Stable decompositions of certain representations of the finite general linear groups

Gan, Wee Liang; Watterlond, John

doi:10.48550/arxiv.1605.08434

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“…In this section we consider representations of the categories VI(q), where q is a power of a prime, and FI m , where m is a positive integer. These categories can be seen discussed in [Gad1,GW,LY,PS,SS].…”

Section: Generalizations and Alterationsmentioning

confidence: 99%

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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