Category: math.RT
Showing 1–20 of 29,847 results
Kyungtak Hong, Alexander Tsymbaliuk
We present the RLL-realization of extended orthosymplectic quantum supergroups for any parity sequence, with R-matrices evaluated in the earlier work arxiv:2408.16720. Our isomorphism is compatible with the internal structure of generalized doubles. We also relate different sign conventions through 2-cocycle twists. Furthermore, we establish a factorization of the reduced R-matrix within the RLL-realization.
Christopher P. Bendel, Daniel K. Nakano, Cornelius Pillen
Let $G$ be a reductive algebraic group scheme defined over ${\mathbb F}_{p}$ and $k$ be an algebraically closed field of characteristic $p$. There are two associated families of finite group schemes, the $r$-th Frobenius kernels, denoted by $G_r$, and the fixed points of the iterated Frobenius map, the finite groups of Lie type, denoted by $G(\mathbb{F}_q).$ Bendel, Nakano and Pillen initiated the investigation of the induction functor $\operatorname{ind}_{G(\mathbb{F}_q)}^G-$. Using filtrations and truncation, large amounts of data coming from the algebraic group and the Frobenius kernels can be transferred to the finite group. This paper looks at connections between a fundamental theorem of Chastkofsky and Jantzen and the induction functor via the cohomology and representation theory of $G$.
Yuan Chai
We use enhanced Langlands parameters to obtain a classification for irreducible representations of twisted $p$-adic general linear groups in unramified principal series. We give the definition of standard representations and prove the twisted Kazhdan-Lusztig conjecture for the multiplicities in the Grothendieck group. We mainly follow Lusztig's work in the connected case using graded Hecke algebra. We show that the parametrization is compatible with the Whittaker-normalized one.
Benjamin Klopsch, Margherita Piccolo, Britta Späth
The representation zeta function of a profinite group $G$ encodes the distribution of continuous irreducible complex representations of $G$ as a function of the dimension. Its abscissa of convergence $α(G)$ describes the polynomial degree of representation growth of $G$. Within the class of quasi-semisimple profinite groups, we characterise those of polynomial representation growth (PRG) and we prove that whether such a group $G$ has PRG or not only depends on its semisimple part $G/\mathrm{Z}(G)$. Moreover, we show that, for quasi-semisimple profinite groups $G$ that have uniformly bounded Lie ranks, the degree of growth satisfies $α(G) = α(G/\mathrm{Z}(G))$. We provide a technique to produce, for any prescribed positive real number $\varrho$, quasi-semisimple profinite groups $G$ with PRG of degree $α(G) = \varrho$. Our method allows for considerable flexibility regarding the inclusion of finite simple groups of Lie type as composition factors of $G$. Furthermore, we can arrange for the groups $G$ of prescribed representation growth to be profinite completions of suitable finitely generated discrete groups $Γ$ so that the group $Γ$ has the same representation zeta function as $G$.
Wille Liu, Wei-Hsuan Hsin, Cheng-Chiang Tsai
We prove that on a semisimple Lie algebra $\mathfrak{g}$ over a finite field of large characteristic, if a complex-valued invariant function $f$ and its Fourier transform $\hat f$ are both supported in the nilpotent cone of $\mathfrak{g}$, then $\hat f = γ^{-1}f$ for an explicit quadratic Gauss sum $γ$. Consequently, we determine a fourth root of unity appearing in various formulae of generalised Gel'fand--Graev characters, known as Lusztig constant, previously known in special cases due to works of Kawanaka, Digne--Lehrer--Michel, Waldspurger and Geck. As consequence, we show the validity of a conjecture of Letellier on the compatibility of Fourier transform with Deligne--Lusztig induction.
Karin Erdmann, Kay Jin Lim
Schocker classified the representation type of the descent algebra of type $\mathbb{A}$ over any field of characteristic zero. In an earlier paper, the authors extended this classification for type $\mathbb{A}$ to fields of positive characteristic. In this paper, we complete the classification for all other types except for $\mathbb{E}_8$. The proof for type $\mathbb{B}$ is entirely theoretical, while some small cases in type $\mathbb{D}$ and the exceptional types require computer computation to determine their Ext-quivers.
Filippo Ambrosio, Lewis Topley, Matthew Westaway
Let $G$ be a reductive group over an algebraically closed field of positive characteristic $p$, good for the root system of $G$. The closures of $G$-orbits in the Hilbert nullcone of the coadjoint representation are conical affine Poisson varieties, generically of full rank, known as {\em nilpotent coadjoint orbits}. In this paper, we classify the filtered Hamiltonian quantizations of these orbit closures for $G = GL_N$ and any $p > 0$. Our main new technique is a construction of quantizations from certain primitive quotients of the enveloping algebra, inducing them from the stabiliser in $G$ of the Frobenius twisted $p$-character.
