiten quiver.

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math.QA 2026-06-26

Chiral Poisson cohomology controls quantizations of vertex Poisson algebras

by Dylan Butson, Sujay Nair

On the deformation theory of chiral quantizations

Obstructions reduce to de Rham cohomology for affine symplectic varieties and prove rigidity of boundary Virasoro minimal models.

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We give an operadic approach to deformation quantization of vertex Poisson algebras, a chiral analogue of the traditional problem of deformation quantization of Poisson algebras. Our main result is an order-by-order deformation-obstruction theory for such quantizations, controlled by the chiral analogue of Poisson cohomology. In the special case of chiral quantizations of affine symplectic varieties, quantizations of the vertex Poisson algebras of functions on their arc spaces, we prove that this deformation-obstruction theory is controlled by their de Rham cohomology. As another application, we prove that the boundary Virasoro minimal models are rigid under deformations.

0

math.RT 2026-06-26

Dynkin subgraphs label new components in L(ρ) ⊗ L(ρ)

by Rekha Biswal, Patrick Polo

On some components of L(rho)otimes L(rho) associated with rooted trees for symmetrizable Kac-Moody algebras

Elements π_{D,I} of weight λ_{D,I} − ρ are shown to be ρ-dominant in B(ρ) when I has simple bonds and no long cycles, adding explicit famili

abstract click to expand

Let $\mathfrak{g}$ be a symmetrizable Kac-Moody algebra over $\mathbb{C}$ and let $L(\rho)$ be the irreducible integrable $\mathfrak{g}$-module with highest weight $\rho$. Let $I$ be a subgraph of the Dynkin diagram of $\mathfrak{g}$ which has only simple bonds and no cycle of length $\geq 3$. For every subset $D$ of $I$, denote by $\beta_D$ the sum of the simple roots corresponding to $D$. To every $D \subset I$ such that $\lambda_{D,I} = 2\rho - \beta_I - \beta_D$ is dominant, we associate certain elements $\pi_{D,I}$ of weight $\lambda_{D,I} {-} \rho$ in the crystal $B(\rho)$, which depend on the choice of a root vertex in each connected component of $I$. Then we prove that our elements are $\rho$-dominant elements of $B(\rho)$, hence provide new families of components of the tensor product $L(\rho)\otimes L(\rho)$.

0

math.RT 2026-06-26

Brick-infinite algebras admit infinitely many non-τ-rigid bricks

by Kaveh Mousavand, Charles Paquette

Brick infinite algebras admit infinitely many non-τ-rigid bricks

Proves that only finitely many non-τ-rigid bricks forces the algebra to have only finitely many bricks total.

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Let $A$ be a finite dimensional algebra over an algebraically closed field. Motivated by some foundational interactions between bricks and $\tau$-rigid modules, we prove, in full generality, that if all but finitely many bricks of the algebra $A$ are $\tau$-rigid, then $A$ is brick-finite. Equivalently, any brick-infinite algebra admits infinitely many bricks which are not $\tau$-rigid. Because $\tau$-rigidity implies rigidity, our result verifies a weaker version of an open conjecture which states that if (almost) all bricks over $A$ are rigid, then $A$ should be brick-finite. In retrospect, this work strengthens some previous results and contributes to the recent studies of a series of challenging problems, all tied to the $2$nd brick-Brauer-Thrall conjecture. More specifically, without any tameness assumption, we settle a question that was previously known only for $E$-tame algebras.

0

math.QA 2026-06-26

Artin monoid maps give many twin homomorphism solutions

by Arkady Berenstein, Jacob Greenstein +1 more

Artin monoids, their homomorphisms and twins

Optimal maps injective on generators and their compositions solve the twin problem for Coxeter groups and Hecke monoids.

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Motivated by the twin homomorphism problem for Coxeter groups and the corresponding Hecke monoids, we find a large class of its solutions originating from standard homomorphisms of Artin monoids and their compositions. These homomorphisms are expected to be injective when they are optimal and injective on generators, which generalizes the homogeneous homomorphisms and the famous Tits conjecture settled by Crisp and Paris. We classify disjoint standard homomorphisms and conjecture the complete classification when the domain is of rank two.

