In this paper, we present two methods, induction and restriction procedures, to construct new stable equivalences of Morita type. Suppose that a stable equivalence of Morita type between two algebras $A$ and $B$ is defined by a $B$-$A$-bimodule $N$. Then, for any finite admissible set $Φ$ and any generator $X$ of the $A$-module category, the $Φ$-Auslander-Yoneda algebras of $X$ and $N\otimes_AX$ are stably equivalent of Morita type. Moreover, under certain conditions, we transfer stable equivalences of Morita type between $A$ and $B$ to ones between $eAe$ and $fBf$, where $e$ and $f$ are idempotent elements in $A$ and $B$, respectively. Consequently, for self-injective algebras $A$ and $B$ over a field without semisimple direct summands, and for any $A$-module $X$ and $B$-module $Y$, if the $Φ$-Auslander-Yoneda algebras of $A\oplus X$ and $B\oplus Y$ are stably equivalent of Morita type for one finite admissible set $Φ$, then so are the $Ψ$-Auslander-Yoneda algebras of $A\oplus X$ and $B\oplus Y$ for {\it every} finite admissible set $Ψ$. Moreover, two representation-finite algebras over a field without semisimple direct summands are stably equivalent of Morita type if and only if so are their Auslander algebras. As another consequence, we construct an infinite family of algebras of the same dimension and the same dominant dimension such that they are pairwise derived equivalent, but not stably equivalent of Morita type. This answers a question by Thorsten Holm.
The purpose of this paper is to study induction and restriction of two-variable Hecke algebras. First, we provide the explicit form of the Mackey decomposition formula. And then, we elucidate how (anti-)involutions interact with induction product and restriction.
We study the algebras generated by restriction and induction operations on complex modules over dihedral groups. In the case where the orders of all dihedral groups involved are not divisible by four, we describe the relations, a basis, the center, and a decomposition into indecomposables of these algebras.
We use geometric parabolic induction functors and the adjoint functors for the supergroups Osp(2m+1,2n) (where m and n vary) to categorify the action of the infinite-dimensional Clifford algebra on the Fock space of semi-infinite forms.
It is proved the the restriction of any supercharacter of a finite group of triangular type on its subgroup is a sum of supercharacters with nonnegative integer coefficients. We define superinduction and prove the analog of the Frobenius formula for supercharacters.
We define the induction and restriction functors for cyclotomic q-Schur algebras, and study some properties of them. As an application, we categorify a higher level Fock space by using the module categories of cyclotomic q-Schur algebras.
Lusztig proved the compatibility of induction functors and restriction functors for Lusztig's perverse sheaves. Fang-Lan-Xiao established a categorification of Green's formula and gave a sheaf-level proof of this compatibility for all semisimple complexes. As an application, we study the relation between induction functors and derivation functors, which is a kind of special restriction functors.
We prove a geometric criterion for the bounded multiplicity property of "small" infinite-dimensional representations of real reductive Lie groupsin both induction and restrictions. Applying the criterion to symmetric pairs, we give a full description of the triples $H \subset G \supset G'$ such that any irreducible admissible representations of $G$ with $H$-distinguished vectors have the bounded multiplicity property when restricted to the subgroup $G'$. This article also completes the proof of the general results announced in the previous paper [Adv. Math. 2021, Section 7].
We construct the polynomial induction functor, which is the right adjoint to the restriction functor from the category of polynomial representations of a general linear group to the category of representations of its Weyl group. This construction leads to a representation-theoretic proof of Littlewood's plethystic formula for the multiplicity of an irreducible representation of the symmetric group in such a restriction. The unimodality of certain bipartite partition functions follows.
We find upper and lower bounds of the multiplicities of irreducible admissible representations $π$ of a semisimple Lie group $G$ occurring in the induced representations $Ind_H^Gτ$ from irreducible representations $τ$ of a closed subgroup $H$. As corollaries, we establish geometric criteria for finiteness of the dimension of $Hom_G(π,Ind_H^G τ)$ (induction) and of $Hom_H(π|_H,τ)$ (restriction) by means of the real flag variety $G/P$, and discover that uniform boundedness property of these multiplicities is independent of real forms and characterized by means of the complex flag variety.
