Stability Subgroup Research Articles

The $p$-colorable subgroup of Thompson’s group

V. F. R. Jones introduced a method of constructing knots and links from elements of Thompson’s group F by using its unitary representations. He also defined several subgroups of F as the stabilizer subgroups and some researchers studied them algebraically. One of the subgroups is called the 3 -colorable subgroup \mathcal{F} , and the authors proved that all knots and links obtained from non-trivial elements of \mathcal{F} are 3 -colorable. In this paper, for any odd integer p greater than two, we define the p -colorable subgroup of F whose non-trivial elements yield p -colorable knots and links and show it is isomorphic to a certain Brown–Thompson group.

Actions of large finite groups on manifolds

In this paper, we survey some recent results on actions of finite groups on topological manifolds. Given an action of a finite group [Formula: see text] on a manifold [Formula: see text], these results provide information on the restriction of the action to a subgroup of [Formula: see text] of index bounded above by a number depending only on [Formula: see text]. Some of these results refer to the algebraic structure of the group, such as being abelian or nilpotent or admitting a generating subset of controlled size; other results refer to the geometry of the action, e.g. to the existence of fixed points, to the collection of stabilizer subgroups or to the action on cohomology.

Clifford orbits from cayley graph quotients

We describe the structure of the $n$-qubit Clifford group $\mathcal{C}_n$ via Cayley graphs, whose vertices represent group elements and edges represent generators. In order to obtain the action of Clifford gates on a given quantum state, we introduce a quotient procedure. Quotienting the Cayley graph by the stabilizer subgroup of a state gives a reduced graph which depicts the state's Clifford orbit. Using this protocol for $\mathcal{C}_2$, we reproduce and generalize the reachability graphs introduced in \cite{Keeler2022}. Since the procedure is state-independent, we extend our study to non-stabilizer states, including the W and Dicke states. Our new construction provides a more precise understanding of state evolution under Clifford circuit action.

Quantum symmetric conjugacy classes of non-exceptional groups

We evaluate one-dimensional representations of quantum symmetric conjugacy classes of classical matrix groups along with their quantum stabilizer subgroups.

Left multiplication operators on the Riordan group

In this paper, we introduce the left multiplication operators on the Riordan group, and carry out a systematic study on them. By using these operators, we obtain a family H of infinite Riordan subgroups. This family includes not only some well-known subgroups, but also the subgroup families introduced recently in literature. Fundamental properties of the left multiplication operators and the subgroup family H are presented. The Riordan involutions and pseudo-involutions, the stabilizer subgroups in family H, and the n-th commutator subgroup of any subgroup in family H are studied. Some further applications of the left multiplication operators are also provided. In particular, we study a class of left multiplication operators which are closely related to the diagonal translation operator and the [m]-complementary operator, and enrich the theory of complementary Riordan arrays. Moreover, we introduce and study the higher-level lifts of Riordan arrays as well as the vertical and horizontal 1/m Riordan arrays. As a result, it can be found that the left multiplication operators provide a unified approach to numerous subgroups, operators and transformations of the Riordan group.

Burst error-correcting quantum stabilizer codes designed from idempotents

Certain classical codes can be viewed isomorphically as ideals of group algebras, while studying their algebraic structures help extracting the code properties. Research has shown that this was remarkably efficient in the case when the code generators are idempotents. In quantum error correction, the theory of stabilizer formalism requires classical self-orthogonal additive codes over the finite field GF(4), which, via the lens of group algebras, are essentially F_2-submodules over GF(4). Therefore, this paper provides a classification on idempotents in commutative group algebra GF(4)G, followed by a criterion that allows idempotents to generate stabilizer subgroups. Later, the construction of quantum stabilizer codes is done in the case when G is a cyclic group C_n, for n=2^m-1 and n=2^m+1. Quantum bounds on their burst error minimum distance are subsequently determined.

On parabolic subgroups of symplectic reflection groups

AbstractUsing Cohen’s classification of symplectic reflection groups, we prove that the parabolic subgroups, that is, stabilizer subgroups, of a finite symplectic reflection group, are themselves symplectic reflection groups. This is the symplectic analog of Steinberg’s Theorem for complex reflection groups.Using computational results required in the proof, we show the nonexistence of symplectic resolutions for symplectic quotient singularities corresponding to three exceptional symplectic reflection groups, thus reducing further the number of cases for which the existence question remains open.Another immediate consequence of our result is that the singular locus of the symplectic quotient singularity associated to a symplectic reflection group is pure of codimension two.

Steinberg's fixed point theorem for crystallographic complex reflection groups

Steinberg's fixed point theorem states that given a finite complex reflection group the stabilizer subgroup of a point is generated by reflections that fix this point. This statement is also true for affine Weyl groups. Of the infinite discrete complex reflection groups, it was shown that there are some infinite complex reflections groups that have non-trivial stabilizers that do not contain a single reflection, and therefore, these groups cannot satisfy the fixed point theorem. We thus classify the infinite discrete irreducible complex reflection groups of the infinite family which satisfy the statement of the fixed point theorem.