0

math.AG 2026-06-26

Parabolic stratification extends to all reductive groups for motivic formulas

by Alfonso Zamora

e-polynomials of character varieties

Lecture notes show explicit expressions for character varieties and reduction of mirror symmetry conjectures.

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These lecture notes contain the material presented at one of the mini-courses of the Workshop on Character Varieties and Higgs Bundles held in Liberia, Guanacaste, Costa Rica, in August 2025. They also contain some exercises for the students attending the conference. This manuscript contains the basic ideas and constructions about $e$-polynomials in character varieties and the state of the art of certain research in the field, plus some new further directions. We introduce mixed Hodge structures and $e$-polynomials, together with a series of arithmetic (counting points over finite fields) and geometric (stratification into parabolic types) techniques to compute them. We include a complete example of the calculation of the $e$-polynomial for the ${\rm GL}_3$-character variety of the free group. Finally, we extend the geometric stratification into parabolic types to a general reductive group $G$ to obtain explicit motivic expressions for the $G$-character varieties, and reduce certain topological mirror symmetry conjectures for these moduli spaces.

0

math.RT 2026-06-25

Quantum Harish-Chandra bimodules relate to affine Soergel at roots of unity

by Trung Vu

Quantum Harish-Chandra bimodules at roots of unity and affine Hecke category

The connection at odd order roots of unity also reaches the non-commutative Springer resolution.

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The category of Harish-Chandra bimodules for quantum groups was first appeared in the works about topological quantum field theory of surfaces. In this paper, we study this category when the quantum parameter q is an odd order root of unity. We relate the category to the category of affine Soergel bimodules and to non-commutative Springer resolution.

0

math.RT 2026-06-25

Graded Euler characteristic fixes all Hochschild homology degrees from center and HH_0

by Jon Wallem Anundsen, Mads Hustad Sand{o}y

Hochschild (co)homology and cyclic homology via a graded Euler characteristic with applications to higher preprojective algebras

The method works for higher preprojective algebras of tensor products of type A hereditary algebras.

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Computing the structure of the Hochschild (co)homology and the cyclic homology of an algebra can be hard work, but Etingof and Eu showed that it can be done surprisingly easily for preprojective algebras of ADE Dynkin type, at least if one only wants to know the graded vector space structure of each Hochschild cohomology group. Their method is based on exploiting strong structural features of such a preprojective algebra via a graded Euler characteristic that can computed using the algebra's graded Cartan matrix. In this paper, we present a generalization of the method used by Etingof and Eu to higher preprojective algebras. We also apply our generalization to the higher preprojective algebras of the 2-representation finite algebras that arise as tensor products of representation finite hereditary algebras of type A. For this, it turns out to be enough to know the graded vector space structure of the center and the zeroth Hochschild homology to be able to deduce the graded vector space structure of the Hochschild (co)homology and the cyclic homology in all other degrees.

0

cs.CC 2026-06-25

Separating modules match Weisfeiler-Leman power for graphs

by Joshua A. Grochow, Jacob Urisman

Graph Isomorphism and Representation Theory

Polynomial spaces of symmetric circuit size n^Θ(k) distinguish non-isomorphic graphs exactly as Θ(k)-WL does.

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We introduce an approach to distinguishing isomorphism types of graphs based on vector spaces of polynomials that are set-wise invariant under permutations ("separating modules," which are representations of the symmetric group), inspired by the Geometric Complexity Theory approach to separating complexity classes (Mulmuley & Sohoni, SIAM J. Comput., 2001). We characterize the power of this method for distinguishing non-isomorphic graphs under several different complexity measures: - We show that separating modules of "support-degree" $k$ (each monomial touches at most $k$ vertices) are equivalent to the counts of $O(k)$-vertex subgraphs. This is strictly weaker than $O(k)$-dimensional Weisfeiler--Leman (F\"urer, ICALP '01). - We show that separating modules of symmetric circuit size $n^{\Theta(k)}$ are equivalent to $\Theta(k)$-WL. This generalizes and strengthens a result of Dawar & Wilsenach (CSL '18; ICALP '20; ACM Trans. Comput. Log., 2022; Theory Comput., 2025): they proved one direction of this equivalence for invariant polynomials; we generalize to separating modules and prove both directions. - When considering only the multiplicities of separating modules (as was proposed in GCT by Mulmuley & Sohoni, ibid., rather than the polynomials themselves), we show that two graphs are separated by multiplicities if and only if their automorphism groups have different cycle indices. The latter result is notable in the analogy with GCT, as it is the only result we are aware of in which the multiplicity approach to separating isomorphism types of objects has been given an "intrinsic" characterization in terms of the objects themselves. We use this to show that for graphs, multiplicity obstructions are stronger than occurrence obstructions. We also connect invariant polynomials to the Graph Reconstruction Conjectures and Forman's "invariants of finite type" (Adv. Math., 2004).