Let $\mathbb{C}\mathsf{A}_n = \mathbb{C}[S_2\wr S_2 \wr\cdots \wr S_2]$ be the group algebra of $n$-step iterated wreath product. We prove some structural properties of $\mathsf{A}_n$ such as their centers, centralizers, right and double cosets. We apply these results to explicitly write down Mackey theorem for groups $\mathsf{A}_n$ and give a partial description of the natural transformations between induction and restriction functors on the representations of the iterated wreath product tower by computing certain hom spaces of the category of $\displaystyle \bigoplus_{m\geq 0}(\mathsf{A}_m, …
This paper is about the reduced group C*-algebras of real reductive groups, and about Hilbert C*-modules over these C*-algebras. We shall do three things. First we shall apply theorems from the tempered representation theory of reductive groups to determine the structure of the reduced C*-algebra (the result has been known for some time, but it is difficult to assemble a full treatment from the existing literature). Second, we shall use the structure of the reduced C*-algebra to determine the structure of the Hilbert C*-bimodule that represents the functor of parabolic induction. Third, we sha…
Cellular categories are a generalization of cellular algebras, which include a number of important categories such as (affine)Temperley-Lieb categories, Brauer diagram categories, partition categories, the categories of invariant tensors for certain quantised enveloping algebras and their highest weight representations, Hecke categories and so on. The common feather is that, for most of the examples, the endomorphism algebras of the categories form a tower of algebras. In this paper, we give an axiomatic framework for the cellular categories related to the quasi-hereditary tower and then study…
Let G be a reductive group and L a Levi subgroup. Parabolic induction and restriction are a pair of adjoint functors between Ad-equivariant derived categories of either constructible sheaves or (not necessarily holonomic) D-modules on G and L, respectively. Bezrukavnikov and Yom Din proved, generalizing a classic result of Lusztig, that these functors are exact. In this paper, we consider a special case where L=T is a maximal torus. We give explicit formulas for parabolic induction and restriction in terms of the Harish-Chandra D-module on G x T. We show that this module is flat over D(T), whi…
In this paper we study a toy categorical version of Lusztig's induction and restriction functors for character sheaves, but in the abstract setting of multifusion categories. Let $\mathscr{C}$ be an indecomposable multifusion category and let $\mathscr{M}$ be an invertible $\mathscr{C}$-bimodule category. Then the center $\mathscr{Z}_{\mathscr{C}}(\mathscr{M})$ of $\mathscr{M}$ with respect to $\mathscr{C}$ is an invertible module category over the Drinfeld center $\mathscr{Z}(\mathscr{C})$ which is a braided fusion category. Let $ζ_{\mathscr{M}}:\mathscr{Z}_{\mathscr{C}}(\mathscr{M})\longrigh…
To a finite group G one can associate a tower of wreath products S_n[G]. It is well known that the graded direct sum of the Grothendieck groups of the categories of finite dimensional complex representations of these groups can be given the structure of a graded Hopf algebra, and in fact a positive self-adjoint Hopf algebra in the sense of Zelevinsky [1], using the induction product and restriction coproduct. This paper introduces and explores an analogously defined algebra/coalgebra structure associated to a more general class of towers of groups, obtained as a certain family of subgroups of …
Both the original Temperley-Lieb algebras $\mathsf{TL}_{n}$ and their dilute counterparts $\mathsf{dTL}_{n}$ form families of filtered algebras: $\mathsf{TL}_{n}\subset \mathsf{TL}_{n+1}$ and $\mathsf{dTL}_{n}\subset\mathsf{dTL}_{n+1}$, for all $n\geq 0$. For each such inclusion, the restriction and induction of every finite-dimensional indecomposable module over $\mathsf{TL}_{n}$ (or $\mathsf{dTL}_{n}$) is computed. To accomplish this, a thorough description of each indecomposable is given, including its projective cover and injective hull, some short exact sequences in which it appears, its …
We introduce parabolic induction and restriction functors for rational Cherednik algebras, and study their basic properties. Then we discuss applications of these functors to representation theory of rational Cherednik algebras. In particular, we prove the Gordon-Stafford theorem about Morita equivalence of the rational Cherednik algebra for type A and its spherical subalgebra, without the assumption that c is not a half-integer, which was required up to now. Also, we classify representations from category O over the rational Cherednik algebras of type A which do not contain an S_n-invariant v…
Our investigation in the present paper is based on three important results. (1) In [12], Ringel introduced Hall algebra for representations of a quiver over finite fields and proved the elements corresponding to simple representations satisfy the quantum Serre relation. This gives a realization of the nilpotent part of quantum group if the quiver is of finite type. (2) In [4], Green found a homological formula for the representation category of the quiver and equipped Ringel's Hall algebra with a comultiplication. The generic form of the composition subalgebra of Hall algebra generated by simp…
We define twisted Frobenius extensions of graded superrings. We develop equivalent definitions in terms of bimodule isomorphisms, trace maps, bilinear forms, and dual sets of generators. The motivation for our study comes from categorification, where one is often interested in the adjointness properties of induction and restriction functors. We show that $A$ is a twisted Frobenius extension of $B$ if and only if induction of $B$-modules to $A$-modules is twisted shifted right adjoint to restriction of $A$-modules to $B$-modules. A large (non-exhaustive) class of examples is given by the fact t…
Let $G$ be a finite group. To any family $\mathscr{F}$ of subgroups of $G$, we associate a thick $\otimes$-ideal $\mathscr{F}^{\mathrm{Nil}}$ of the category of $G$-spectra with the property that every $G$-spectrum in $\mathscr{F}^{\mathrm{Nil}}$ (which we call $\mathscr{F}$-nilpotent) can be reconstructed from its underlying $H$-spectra as $H$ varies over $\mathscr{F}$. A similar result holds for calculating $G$-equivariant homotopy classes of maps into such spectra via an appropriate homotopy limit spectral sequence. In general, the condition $E\in \mathscr{F}^{\mathrm{Nil}}$ implies strong …
Harish-Chandra induction and restriction functors play a key role in the representation theory of reductive groups over finite fields. In this paper, extending earlier work of Dat, we introduce and study generalisations of these functors which apply to a wide range of finite and profinite groups, typical examples being compact open subgroups of reductive groups over non-archimedean local fields. We prove that these generalisations are compatible with two of the tools commonly used to study the (smooth, complex) representations of such groups, namely Clifford theory and the orbit method. As a t…
In this paper, we shall establish the Mackey formulas in the following two set ups: (i) on the tensor induction and restriction functors on the modules over cyclotomic Hecke algebras (Ariki-Koike algebras) and their standard subalgebras of parabolic subgroups. (ii) on the Bezrukavnikov-Etingof induction and restriction functors among categories O of rational Cherednik algebras for the complex reflection group of type G(r, 1, n) and their parabolic subgroups.