Boundary maps and covariant representations

We extend applications of Furstenberg boundary theory to the study of C ∗ $C^*$ -algebras associated to minimal actions Γ ↷ X $\Gamma \!\curvearrowright \!X$ of discrete groups Γ $\Gamma$ on locally compact spaces X $X$ . We introduce boundary maps on ( Γ , X ) $(\Gamma ,X)$ - C ∗ $C^*$ -algebras and investigate their applications in this context. Among other results, we completely determine when C ∗ $C^*$ -algebras generated by covariant representations arising from stabilizer subgroups are simple. We also characterize the intersection property of locally compact Γ $\Gamma$ -spaces and simplicity of their associated crossed products.

Spinning particles, coadjoint orbits and Hamiltonian formalism

The extensive analysis of the dynamics of relativistic spinning particles is presented. Using the coadjoint orbits method the Hamiltonian dynamics is explicitly described. The main technical tool is the factorization of general Lorentz transformation into pure boost and rotation. The equivalent constrained dynamics on Poincare group (viewed as configuration space) is derived and complete classification of constraints is performed. It is shown that the first class constraints generate local symmetry corresponding to the stability subgroup of some point on coadjoint orbit. The Dirac brackets for second class constraints are computed. Finally, canonical quantization is performed leading to infinitesimal form of irreducible representations of Poincare group.

Triple transitivity, maximal and rigid von Neumann subalgebras

Given any faithful 3-transitive action of a countable discrete group G on an infinite set X. Assume either the action is sharply 3-transitive or the stabilizer subgroup of any two distinct points is I.C.C. We show that the stabilizer subgroup of any point in X generates a maximal and rigid von Neumann subalgebra in LG. This extends a previous result by Jiang [4], who proved the same result but under stronger assumption of 4-transitivity.

Maximal von Neumann subalgebras arising from maximal subgroups

Ge (2003) asked the question whether LF∞ can be embedded in to LF2 as a maximal subfactor. We answer it affirmatively in three different approaches, all containing the same key ingredient: the existence of maximal subgroups with infinite index. We also show that point stabilizer subgroups for every faithful, 4-transitive action on an infinite set give rise to maximal von Neumann subalgebras. By combining this with the known results on constructing faithful, highly transitive actions, we get many maximal von Neumann subalgebras arising from maximal subgroups with infinite index.

Higher spin mapping class groups and strata of Abelian differentials over Teichmüller space

For g≥5, we give a complete classification of the connected components of strata of abelian differentials over Teichmüller space, establishing an analogue of Kontsevich and Zorich's classification of their components over moduli space. Building on work of the first author [2], we find that the non-hyperelliptic components are classified by an invariant known as an r–spin structure. This is accomplished by computing a certain monodromy group valued in the mapping class group. To do this, we determine explicit finite generating sets for all r–spin stabilizer subgroups of the mapping class group, completing a project begun by the second author in [18]. Some corollaries in flat geometry and toric geometry are obtained from these results.

Distinct genomic landscapes of gastroesophageal adenocarcinoma depending on PD-L1 expression identify mutations in RAS–MAPK pathway and TP53 as potential predictors of immunotherapy efficacy

The impact of molecular alterations on programmed death-ligand 1 (PD-L1) combined positive score (CPS) is not well studied in gastroesophageal adenocarcinomas (GEAs). We aimed to characterize genomic features of tumors with different CPSs in GEAs. Genomic alterations of 2518 GEAs were compared in three groups (PD-L1 CPS ≥ 10, high; CPS= 1-9, intermediate; CPS < 1, low) using next-generation sequencing. We assessed the impact of gene mutations on the efficacy of immune checkpoint inhibitors (ICIs) and tumor immune environment based on the Memorial Sloan Kettering Cancer Center and The Cancer Genome Atlas databases. High, intermediate, and low CPSs were seen in 18%, 54% and 28% of GEAs, respectively. PD-L1 positivity was less prevalent in women and in tissues derived from metastatic sites. PD-L1 CPS was positively associated with mismatch repair deficiency/microsatellite instability-high, but independent of tumor mutation burden distribution. Tumors with mutations in KRAS, TP53, and RAS-mitogen-activated protein kinase (MAPK) pathway were associated with higher PD-L1 CPSs in the mismatch repair proficiency and microsatellite stability (pMMR&MSS) subgroup. Patients with RAS-MAPK pathway alterations had longer overall survival (OS) from ICIs compared to wildtype (WT) patients [27 versus 13 months, hazard ratio (HR)= 0.36, 95% confidence interval (CI): 0.19-0.7, P= 0.016] and a similar trend was observed in the MSS subgroup (P= 0.11). In contrast, patients with TP53 mutations had worse OS from ICIs compared to TP53-WT patients in the MSS subgroup (5 versus 21 months, HR= 2.39, 95% CI: 1.24-4.61, P= 0.016). This is the largest study to investigate the distinct genomic landscapes of GEAs with different PD-L1 CPSs. Our data may provide novel insights for patient selection using mutations in TP53 and RAS-MAPK pathway and for the development of rational combination immunotherapies in GEAs.