0

math.AT 2026-06-25

Smooth Artin motives match Bredon modules for étale fundamental group

by Yorick Fuhrmann

Profinite Borel completeness and smooth Artin motives

Nisnevich case extends Voevodsky theorem; étale case equates completeness notions with sheaves versus hypersheaves.

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The purpose of this paper is twofold. In the first part, we revisit the description of the $\infty$-category of Borel complete equivariant spectra for a finite group given by Mathew-Naumann-Noel, introduce a version with coefficients, and then consider Borel equivariance for profinite groups. Here we identify two generally differing notions: levelwise Borel completeness and the hypercompletion thereof. In the second part, we study variants of smooth Artin motives, which are subcategories of the $\infty$-categories of effective Nisnevich and \'etale Voevodsky motives over a base scheme $S$ that are controlled by the \'etale fundamental group $\pi_1^{\mathrm{\'et}}(S)$. In the Nisnevich case, we extend a theorem of Voevodsky and identify smooth Artin motives with modules over the Bredon cohomology spectrum for the profinite group $\pi_1^{\mathrm{\'et}}(S)$. In the \'etale case, we show that the difference between our two notions of profinite Borel completeness is precisely the difference between \'etale sheaves and hypersheaves on finite \'etale schemes.

0

math.RT 2026-06-25

Refined algorithm lists generators of cluster automorphism groups

by Jindong Zhao, Haiyan Zhu

Effective Computation of Mutation Paths and Generators of Cluster Automorphism Groups

Improved marked-vertex method enumerates all mutation paths and produces explicit generators for every finite-mutation-type rank-4 case.

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In this paper, we improve the marked-vertex strategy introduced by Fu and Liang, and design the algorithm to compute all mutation paths and generator elements of cluster automorphism groups efficiently. As an application, we get generators of cluster automorphism groups of cluster algebras of finite mutation type of rank 4.

0

math.DG 2026-06-25

Condition for trivializing deformations of Lie subalgebras

by Ilias Ermeidis

On the Moser trick for Lie subalgebras and foliations

Necessary and sufficient criterion given via direct Moser proof for foliations, extending to subalgebroids

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Given a smooth deformation of a Lie subalgebra, we establish a necessary and sufficient condition for its smooth triviality and derive an analogous criterion for Lie ideals. We then give a direct proof of the Moser trick for foliations, which forms the basis for extending this result to general Lie subalgebroids.

0

math.RT 2026-06-25

Orbit construction defines preprojective algebras for hereditary algebras

by Aaron Chan, Osamu Iyama +1 more

Preprojective algebras and generalisations: A short survey

Contracted and total generalizations raise open questions about orbit algebras of H-modules.

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The preprojective algebra of a hereditary algebra $H$ can be defined as a certain orbit construction of the regular representation generated by the Auslander-Reiten translation. In this short survey, we will look at two important generalisations, namely, the contracted preprojective algebra and the total preprojective algebra. We will include several open problems and questions motivated by examples in the hope to stimulate future research on general orbit algebras of $H$-modules.

0

math.RT 2026-06-25

G wr PT_n monoid algebra has global dimension n-1

by Itamar Stein

The representation theory of the wreath product of a finite group with the monoid of all partial functions on a finite set as an EI-category algebra

Quiver follows from multiplicities of simple G-modules in tensor products via the Ehresmann EI-category.

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Let $G$ be a finite group. We provide a description of the ordinary quiver of the complex monoid algebra of the wreath product $G \wr \mathrm{PT}_n$, where $\mathrm{PT}_n$ denotes the monoid of all partial functions on an $n$-element set. This description depends on the multiplicities of simple $G$-modules appearing in the decomposition of tensor products of simple $G$-modules. We also prove that the global dimension of this algebra is $n-1$. Both results are obtained by analyzing the associated Ehresmann EI-category related to the monoid. Finally, we describe the quiver of the algebra of the wreath product of $G$ with the submonoid of all order-preserving partial functions.

