Homological invariants of FI-modules and FI -modules

Ramos, Eric

doi:10.1016/j.jalgebra.2017.12.037

Cited by 20 publications

(11 citation statements)

References 21 publications

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“…Proof. The first statement is an immediate corollary of [CE, Theorem A] (or see [Ram2,Theorem B] for more details) and the second statement is [Ram1,Theorem B].…”

Section: Preliminaries On Fi-modulesmentioning

confidence: 99%

“…We say that an FI-module is semi-induced 1 if it admits a finite length filtration where the quotients are induced modules. The following is a useful property of semi-induced modules which holds not only for FI-modules but also many other similar functor categories; see for example [Ram1,Remark 2.33] and [Nag2,Corollary 4.23].…”

Section: Preliminaries On Fi-modulesmentioning

confidence: 99%

“…We note that stable degree and local degree have respectively been previously studied in [Ram1] and [DV] under different names.…”

Section: Introductionmentioning

confidence: 99%

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Linear and quadratic ranges in representation stability

Church

Miller

Nagpal

et al. 2018

Advances in Mathematics

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We prove two general results concerning spectral sequences of FI-modules. These results can be used to significantly improve stable ranges in a large portion of the stability theorems for FImodules currently in the literature. We work this out in detail for the cohomology of configuration spaces where we prove a linear stable range and the homology of congruence subgroups of general linear groups where we prove a quadratic stable range. Previously, the best stable ranges known in these examples were exponential. Up to an additive constant, our work on congruence subgroups verifies a conjecture of Djament.

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“…Proof. The first statement is an immediate corollary of [CE, Theorem A] (or see [Ram2,Theorem B] for more details) and the second statement is [Ram1,Theorem B].…”

Section: Preliminaries On Fi-modulesmentioning

confidence: 99%

Section: Preliminaries On Fi-modulesmentioning

confidence: 99%

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Linear and quadratic ranges in representation stability

Church

Miller

Nagpal

et al. 2018

Advances in Mathematics

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“…It is sufficient to construct a natural isomorphism between ΣD and DΣ. This has been done by Ramos in [13,Lemma 3.5]. Note that in the setting of that paper, the self-embedding functor is defined in a way different from ours; see [13,Definition 2.20].…”

Section: Proof Statementsmentioning

confidence: 96%

Filtrations and homological degrees of FI-modules

2017

Journal of Algebra

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Abstract. Let be a commutative Noetherian ring. In this paper we consider ♯-filtered modules of the category FI firstly introduced in [12]. We show that a finitely generated FI-module V is ♯-filtered if and only if its higher homologies all vanish, and if and only if a certain homology vanishes. Using this homological characterization, we characterize finitely generated C-modules V whose projective dimension pd(V ) is finite, and describe an upper bound for pd(V ). Furthermore, we give a new proof for the fact that V induces a finite complex of ♯-filtered modules, and use it as well as a result of Church and Ellenberg in [1] to obtain another upper bound for homological degrees of V .

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“…The representation theory of FI is currently under active research which was initiated in [4]. There is also research on the representation theory of the wreath product F ≀ FI (see [8,11,13]). We will be interested here in the finite version of this category.…”

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confidence: 99%

The Littlewood-Richardson rule for wreath products with symmetric groups and the quiver of the categoryF≀FI_n

Stein

2016

Communications in Algebra

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We give a new proof for the Littlewood-Richardson rule for the wreath product F ≀Sn where F is a finite group. Our proof does not use symmetric functions but more elementary representation theoretic tools. We also derive a branching rule for inducing the natural embedding of F ≀ Sn to F ≀ Sn+1. We then apply the generalized Littlewood-Richardson rule for computing the ordinary quiver of the category F ≀ FIn where FIn is the category of all injective functions between subsets of an n-element set. Frobenius reciprocity, answering one of these questions essentially answers the other one. Moreover, since both induction and restriction are additive, it is enough to consider the case where U or V are irreducible representations. Even considering this reduction, the question, in general, is very difficult. If G = S n the answer is known for certain natural choices of H and these solutions are often called branching rules. The most classical case is where H = S n−1 viewed as the subgroup of all permutations that fix n. An important generalization is the Littlewood-Richardson rule which gives the answer for the case H = S k × S n−k .Let F and G be finite groups such that G acts on the left of a finite set X. We denote by F ≀ X G the wreath product of F and G. The representation theory of F ≀ X G is a well-studied subject (see [3] and [6, Chapter 4]) and the case G = S n with the natural action on {1, . . . , n} is of special importance. Finding generalizations for the branching rules is a natural question. The "classical" branching rule for inducing from F ≀ S n to F ≀ S n+1 was found by Pushkarev [10]. In this paper we generalize the Littlewood-Richardson rule to the group F ≀ S n . After the present paper was already circulating, we became aware of the paper [5] by Ingram, Jing and Stitzinger, where the same result was obtained using symmetric functions. However, our approach is different. We use only elementary representation theoretic tools and base our proof on the explicit description of the irreducible representations of F ≀ S n . In Section §5 we use the generalized Littlewood-Richardson rule to retrieve Pushkarev's result.Then we turn to give an application to the representation theory of a natural family of categories. Denote by FI the category of finite sets and injective functions. The representation theory of FI is currently under active research which was initiated in [4]. There is also research on the representation theory of the wreath product F ≀ FI (see [8,11,13]). We will be interested here in the finite version of this category. We denote by FI n the category of all subsets of {1, . . . , n} and injective functions. In Section §6 we will give a description of the ordinary quiver of the algebra of F ≀ FI n . The case where F is the trivial group was originally done by [2] and a simple proof was later given in [9]. For the general case, we imitate the method of [9] but where they use usual branching rule for S n we will use the generalization for F ≀ S n .

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Homological invariants of FI-modules and FI -modules

Cited by 20 publications

References 21 publications

Linear and quadratic ranges in representation stability

Linear and quadratic ranges in representation stability

Filtrations and homological degrees of FI-modules

The Littlewood-Richardson rule for wreath products with symmetric groups and the quiver of the categoryF≀FI_n

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Homological invariants of FI-modules and FI -modules

Cited by 20 publications

References 21 publications

Linear and quadratic ranges in representation stability

Linear and quadratic ranges in representation stability

Filtrations and homological degrees of FI-modules

The Littlewood-Richardson rule for wreath products with symmetric groups and the quiver of the categoryF≀FIn

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Product

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The Littlewood-Richardson rule for wreath products with symmetric groups and the quiver of the categoryF≀FI_n