Rational Cone of Norm-Invariant Vectors Under a Matrix Action

2021 ◽  
pp. 394-409
Author(s):  
Juan Tolosa
Keyword(s):  
Invariant Vectors

scholarly journals On the Haagerup and Kazhdan properties of R. Thompson’s groups

2019 ◽  
Vol 22 (5) ◽  
pp. 795-807 ◽  
Author(s):  
Arnaud Brothier ◽  
Vaughan F. R. Jones
Keyword(s):  
Commutator Subgroup ◽  
Invariant Vector ◽  
Haagerup Property ◽  
Thompson Groups ◽  
Thompson's Groups ◽  
Intermediate Subgroup ◽  
Regular Representations ◽  
Left Regular ◽  
Invariant Vectors ◽  
Property T

Abstract A machinery developed by the second author produces a rich family of unitary representations of the Thompson groups F, T and V. We use it to give direct proofs of two previously known results. First, we exhibit a unitary representation of V that has an almost invariant vector but no nonzero {[F,F]} -invariant vectors reproving and extending Reznikoff’s result that any intermediate subgroup between the commutator subgroup of F and V does not have Kazhdan’s property (T) (though Reznikoff proved it for subgroups of T). Second, we construct a one parameter family interpolating between the trivial and the left regular representations of V. We exhibit a net of coefficients for those representations which vanish at infinity on T and converge to 1 thus reproving that T has the Haagerup property after Farley who further proved that V has this property.

scholarly journals Decay property of stopped Markovian bulk-arriving queues

2008 ◽  
Vol 40 (1) ◽  
pp. 95-121 ◽  
Author(s):  
Junping Li ◽  
Anyue Chen
Keyword(s):  
Generating Functions ◽  
Busy Period ◽  
Invariant Measures ◽  
Geometric Interpretation ◽  
Decay Property ◽  
Decay Parameter ◽  
Decay Properties ◽  
Invariant Vectors ◽  
Quasistationary Distributions

We consider decay properties including the decay parameter, invariant measures, invariant vectors, and quasistationary distributions of a Markovian bulk-arriving queue that stops immediately after hitting the zero state. Investigating such behavior is crucial in realizing the busy period and some other related properties of Markovian bulk-arriving queues. The exact value of the decay parameter λC is obtained and expressed explicitly. The invariant measures, invariant vectors, and quasistationary distributions are then presented. We show that there exists a family of invariant measures indexed by λ ∈ [0, λC]. We then show that, under some conditions, there exists a family of quasistationary distributions, also indexed by λ ∈ [0, λC]. The generating functions of these invariant measures and quasistationary distributions are presented. We further show that a stopped Markovian bulk-arriving queue is always λC-transient and some deep properties are revealed. The clear geometric interpretation of the decay parameter is explained. A few examples are then provided to illustrate the results obtained in this paper.

Uniqueness and Decay Properties of Markov Branching Processes with Disasters

2014 ◽  
Vol 51 (03) ◽  
pp. 613-624 ◽  
Author(s):  
Anyue Chen ◽  
Kai Wang Ng ◽  
Hanjun Zhang
Keyword(s):  
Branching Process ◽  
Invariant Measures ◽  
Branching Processes ◽  
Decay Parameter ◽  
Markov Branching Process ◽  
Decay Properties ◽  
Invariant Vectors ◽  
Quasistationary Distributions

In this paper we discuss the decay properties of Markov branching processes with disasters, including the decay parameter, invariant measures, and quasistationary distributions. After showing that the corresponding q-matrix Q is always regular and, thus, that the Feller minimal Q-process is honest, we obtain the exact value of the decay parameter λ C . We show that the decay parameter can be easily expressed explicitly. We further show that the Markov branching process with disaster is always λ C -positive. The invariant vectors, the invariant measures, and the quasidistributions are given explicitly.

scholarly journals Buildings, spiders, and geometric Satake

2013 ◽  
Vol 149 (11) ◽  
pp. 1871-1912 ◽  
Author(s):  
Bruce Fontaine ◽  
Joel Kamnitzer ◽  
Greg Kuperberg
Keyword(s):  
Algebraic Group ◽  
Configuration Space ◽  
Tensor Products ◽  
Configuration Spaces ◽  
Affine Buildings ◽  
Simple Algebraic Group ◽  
Trivalent Graphs ◽  
Invariant Vectors ◽  
Invariant Spaces ◽  
The Web

AbstractLet$G$be a simple algebraic group. Labelled trivalent graphs called webs can be used to produce invariants in tensor products of minuscule representations. For each web, we construct a configuration space of points in the affine Grassmannian. Via the geometric Satake correspondence, we relate these configuration spaces to the invariant vectors coming from webs. In the case of$G= \mathrm{SL} (3)$, non-elliptic webs yield a basis for the invariant spaces. The non-elliptic condition, which is equivalent to the condition that the dual diskoid of the web is$\mathrm{CAT} (0)$, is explained by the fact that affine buildings are$\mathrm{CAT} (0)$.

