Stable Decompositions of Certain Representations of the Finite General Linear Groups

Gan, Wee Liang; Watterlond, John

doi:10.1007/s00031-017-9440-y

Transformation Groups

2017

DOI: 10.1007/s00031-017-9440-y

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Stable Decompositions of Certain Representations of the Finite General Linear Groups

Wee Liang Gan

John Watterlond

Abstract: Abstract. We prove that the irreducible decomposition of the permutation representation of GLn(Fq) on GLn(Fq)/GLn−m(Fq) stabilizes for large n. We deduce, as a consequence, a representation stability theorem for finitely generated VIC-modules.

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Cited by 4 publications

References 8 publications

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Families of nested graphs with compatible symmetric-group actions

Ramos

White

2019

Sel. Math. New Ser.

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For each n, r ≥ 0, let KG(n, r) denote the Kneser Graph; that whose vertices are labeled by r-element subsets of n, and whose edges indicate that the corresponding subsets are disjoint. Fixing r and allowing n to vary, one obtains a family of nested graphs, each equipped with a natural action by a symmetric group Sn, such that these actions are compatible. Collections of graphs of this type are common in algebraic combinatorics and include families such as the Johnson Graphs, Crown Graphs and Rook Graphs. In previous work [RW], the authors systematically studied families of this type using the language of representation stability and FI-modules. In that work, it is shown that such families of graphs exhibit a large variety of asymptotic regular behaviors.The present work applies the theory developed in [RW], later refined in [RSW], to study random walks on the graphs of such families. We show that the moments of hitting times exhibit rational function behavior asymptotically. By consequence we conclude similar facts about the entries of the discrete Green's functions, as defined by Chung and Yau [CY]. Finally, we illustrate how the algebro-combinatorial structure of the graphs in these families give bounds on the mixing times of random walks on those graphs. We suggest some possible directions for future study, including of the appearance, or not, of the cut-off phenomenon, originally presented by Diaconis [D].

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Families of nested graphs with compatible symmetric-group actions

Ramos

White

2019

Sel. Math. New Ser.

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Representation stability for filtrations of Torelli groups

Patzt

2018

Math. Ann.

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We show, finitely generated rational VIC Q -modules and SI Q -modules are uniformly representation stable and all their submodules are finitely generated. We use this to prove two conjectures of Church and Farb, which state that the quotients of the lower central series of the Torelli subgroups of Aut (Fn) and Mod(Σ g,1 ) are uniformly representation stable as sequences of representations of the general linear groups and the symplectic groups, respectively. Furthermore we prove an analogous statement for their Johnson filtrations. PETER PATZTInjectivity: The map φ n : V n → V n+1 is injective for all large enough n ∈ N.Multiplicity stability: We can write. . ) and cλ ,n is independent of n for all large enough n ∈ N.A consistent sequence is called uniformly representation stable if the multiplicities cλ ,n stabilize uniformly.Functors from a category C to the category Q−mod of vector spaces over Q are called C-modules. Every FI-module V : FI → Q−mod gives rise to a consistent sequence, by takingThe connection to representation stability was provided by Church-Ellenberg-Farb in the following theorem.Theorem (Church-Ellenberg-Farb [CEF15, Thm 1.13]). An FI-module V is finitely generated if and only if its consistent sequence is uniformly representation stable and V n is finite dimensional for all n ∈ N.This theorem depends on the following noetherian property of FI-modules.Theorem (Church-Ellenberg-Farb [CEF15, Thm 1.3]). Every submodule of a finitely generated FI-module is finitely generated.Analogous theorems for the hyperoctahedral groups were proved by Wilson [Wil14, Thm 4.21 + Thm 4.22].Representation stability over the general linear groups and symplectic groups. The rational representation theory for both GL n Q and Sp 2n Q is semisimple and the irreducibles are indexed by pairs of partitions (λ + , λ − ) such that the lengths ℓ(λ + ) + ℓ(λ − ) ≤ n and by partitions λ whose length ℓ(λ) ≤ n, respectively. We respectively denote these irreducibles by GL n (λ + , λ − ) and Sp 2n (λ).For a consistent sequence of rational representations of the general linear groups or the symplectic groups Church-Farb [CF13, Def 2.3] define (uniform) representation stability analogously to the symmetric groups-only the analogue of the multiplicity stability condition is easier to state: ematical School, the SFB Raum-Zeit-Materie and the Dahlem Research School. The author also wants to thank for helpful conversations. Special thanks to Steven Sam for his extensive help with the modification rules, and to Kevin Casto for pointing out the conjectures to the author. Rational representation theory of the general linear groups and the symplectic groupsLet us start by shortly recalling the rational (or algebraic) representation theory of the algebraic groups GL n Q and Sp 2n Q. More elaboration can be found in the

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Asymptotic behavior of representations of graded categories with inductive functors

Gan

2019

Journal of Pure and Applied Algebra

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Stable Decompositions of Certain Representations of the Finite General Linear Groups

Abstract: Abstract. We prove that the irreducible decomposition of the permutation representation of GLn(Fq) on GLn(Fq)/GLn−m(Fq) stabilizes for large n. We deduce, as a consequence, a representation stability theorem for finitely generated VIC-modules.

Cited by 4 publications

References 8 publications

Families of nested graphs with compatible symmetric-group actions

Families of nested graphs with compatible symmetric-group actions

Representation stability for filtrations of Torelli groups

Asymptotic behavior of representations of graded categories with inductive functors

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