We show that induction and restriction functors for inclusions of nilCoxeter algebras provide a categorical realization of the algebra of polynomial differential operators in one variable.
We define the i-restriction and i-induction functors on the category O of the cyclotomic rational double affine Hecke algebras. This yields a crystal on the set of isomorphism classes of simple modules, which is isomorphic to the crystal of a Fock space.
A key result in a 2004 paper by S. Arkhipov, R. Bezrukavnikov, and V. Ginzburg (ABG) gives an equivalence of the bounded derived category of finite dimensional modules for the principal block of a Lusztig quantum algebra at an $\ell^{th}$ root of unity, with am explicit full subcategory of the bounded derived category of integrable type 1 modules for a Borel part of the quantum algebra. Some restrictions on $\ell$ are required; in particular, it is assumed $\ell > h$, the Coxeter number. The same paper suggests there is an analogous result for representations of semisimple algebraic groups in …
Lie bialgebras occur as the principal objects in the infinitesimalisation of the theory of quantum groups - the semi-classical theory. Their relationship with the quantum theory has made available some new tools that we can apply to classical questions. In this paper, we study the simple complex Lie algebras using the double-bosonisation construction of Majid. This construction expresses algebraically the induction process given by adding and removing nodes in Dynkin diagrams, which we call Lie induction. We first analyse the deletion of nodes, corresponding to the restriction of adjoint repre…
The tensor ideal localising subcategories of the stable module category of all, including infinite dimensional, representations of a finite group scheme over a field of positive characteristic are classified. Various applications concerning the structure of the stable module category and the behavior of support and cosupport under restriction and induction are presented.
The basic properties of the Temperley-Lieb algebra $TL_n$ with parameter $β= q + q^{-1}$, for $q$ any non-zero complex number, are reviewed in a pedagogical way. The link and standard (cell) modules that appear in numerous physical applications are defined and a natural bilinear form on the standard modules is used to characterize their maximal submodules. When this bilinear form has a non-trivial radical, some of the standard modules are reducible and $TL_n$ is non-semisimple. This happens only when $q$ is a root of unity. Use of restriction and induction allows for a finer description of the…
This survey is about old and new results about the modular representation theory of finite reductive groups with a strong emphasis on local methods. This includes subpairs, Brauer's Main Theorems, fusion, Rickard equivalences. In the defining characteristic we describe the relation between $p$-local subgroups and parabolic subgroups, then give classical consequences on simple modules and blocks, including the Alperin weight conjecture in that case. In the non-defining characteristics, we sketch a picture of the local methods pioneered by Fong-Srinivasan in the determination of blocks and their…
We consider a finite dimensional strongly $G$-graded algebra $A$ with { self-injective} $1$-component $B$, and in our main result we prove that the induction from $B$ to $A$ of a basic support $τ$-tilting pair of $B$-modules is a support $τ$-tilting pair $(M,P)$ of $A$-modules if and only if $M$ is $G$-invariant. A similar statement holds for the restriction from $A$ to $B$, so our results may be viewed as Clifford and Maschke type theorems for $2$-term silting complexes. We also give applications to semibricks and the associated wide subcategories.
We define a category of planar diagrams whose Grothendieck group contains an integral version of the infinite rank Heisenberg algebra, thus yielding a categorification of this algebra. Our category, which is a q-deformation of one defined by Khovanov, acts naturally on the categories of modules for Hecke algebras of type A and finite general linear groups. In this way, we obtain a categorification of the bosonic Fock space. We also develop the theory of parabolic induction and restriction functors for finite groups and prove general results on biadjointness and cyclicity in this setting.
We define the generalized Burnside algebra $HB(W_{n})$ for $B_{n}$-type Coxeter group $W_{n}$ and construct an surjective algebra morphism between Mantaci-Reutenauer algebra ${\sum}'(W_{n})$ and $HB(W_{n})$. Then, by obtaining the primitive idempotents $(e_λ)_{λ\in \mathcal{DP}(n)}$ of $HB(W_{n})$, we consider the image $\textrm{res}_{W_{A}}^{W_{n}}e_{B}$ and $\textrm{ind}_{W_{A}}^{W_{n}}e_{B}^{A}$ under restriction and induction map between generalized Burnside algebras. We give an alternative formula to compute the elements number of conjugate classes of $W_{n}$. We also obtain an effective …