Santiago Chaves, Andreas Debrouwere, Alberto Hernández Alvarado, Jasson Vindas, Rafael Zamora
We establish new strong factorization properties for the smooth vectors of representations of exponential solvable Lie groups on Fréchet spaces. In particular, our results improve upon the Dixmier-Malliavin factorization theorem for simply connected nilpotent Lie groups.
Thomas Creutzig, Volodymyr Kovalchuk, Andrew R. Linshaw, Arim Song, Uhi Rinn Suh
The universal $2$-parameter vertex algebra $\mathcal{W}_{\infty}$ of type $\mathcal{W}(2,3,\dots)$ is a classifying object for vertex algebras of type $\mathcal{W}(2,3,\dots,N)$ for some $N$; under mild hypotheses, all such vertex algebras arise as quotients of $\mathcal{W}_{\infty}$. In 2017, Gaiotto and Rapčák introduced a family of such vertex algebras called $Y$-algebras, and conjectured that they fall into groups of three that are mutually isomorphic. This is a common generalization of both Feigin-Frenkel duality and the coset realization of principal $\mathcal{W}$-algebras in type $A$, and was proven in 2021 for the simple $Y$-algebras (i.e., one label is zero) by the first and third authors. In this paper, we extend this entire story to the $\mathcal{N}=2$ superconformal setting. First, we prove the 2013 conjecture of Gaberdiel and Candu that there exists a universal $2$-parameter vertex algebra $\mathcal{W}^{\mathcal{N}=2}_{\infty}$ which is an extension of the $\mathcal{N}=2$ superconformal algebra, and has four additional generators in weights $i, i + \frac{1}{2}, i + \frac{1}{2}, i+1$, for each integer $i > 1$. This admits many $1$-parameter quotients which we call $\mathcal{N}=2$ supersymmetric $Y$-algebras, and we prove the dualities among these algebras which were conjectured in 2018 by Prochazka and Rapčák. A special case is the coset realization of the principal $\mathcal{W}$-algebra $\mathcal{W}^k(\mathfrak{sl}_{n+1|n})$ which was conjectured in 1992 by Ito. As a corollary, we obtain the strong rationality of $\mathcal{W}_k(\mathfrak{sl}_{n+1|n})$ for $k = -1 + \frac{1}{n+a+1}$ for all positive integers $n,a$, and we describe its module category. This generalizes Adamović's 1999 result on $\mathcal{N}=2$ minimal models, which is the case $n=1$.
Kenta Ueyama
Let $S$ be an $\mathbb N$-graded Koszul Artin-Schelter regular algebra and let $σ$ be a graded algebra automorphism of $S$. We study the stable category of graded maximal Cohen-Macaulay modules over the trivial extension algebra $S\ltimes S_σ(-1)$. We show that this category is triangle equivalent to the bounded derived category of finitely generated (ungraded) modules over the Koszul dual algebra of the Zhang twist $S^{σ^{-1}}$. In the connected graded case, we also obtain a criterion for when two such stable categories are triangle equivalent, and show that such an equivalence induces an equivalence between the categories of graded modules over the original algebras.
Marco Mackaay, Vanessa Miemietz, Pedro Vaz
In this article, we develop a generalization of finitary birepresentation theory applicable to Soergel bimodules for infinite Coxeter groups. We establish a reduction process for the classification of simple birepresentations of almost finitary bicategories, and consider in detail the case of Soergel bimodules in extended affine type A.
Pavle Pandžić, Ana Prlić, Vladimír Souček, Vít Tuček
We classify unitary highest weight modules with a given integral infinitesimal character for the real Lie algebras $\mathfrak{su}(p,q)$ and $\mathfrak{so}^*(2n)$. We treat both regular and singular cases. For $\mathfrak{su}(p,q)$ we identify the unitarizable modules in the Hasse diagrams of the highest weight orbit. Analogous results for the other Hermitian Lie algebras were given in our earlier publications.
Markus Klemetti, Ran Levi, Henri Riihimaki, Daniel Solch
Let $C_\bullet$ be a simplicial object in the category $Cat$ of small categories. For a field $k$, taking the Grothendieck groups of isomorphism classes of $kC_n$-modules gives rise to a cochain complex, whose cohomology, which we refer to as representation cohomology, is the object studied in this article. In particular, to any small category $C$, we associate a simplicial object in $Cat$, where for each $n\ge 0$ the objects of the level $n$ category are the simplices of the nerve of $C$. The basic properties of the resulting representation cohomology of these simplicial objects and certain subobjects are then studied in detail. We present some general theoretical computations in favourable cases.
Petter Andreas Bergh
It has been conjectured that finite tensor categories have finitely generated cohomology. We show that this is equivalent to finitely generated Hochschild cohomology for the endomorphism algebras of the projective generators.