0

math.RT 2026-06-25

Cleft extension restriction functors match on singular equivalences under equivariance

by Miltiadis Karakikes

Equivariant Cleft Extensions and Singular Equivalences

The ordinary and equivariant versions induce singular equivalences together once the cleft extension lifts to the group action setting.

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We study the equivariant lifting of cleft extensions of abelian categories and its impact on singularity categories. Specifically, we establish the necessary framework for lifting a cleft extension to a G-equivariant cleft extension. Furthermore, we prove that a restriction functor associated to a cleft extension induces a singular equivalence if and only if its equivariant counterpart does. As a concrete application, we demonstrate that the skew group ring of a $G$-equivariant $\theta$-extension is isomorphic to a $\widehat{\theta}$-extension of the base skew group ring, allowing us to lift singular equivalences for these structures.

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math.RT 2026-06-25

Total positivity yields cell decomposition for symmetric spaces

by Huanchen Bao

Total positivity and symmetric spaces

Hausdorff closure of G>0 in G/K carries explicit positive parametrizations with subtraction-free transitions between two natural families.

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We define a notion of total positivity for the symmetric space $G/K$ by taking the Hausdorff closure of the image of Lusztig's totally positive part $G_{>0}$ in $G/K$. We introduce double Bruhat cells for the symmetric space and define their totally positive pieces. We prove a cell decomposition of the totally nonnegative symmetric space, give explicit positive parametrizations of all cells, establish closure relations, and show that the transition maps between the two natural families of parametrizations are subtraction-free.

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math.RT 2026-06-25

Whittaker averaging kernel characterized microlocally

by Jeremy Taylor

The tilting property of Whittaker averaged central sheaves

The description generalizes the anti-temperedness equivalence for automorphic sheaves and extends tilting of central sheaves to integer coef

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We characterize the kernel of Iwahori-Whittaker averaging in microlocal terms. Applying this to automorphic sheaves, we generalize Faergeman and Raskin's theorem that anti-temperedness is equivalent to having irregular singular support. Moreover, using a Radon transform argument of Bezrukavnikov and Morton-Ferguson, we extend the tilting property of Whittaker averaged central sheaves to integer coefficients.

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math.GR 2026-06-24

Definition proposed for representations of skew left braces

by A Ballester-Bolinches, R. Esteban-Romero +1 more

Representations of finite skew braces

Trifactorised groups associated with the braces are shown to play a fundamental role

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One of the classical open problems in the theory of skew left braces is the study of their representation theory. We propose in this paper a definition of representation of a skew left brace and study its properties. Representations of the trifactorised groups associated with skew left braces play a fundamental role.

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math.RT 2026-06-24

One-parameter family makes Schur reps a parabolic O subcategory

by Addison Day, Jonathan R. Kujawa

Interpolating Schur Algebras

The algebras quotient to Schur algebras at nonnegative integers while preserving based quasi-hereditary and highest weight structure.

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We introduce and study a one-parameter family of algebras that naturally generalize the Schur algebras. We show the Schur algebra is canonically a quotient when the parameter is a nonnegative integer, characterize when they are semisimple, show they are based quasi-hereditary, and that their category of representations is a highest weight category that can be identified as a subcategory of parabolic category $\mathcal{O}$ for the general linear Lie algebra.

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math.QA 2026-06-24

Sheaf of vertex superalgebras recovers singular hypertoric variety

by Tomoyuki Arakawa, Andrea E. V. Ferrari +1 more

Vertex Superalgebras for Hypertoric Varieties and 3d Abelian Gauge Theories

Global sections of the ħ-adic construction equal the A-twisted boundary and prove the Higgs branch conjecture for abelian 3d theories.