Vanishing of the first-order cohomology on Olshanski spherical pairs

2021 ◽  
pp. 2150030
Author(s):  
Marouane Rabaoui
Keyword(s):  
Direct Limit ◽  
Unitary Representations ◽  
Limit Groups ◽  
First Order ◽  
Cohomology Space ◽  
Invariant Vectors

In this paper, we study the first-order cohomology space of countable direct limit groups related to Olshanski spherical pairs, relatively to unitary representations which do not have almost invariant vectors. In particular, we prove a variant of Delorme’s vanishing result of the first-order cohomology space for spherical representations of Olshanski spherical pairs.

scholarly journals Lie theory of finite simple groups and the Roth property

2017 ◽  
Vol 163 (2) ◽  
pp. 301-340 ◽  
Author(s):  
J. LÓPEZ PEÑA ◽  
S. MAJID ◽  
K. RIETSCH
Keyword(s):  
Finite Group ◽  
Dimensional Subspace ◽  
Conjugacy Classes ◽  
Simple Groups ◽  
Finite Simple Groups ◽  
Killing Form ◽  
Dihedral Groups ◽  
Invariant Vectors ◽  
A Finite Group ◽  
Conjugation Representation

AbstractIn noncommutative geometry a ‘Lie algebra’ or bidirectional bicovariant differential calculus on a finite group is provided by a choice of an ad-stable generating subset $\mathcal{C}$ stable under inversion. We study the associated Killing form K. For the universal calculus associated to $\mathcal{C}$ = G \ {e} we show that the magnitude $\mu=\sum_{a,b\in\mathcal{C}}(K^{-1})_{a,b}$ of the Killing form is defined for all finite groups (even when K is not invertible) and that a finite group is Roth, meaning its conjugation representation contains every irreducible, iff μ ≠ 1/(N − 1) where N is the number of conjugacy classes. We show further that the Killing form is invertible in the Roth case, and that the Killing form restricted to the (N − 1)-dimensional subspace of invariant vectors is invertible iff the finite group is an almost-Roth group (meaning its conjugation representation has at most one missing irreducible). It is known [9, 10] that most nonabelian finite simple groups are Roth and that all are almost Roth. At the other extreme from the universal calculus we prove that the 2-cycles conjugacy class in any Sn has invertible Killing form, and the same for the generating conjugacy classes in the case of the dihedral groups D2n with n odd. We verify invertibility of the Killing forms of all real conjugacy classes in all nonabelian finite simple groups to order 75,000, by computer, and we conjecture this to extend to all nonabelian finite simple groups.

Pro- Iwahori–Hecke algebras are Gorenstein

2013 ◽  
Vol 13 (4) ◽  
pp. 753-809 ◽  
Author(s):  
Rachel Ollivier ◽  
Peter Schneider
Keyword(s):  
Upper Bound ◽  
Reductive Group ◽  
Global Dimension ◽  
Arbitrary Field ◽  
Graded Rings ◽  
Injective Dimension ◽  
Locally Compact ◽  
Invariant Vectors ◽  
Residue Characteristic ◽  
Iwahori Subgroup

AbstractLet$\mathfrak{F}$be a locally compact nonarchimedean field with residue characteristic$p$, and let$\mathrm{G} $be the group of$\mathfrak{F}$-rational points of a connected split reductive group over$\mathfrak{F}$. For$k$an arbitrary field of any characteristic, we study the homological properties of the Iwahori–Hecke$k$-algebra${\mathrm{H} }^{\prime } $and of the pro-$p$Iwahori–Hecke$k$-algebra$\mathrm{H} $of$\mathrm{G} $. We prove that both of these algebras are Gorenstein rings with self-injective dimension bounded above by the rank of$\mathrm{G} $. If$\mathrm{G} $is semisimple, we also show that this upper bound is sharp, that both$\mathrm{H} $and${\mathrm{H} }^{\prime } $are Auslander–Gorenstein, and that there is a duality functor on the finite length modules of$\mathrm{H} $(respectively${\mathrm{H} }^{\prime } $). We obtain the analogous Gorenstein and Auslander–Gorenstein properties for the graded rings associated to$\mathrm{H} $and${\mathrm{H} }^{\prime } $.When$k$has characteristic$p$, we prove that in ‘most’ cases$\mathrm{H} $and${\mathrm{H} }^{\prime } $have infinite global dimension. In particular, we deduce that the category of smooth$k$-representations of$\mathrm{G} = {\mathrm{PGL} }_{2} ({ \mathbb{Q} }_{p} )$generated by their invariant vectors under the pro-$p$Iwahori subgroup has infinite global dimension (at least if$k$is algebraically closed).

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