Francis Brown
Using results of Fayers on the structure of Specht modules, we prove two different formulae for the determinant of matrices which are obtained by amalgamating the entries of two smaller matrices. In particular, this gives formulae for multivariable Vandermonde determinants as a sum of completely factorising terms, each of which is a Vandermonde determinant in fewer variables. As an application, we deduce an elementary proof of the multiplicativity of the transfinite diameter for products of compact sets.
Kun Zhang, Yuanyang Zhou
In this paper, with suitable assumptions, we generalize the work of Külshammer and Puig on extensions of nilpotent blocks to inertial blocks.
Naihuan Jing, Yinlong Liu, Jian Zhang
In this paper, we study the Littlewood theory associated with the quantum super immanants and supersymmetric polynomials, including both the super case and the quantum generalization. In the setting of quantum super Schur-Weyl duality between the quantum superalgebra $U_q(\mathfrak{gl}_{m|n})$ and the Iwahori-Hecke algebra $\mathcal{H}_r$ of type A, we explicitly construct basis vectors of the $(U_q(\mathfrak{gl}_{m|n}), \mathcal{H}_r)$-bimodule on the tensor product space $(\mathbb{C}^{m|n})^{\otimes r}$. Using this construction, we interpret the quantum super immanants via weight spaces of covariant tensor representations of $U_q(\mathfrak{gl}_{m|n})$.
A. M. Bouhada
We study quadratic monomial algebras \(Λ\) and the structure of their graded module categories, with particular emphasis on finitely presented and finitely copresented modules over the Koszul dual algebra \(Λ^{!}\). We prove that, for a finite-dimensional quadratic monomial algebra, the Koszul dual \(Λ^{!}\) is both left coherent and left co-coherent, and that the categories of perfect and coperfect modules coincide with the categories of finitely presented and finitely copresented graded modules, respectively. We further show that quadratic monomial algebras are absolutely Koszul and have global linearity defect at most one. As a consequence, every finitely presented \(Λ\)-module has a rational Poincaré series, and the associated tails and cotails categories are abelian and hereditary. We also establish that the class of finitely presented (resp.\ finitely copresented) modules coincides with the class of modules admitting a linear (resp.\ colinear) truncation. Building on these structural results, we refine the graded derived and singular Koszul dualities for quadratic monomial algebras. In particular, we obtain triangulated equivalences \[ \mathsf{D}^b(Λ^{!}\textup{-Fcp}^{\mathbb{Z}}) \;\simeq\; \mathsf{D}^b(Λ\textup{-gmod}) \;\simeq\; \mathsf{D}^b(Λ^{!}\textup{-Fp}^{\mathbb{Z}}), \qquad \mathsf{D}^b(Λ^{!}\textup{-Fcp}^{\mathbb{Z}}/Λ^{!}\textup{-gmod}) \;\simeq\; \mathsf{D}_{\mathrm{sg}}(Λ\textup{-gmod}). \] Moreover, we show that these equivalences induce natural \(t\)-structures on the bounded derived and singularity categories, whose hearts admit explicit descriptions in terms of linear complexes and shifts of linear modules. Finally, we refine the corresponding ungraded derived and singular Koszul dualities and obtain an ungraded BGG correspondence for quadratic monomial algebras.
Aaron Chan, Osamu Iyama, Rene Marczinzik
We show that the class of twisted fractionally Calabi-Yau algebras of finite global dimension coincides with the stable endomorphism algebras of $d$-cluster tilting modules over $d$-representation-finite algebras. This is an application of our main result stating that an algebra $A$ of finite global dimension is twisted fractionally Calabi-Yau if and only if there exists $i$ such that the replicated algebra $A^{(i)}$ is a higher Auslander algebra if and only if there exist infinitely many $i$ such that $A^{(i)}$ is a higher Auslander algebra. This gives a new connection between the study of higher Auslander-Reiten theory and twisted fractionally Calabi-Yau algebras, and provides a new construction of large classes of higher Auslander algebras and higher representation-finite algebras. We give several applications such as an explicit characterisation of twisted $\frac{n}{2}$-Calabi-Yau algebras, and a triangle equivalence between the bounded derived category of a twisted fractionally Calabi-Yau algebra of finite global dimension and the $\mathbb{Z}$-graded stable module category of an associated higher preprojective algebra.
Jack A. Cook
The goal of this article is to give a proof of a result seemingly absent from the literature characterizing global sections of standard $\mathcal{D}$-modules on the flag variety. This characterization yields a mixture of the Langlands Classification of admissible representations with the Knapp-Zuckerman classification of tempered representations of a real reductive group. We use this result to compute the Cousin-Zuckerman resolution of the trivial representation in terms of standard $(\mathfrak{g},K)$-modules. Further, in the case of $GL(n,\mathbb{H})$ we use this to prove the Lusztig-Vogan bijection for $n=2,3$ and compute the lowest $K$-type map for the zero and principal orbits for general $n$ as well as the image of the trivial representation for even orbits.