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Hypertoric (or toric hyperk\"ahler) varieties are a class of symplectic singularities and their resolutions, obtained as Hamiltonian reductions of a symplectic vector space acted on by a torus. In physics, they appear as Higgs (and Coulomb) branches of 3d $\mathcal{N}=4$ supersymmetric quantum field theories with abelian gauge group. In this work, we construct an $\hbar$-adic (in the sense of microlocalisation) sheaf of vertex operator superalgebras over a given smooth hypertoric variety. Its global sections give the $A$-twisted boundary of the corresponding 3d gauge theory. We use this to prove that the associated affine variety of this hypertoric vertex operator superalgebra recovers the singular hypertoric variety. This proves the 3d Higgs branch conjecture for a large class of boundary vertex operator superalgebras. In particular, these vertex operator superalgebras are quasi-lisse. This is in contrast to the (purely even) hypertoric vertex operator superalgebras (and their $\hbar$-adic localisations) constructed previously by Kuwabara as global sections of sheaves on families of universal Poisson deformations of the hypertoric varieties. These are generally not quasi-lisse. We show that the vertex operator superalgebras defined in this paper are (fermionic) simple-current extensions of those defined by Kuwabara, and investigate the consequences for symplectic duality and characters. We observe that the latter are upgraded from partial (or false) theta functions to quasimodular forms.

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math.RA 2026-06-24

All smooth central Sklyanin quadrics are standard

by Izuru Mori, Kenta Ueyama +1 more

Irreducible noncommutative quadrics

Classifying every singular case shows that smoothness forces the standard form.

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In this paper, we study irreducible noncommutative quadrics $S/(f)$ via noncommutative graded matrix factorizations. We show that the line modules over $S/(f)$ are described by the rulings arising from indecomposable noncommutative linear matrix factorizations of $f$ of rank $2$. We study when Zhang twists of a standard smooth irreducible noncommutative quadric are standard. Finally, by identifying all singular central Sklyanin quadrics, we prove that every smooth central Sklyanin quadric is standard.

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math.GT 2026-06-24

Genus-one differential strata with four or more singularities are never orbifold K(π,1)

by Dawei Chen, Jingyin Huang +2 more

Non-asphericity of strata of genus-one differentials and stability spaces

Result supplies counterexamples to Kontsevich conjecture on quadratic differentials and to conjectures on contractible stability spaces.

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We show that when the number of zeros or poles is at least four, every connected component of the strata of differentials in genus one with prescribed zero and pole orders is not an orbifold $K(\pi,1)$. For quadratic differentials, this provides infinitely many counterexamples to a conjecture attributed to Kontsevich, as well as to a folklore conjecture concerning the contractibility of spaces of Bridgeland stability conditions.

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math.RT 2026-06-24

Polarization reduces invariants of GL(n)

by L. Darondeau, M. Florence +1 more

Weyl's Polarization in Classical Invariant Theory: A Primer, with Worked Examples

Worked examples over the reals show how Weyl's operator generates the full rings from multilinear forms.

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Hermann Weyl's The Classical Groups is a landmark work connecting classical invariant theory with modern representation theory. It shows how polynomial invariants of the general linear, orthogonal, and symplectic groups can be systematically understood through linear representations and tensor methods. The current note is primarily based on a personal reading of the book of Weyl and of the more accessible books Classical Invariant Theory by Kraft and Procesi and by Olver. It is neither exhaustive, nor original, nor state of the art. We focus on a few selected aspects, aiming for an elementary and concrete approach. We work over the field of reals R with the classical groups GL(n), SL(n), O(n), and SO(n). Most of our efforts have been devoted to carefully worked examples, introducing just enough of the general theory to handle them effectively.

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hep-th 2026-06-23

Lagrangians found for mixed-antisymmetric higher-spin fields

by Alexander A. Reshetnyak

General Lagrangian formulations for mixed-antisymmetric tensor fields on flat backgrounds

BRST conversion via so(k,k) Verma modules yields equivalent unconstrained and constrained formulations for k-column Young tableaux in flat s

