<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"><mml:msub><mml:mrow><mml:mi mathvariant="normal">FI</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="script">W</mml:mi></mml:mrow></mml:msub></mml:math>-modules and stability criteria for representations of classical Weyl groups

Wilson, Jennifer C. H.

doi:10.1016/j.jalgebra.2014.08.010

Cited by 40 publications

(43 citation statements)

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“…Church [7] showed representation stability of the rational cohomology of ordered configuration spaces using a Leray spectral sequence and the partition lattice. We generalize his method of using this spectral sequence for other types of arrangements by combining it with our combinatorics and with FI W -module theory developed by Wilson [16,15]. Wilson [16] also showed representation stability for each linear case.…”

Section: Introductionmentioning

confidence: 90%

“…Throughout this section, we let W n denote either the symmetric group S n (type A) or the hyperoctahedral group W n (type B/C). For more details on the theory, we refer the reader to [8,9] for the case of S n (and much more) and [15,16] for the case of W n (as well as the type D Weyl group).…”

Section: Representation Stabilitymentioning

confidence: 99%

“…We say that V has weight ≤ d if for all n, every irreducible representation V (λ ) n appearing with nonzero multiplicity in V n satisfies |λ | ≤ d (if λ is a partition) or |λ + | + |λ − | ≤ d (if λ = (λ + , λ − ) is a pair of partitions). Again, we refer the reader to [15,16] for more details on these concepts; we state here the main properties and example on which our results rely. We start by stating a proposition on finitely generated FI W -modules, bounding the weight and stability degree for kernels, cokernels, and extensions.…”

Section: Representation Stabilitymentioning

confidence: 99%

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Representation stability for the cohomology of arrangements associated to root systems

Bibby

2017

J Algebr Comb

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ABSTRACT. From a root system, one may consider the arrangement of reflecting hyperplanes, as well as its toric and elliptic analogues. The corresponding Weyl group acts on the complement of the arrangement and hence on its cohomology. We consider a sequence of linear, toric, or elliptic arrangements which arise from a family of root systems of type A, B, C, or D, and we show that the rational cohomology stabilizes as a sequence of Weyl group representations. Our techniques combine a Leray spectral sequence argument similar to that of Church in the type A case along with FI W -module theory which Wilson developed and used in the linear case. A key to the proof relies on a combinatorial description, using labelled partitions, of the poset of connected components of intersections of subvarieties in the arrangement.

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Section: Introductionmentioning

confidence: 90%

Section: Representation Stabilitymentioning

confidence: 99%

Section: Representation Stabilitymentioning

confidence: 99%

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Representation stability for the cohomology of arrangements associated to root systems

Bibby

2017

J Algebr Comb

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“…Historically, the Noetherian property was proven for FI-modules over a field of characteristic 0 in [CEF,Theorem 1.3] and independently by Snowden in [S,Theorem 2.3], and over a general Noetherian ring in [CEFN, Theorem A]. The case G = Z/2Z was proven in [JW,Theorem 4.21]. The paper [SS2] proves the theorem for all polycyclic-by-finite groups G.…”

Section: Proofmentioning

confidence: 98%

“…The work in this paper therefore provides a new proof of the bounds given in [CEF,Theorem 3.3.4] and [JW,Theorem 4.16], while providing a novel bound in the cases where k is a general field or where G = 1, Z/2Z. Definition 2.1.…”

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confidence: 99%

Homological invariants of FI-modules and FI -modules

Ramos

2018

Journal of Algebra

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We explore a theory of depth for finitely generated FI G -modules. Using this theory, we prove results about the regularity, and provide novel bounds on stable ranges of FI-modules, making effective a theorem of Nagpal and thereby refining the stable range in results of Church, Ellenberg, and Farb. IntroductionLet G be a group. The category FI G , introduced in [SS2], is that whose objects are finite sets, and whose morphisms are pairs (f, g) : S → T such that f : S → T is an injection, and g : S → G is a map of sets. If G = 1 is the trivial group, then FI G is equivalent to the category FI of finite sets and injections. The full subcategory of FI G whose objects are the sets [n] := {1, . . . , n} is equivalent to FI G , and we therefore often identify the two. For any commutative ring k, an FI G -module over k is a covariant functor V : FI G → Mod k . We will often write V n := V ([n]).In the present paper we study various homological invariants of FI G -modules, and show how they relate to concrete questions about stability. In particular, we generalize the bounds on Castelnuovo-Mumford regularity in [CE, Theorem A], and provide explicit bounds on results from [CEFN, Theorem B] and [NS]. If V is an FI G -module, then we define H 0 (V ) on any finite set [n] to be the quotient of V n by the images of all [n]) and m < n. The functor V → H 0 (V ) is right exact, and we define its right derived functors, H i , to be the homology functors. The paper [CE] studied these functors in the case of FI-modules, and showed various applications to the homology of congruence subgroups. Subsequently, Calegari and Emerton used the results of Church and Ellenberg in studying the homology of arithmetic groups [CaE, Theorem 5.2].After reviewing some preliminary topics, we next turn our attention to bounding the regularity of FI Gmodules. We define the degree of an FI G -module to be the largest integer n for which V n = 0, while V r = 0 whenever r > n. We say that V is generated in degree ≤ m if deg(H 0 (V )) ≤ m (See Definition 2.7). Similarly, we say that V has first homological degree ≤ r if deg(H 1 (V )) ≤ r (See Definition 2.7, and Remark 2.16 for more on this definition). If a module has finite generating and relation degrees, then it is said to be presented in finite degree. The regularity of V is the smallest integer N such that deg(H i (V )) − i ≤ N for all i ≥ 1 (see Definition 2.15). It was proven by Sam and Snowden in [SS3, Corollary 6.3.5] that finitely generated FI-modules in characteristic zero have finite regularity. Following this, Church and Ellenberg proved that FI-modules which are presented in finite degree have finite regularity over any ring, and they provided a bound on this regularity [CE, Theorem A]. More recently, Li and Yu have provided different bounds on the regularity of FI-modules [LY, Theorem 1.8]. One of the goals of this paper is to prove that similar bounds exist for FI G -modules. Indeed, we will find that the regularity of a module V , which is presented in finite degree, can be bounded in term...

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$${{\,\mathrm{\texttt {VIC}}\,}}$$-modules over noncommutative rings

Putman

Sam

2022

Sel. Math. New Ser.

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FIW-modules and stability criteria for representations of classical Weyl groups

Cited by 40 publications

References 34 publications

Representation stability for the cohomology of arrangements associated to root systems

Representation stability for the cohomology of arrangements associated to root systems

Homological invariants of FI-modules and FI -modules

$${{\,\mathrm{\texttt {VIC}}\,}}$$-modules over noncommutative rings

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