Linear and quadratic ranges in representation stability

Church, Thomas M.; Nagpal, Rohit; Reinhold, Jens

doi:10.48550/arxiv.1706.03845

Cited by 4 publications

(4 citation statements)

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“…This is an invariant associated to VI-module with several useful properties that we prove below. An invariant with the same name, but for FI-modules, is discussed in [CMNR,§2].…”

Section: Some Consequences Of the Shift Theoremmentioning

confidence: 99%

$\mathrm{VI}$ modules in non-describing characteristic, Part I

Nagpal

2017

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Let VI be the category of finite dimensional Fq-vector spaces whose morphisms are injective linear maps, and let k be a noetherian ring. We study the category of functors from VI to k-modules in the case when q is invertible in k. Our results include a structure theorem, finiteness of regularity, and a description of the Hilbert series. These results are crucial in the classification of smooth irreducible GL∞(Fq)-representations in non-describing characterisitic which is contained in Part II of this paper [Nag2]. Contents 1. Introduction 1 2. Overview of VI-modules 5 3. Induced and semi-induced VI-modules 8 4. The shift theorem 11 5. Some consequences of the shift theorem 25 References 33

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“…This is an invariant associated to VI-module with several useful properties that we prove below. An invariant with the same name, but for FI-modules, is discussed in [CMNR,§2].…”

Section: Some Consequences Of the Shift Theoremmentioning

confidence: 99%

$\mathrm{VI}$ modules in non-describing characteristic, Part I

Nagpal

2017

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“…The dimension functions of FI modules are eventually polynomial, and there has been much work in studying the invariants of FI modules which control when this occurs. (For recent applications see [4]). Which invariants of FS op modules control when the Hilbert series of a finitely generated FS op module agrees with a sum of polynomials times exponential?…”

Section: Proof Of Finite Generationmentioning

confidence: 99%

Stability in the homology of Deligne-Mumford compactifications

Tosteson¹

2018

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Using the the theory of FS op modules, we study the asymptotic behavior of the homology of M g,n , the Deligne-Mumford compactification of the moduli space of curves, for n ≫ 0. An FS op module is a contravariant functor from the category of finite sets and surjections to vector spaces. Via maps that glue on marked P 1 's, we give the homology of M g,n the structure of an FS op module and bound its degree of generation. As a consequence, we prove that the generating function n dim(H i (M g,n ))t n is rational, and its denominator has roots in the set {1, 1/2, . . . , 1/p(g, i)} where p(g, i) is a polynomial of order O(g 2 i 2 ). We also obtain restrictions on the decomposition of the homology of M g,n into irreducible S n representations.

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“…It follows immediately from [Nag,Corollary 5.2] and [GL,Theorem 1.1(1)]. A more detailed argument in the FI-module case is provided in [CMNR,Theorem 2.10(4)].…”

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confidence: 93%