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Lagrangian formulations for (ir)reducible integer higher-spin massless and massive Poincare group representations subject to Young tableau with $k$ columns $Y[\hat{s}_1,\hat{s}_2,...,\hat{s}_k]$ in $d$-dimensional Minkowski space-time are firstly presented. The particles are described in a metric-like formulation by tensor fields with $k$ groups of antisymmetric Lorentz indices $\Phi_{\mu^1[{\hat{s}_1}],\mu^2[{\hat{s}_2}],..., \mu^k[{\hat{s}_k}]}$ by means of the BRST procedure with complete, $Q$, and incomplete, $Q_c$, BRST operators. Starting from a description of bosonic mixed-antisymmetric higher-spin fields in terms of an auxiliary Fock space associated with a special Poincare module, we realize a conversion of the initial operator constraint system into a system of first-class operator constraints. To this aim, we find, in first time, by means of Verma module the auxiliary representations of the constraint subalgebra, to be isomorphic due to Howe duality to $so(k,k)$ algebra, and containing the subsystem of second-class operators in terms of new oscillator variables forming the Fock module. An unconstrained (with $Q$) and constrained (with $Q_c$ and BRST invariant algebraic constraints) gauge Lagrangian formulations with equivalent dynamics, but different configuration spaces are found. Concept of consistent interactions are suggested.

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math.RT 2026-06-23

Module construction completes Humphreys-Verma for all rank-2 groups

by Stephen Donkin, Haralampos Geranios

On the Humphreys-Verma Conjecture for semisimple algebraic groups of rank 2

For G2 in characteristic 2, an explicit G-module restricts to Q1(0) on G1, the last open case and the first not coming from a tilting module

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Let $G$ be a connected, semisimple, simply connected algebraic group over an algebraically closed field of positive characteristic. For each restricted dominant weight $\lambda$, there is the associated principal indecomposable $G_1$-module $Q_1(\lambda)$, where $G_1$ is the first infinitesimal subgroup of $G$. The assertion that, for every such $\lambda$, there exists a $G$-module whose restriction to $G_1$ is isomorphic to $Q_1(\lambda)$ is known as the Humphreys--Verma Conjecture. For groups of rank $2$, it was shown in \cite{BNPS1} that the Humphreys--Verma Conjecture holds in all cases except one, namely when $G$ is of type $G_2$, the characteristic is $2$, and $\lambda=0$. This case remained completely open. Moreover, in every previously resolved case, the module $Q_1(\lambda)$ could be realized as the restriction of a suitable tilting module. However, in \cite{BNPS2} it was shown that $Q_1(0)$ for $G_2$ in characteristic $2$ cannot arise as the restriction of a tilting module, thereby providing the first counterexample to a conjecture of the first author. In this paper, we construct a $G$-module whose restriction to $G_1$ is $Q_1(0)$, thereby establishing the Humphreys--Verma Conjecture in the last remaining rank $2$ case. Our construction provides the first known example of a $G$-structure on a principal indecomposable $G_1$-module that does not arise from a tilting module. This reveals a new phenomenon in the study of the Humphreys--Verma Conjecture and suggests new directions for understanding $G$-structures on principal indecomposable $G_1$-modules.

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math-ph 2026-06-23

Generalized Cartan matrix yields spectra for non-Coxeter Toda models

by Martin T. Luu

Weyl orbit particles

Particles correspond to orbits under arbitrary regular Weyl group elements, so masses follow from the eigenvector of the extended matrix.

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The mass spectrum of affine Toda theory is known to be expressible in terms of a suitable eigenvector of the relevant Cartan matrix. The particles correspond in a precise manner to the Coxeter element orbits in the set of roots. Recently, variants of affine Toda theory have been constructed for many different Weyl group elements. Again, the particles correspond to orbits in the set of roots and this allows the calculation of the classical mass spectrum. We show how these spectral calculations generalize the affine Toda relation with the Cartan matrix. As an example, we calculate the spectrum for the unique non-Coxeter infinite family $\textrm{D}_{2n}(a_{n-1})$ of primitive regular conjugacy classes in the Weyl groups of complex simple Lie algebras.

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nlin.SI 2026-06-23

Katz extension of disc connection supplies extra symmetry in KdV hierarchy

by Martin T. Luu

The KdV vacuum operator and its Katz extension

The extension, written with Kac coordinates of a Weyl class, identifies the vacuum solution and acts as an additional symmetry.

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We define a connection on the formal disc that can be used to single out the vacuum of the Drinfeld-Sokolov KdV hierarchy associated to a simple complex finite-dimensional Lie algebra. As a connection, it has a canonical Katz extension from the disc to the sphere. We express this Katz extension in terms of the Kac coordinates of a suitable Weyl group conjugacy class. As a consequence, we show that the Katz extension has meaning in the context of the integrable hierarchy: It describes an additional symmetry.

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math.CO 2026-06-23

RC graph pairs count forest polynomial products

by Matthew J. Samuel

A Littlewood-Richardson Rule for Forest Polynomials via the Schubert Bialgebra

The coefficients in the product of two forest polynomials equal the number of pairs whose lift product matches the target code and weight.

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The forest polynomials $\mathfrak{P}_a$ of Nadeau-Tewari form a $\mathbb{Z}$-basis of $\mathbb{Z}[x_1, x_2, \dots]$ whose role for the cohomology of the quasisymmetric flag variety parallels that of Schubert polynomials for the classical flag variety. Nonnegativity of the structure constants $\beta^c_{a,b}$ in $\mathfrak{P}_a \mathfrak{P}_b = \sum_c \beta^c_{a,b} \mathfrak{P}_c$ is known, but no Littlewood-Richardson-style enumerative rule has been available. We give such a rule: $\beta^c_{a,b}$ counts pairs of forest RC graphs of forest-codes $a$ and $b$ whose lift product lands on a forest RC graph of forest-code and weight both equal to $c$. The same rule descends to the cup product on $H^\bullet(QFl_n)$. The proof introduces a Schubert bialgebra $\mathcal{A}$ and lifts the multiplication on its graded dual $\mathcal{D}$ to a product on a free abelian group $\mathcal{B}RC$ of bounded RC graphs; the same machinery yields enumerative LR rules for the dual Schubert, dual key, dual forest, and dual slide bases of $\mathcal{D}$.

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math.AG 2026-06-23

Natural map sends BRST vertex algebra to chiral operators on quiver variety

by Ioana Coman, Myungbo Shim +2 more

Chiralization of Quiver Varieties

The map from V(v,w) to D^ch is injective under stronger assumptions and arises from BRST reduction of beta-gamma and Heisenberg systems.

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Given a quiver Q with gauge dimension $\bf v$ and framing dimension $\bf w$, one can define the extended quiver variety $\widetilde{\mathcal M}(\mathbf v,\mathbf w)$, which is a smooth family of deformations of the Nakajima quiver variety $\mathcal M(\mathbf v,\mathbf w)$. In this paper we discuss two vertex algebras which chiralize the geometry $\widetilde{\mathcal M}(\mathbf v,\mathbf w)$. We construct a sheaf of $\hbar$-adic vertex superalgebras $\mathscr D^{\mathrm{ch}}_{\widetilde{\mathcal M}(\mathbf v,\mathbf w),\hbar}$ on $\widetilde{\mathcal M}(\mathbf v,\mathbf w)$ which quantizes the jet bundle of $\widetilde{\mathcal M}(\mathbf v,\mathbf w)$, and define a vertex algebra $\mathsf D^{\mathrm{ch}}(\widetilde{\mathcal M}(\mathbf v,\mathbf w))$ to be the $\hbar=1$ specialization of the $\mathbb C^{\times}$-finite part of the vector space of global sections $\Gamma(\widetilde{\mathcal M}(\mathbf v,\mathbf w), \mathscr D^{\mathrm{ch}}_{\widetilde{\mathcal M}(\mathbf v,\mathbf w),\hbar})$. We define another vertex superalgebra $\mathcal V(\mathbf v,\mathbf w)$ by BRST reduction of the tensor product of the $\beta\gamma bc$-system and Heisenberg VOA associated to the quiver Q, and show that there exists a natural vertex superalgebra map from $\mathcal V(\mathbf v,\mathbf w)$ to $\mathsf D^{\mathrm{ch}}(\widetilde{\mathcal M}(\mathbf v,\mathbf w))$. Under certain technical assumptions, we prove that the negative degree BRST cohomologies of the tensor product of $\beta\gamma bc$-systems and Heisenberg VOA associated to the quiver Q are zero, and under stronger assumptions, that the aforementioned vertex superalgebra map is injective. Physically, the vertex superalgebra $\mathcal V(\mathbf v,\mathbf w)$ is closely related to the boundary VOA of the H-twisted 3D $\mathcal N=4$ quiver gauge theory associated to the quiver Q with gauge and framing dimension vectors $\bf v$ and $\bf w$.

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