Symmetric association schemes arising from abstract regular polytopes

This article investigates the question of when every double coset of a string $C$-group $G$ relative to its vertex stabilizer subgroup $H$ is represented by an involution. We show that this is the case for every finite string Coxeter group except in the $\{5,3,3\}$ case of type $H_4$, and for the infinite Coxeter groups of Schlafli type $\{4,4\}$ and $\{3,6\}$. From this it is immediate that, for every string $C$-group of these types, the double coset algebra $\mathbb{C}[G/\!\!/H]$ is commutative and all of its characters are realizable over $\mathbb{R}$. In particular, the abstract regular polytopes with these automorphism groups have a polyhedral realization cone.

A general method to construct invariant PDEs on homogeneous manifolds

Let [Formula: see text] be an [Formula: see text]-dimensional homogeneous manifold and [Formula: see text] be the manifold of [Formula: see text]-jets of hypersurfaces of [Formula: see text]. The Lie group [Formula: see text] acts naturally on each [Formula: see text]. A [Formula: see text]-invariant partial differential equation of order [Formula: see text] for hypersurfaces of [Formula: see text] (i.e., with [Formula: see text] independent variables and [Formula: see text] dependent one) is defined as a [Formula: see text]-invariant hypersurface [Formula: see text]. We describe a general method for constructing such invariant partial differential equations for [Formula: see text]. The problem reduces to the description of hypersurfaces, in a certain vector space, which are invariant with respect to the linear action of the stability subgroup [Formula: see text] of the [Formula: see text]-prolonged action of [Formula: see text]. We apply this approach to describe invariant partial differential equations for hypersurfaces in the Euclidean space [Formula: see text] and in the conformal space [Formula: see text]. Our method works under some mild assumptions on the action of [Formula: see text], namely: A1) the group [Formula: see text] must have an open orbit in [Formula: see text], and A2) the stabilizer [Formula: see text] of the fiber [Formula: see text] must factorize via the group of translations of the fiber itself.

Monotileable Amenable Groups: An Application

We study actions of countable discrete groups which are Monotileable amenable groups in the sense that there exists a mean on X which is invariant under the action of G. Assuming that G is nonamenable, we obtain structural results for the stabilizer subgroups of amenable actions which allow us to relate the first l2-Betti number of G with that of the stabilizer subgroups.

Atypical communication characteristics among clinic-referred youth with and without autism spectrum disorder: Stability and associations with clinical correlates.

Atypical communication characteristics (ACCs), such as speech delay, odd pitch, and pragmatic difficulties, are common features of autism spectrum disorder (ASD) as are the symptoms of a wide range of psychiatric disorders. Using a simple retrospective method, this study aimed to better understand the relation and stability of ACCs with a broad range of psychiatric symptoms among large, well-characterized samples of clinic-referred children and adolescents with and without ASD. Youth with ASD had higher rates and a more variable pattern of developmental change in ACCs than the non-ASD diagnostic group. Latent class analysis yielded three ACC stability subgroups within ASD: Stable ACCs, Mostly Current-Only ACCs, and Little Professors. Subgroups exhibited differences in severity of ASD symptomatology, co-occurring psychiatric symptoms, and other correlates. Our findings provide support for the clinical utility of characterizing caregiver-perceived changes in ACCs in identifying children at risk for co-occurring psychopathology and other clinically relevant variables.

Invariant means and the structure of inner amenable groups

We study actions of countable discrete groups which are amenable in the sense that there exists a mean on X which is invariant under the action of G . Assuming that G is nonamenable, we obtain structural results for the stabilizer subgroups of amenable actions which allow us to relate the first ℓ 2 -Betti number of G with that of the stabilizer subgroups. In addition, for any marked finitely generated nonamenable group G we establish a uniform isoperimetric threshold for Schreier graphs G / H of G , beyond which the group H is necessarily weakly normal in G . Even more can be said in the particular case of an atomless mean for the conjugation action—that is, when G is inner amenable. We show that inner amenable groups have fixed price 1 , and we establish cocycle superrigidity for the Bernoulli shift of any nonamenable inner amenable group. In addition, we provide a concrete structure theorem for inner amenable linear groups over an arbitrary field. As a special case of inner amenability, we consider groups which are stable in the sense of Jones and Schmidt, obtaining a complete characterization of linear groups which are stable. Our analysis of stability leads to many new examples of stable groups; notably, all nontrivial countable subgroups of the group H ( R ) , of piecewise-projective homeomorphisms of the line, are stable.

Clinical Significance of Tumor and Immune Cell PD-L1 Expression in Gastric Adenocarcinoma.

The prognostic relevance of programmed cell death ligand-1 (PD-L1) protein expression in gastric cancer (GC) remains controversial. The aims of the present study were to determine the correlations between tumor cell (TC) and immune cell (IC) PD-L1 protein levels with prognosis, and to determine the correlation between PD-L1 expression and different molecular GC subtypes. TC and IC PD-L1 protein levels were measured in 286 GC patients. The patients were classified according to the Cancer Genome Atlas and Asian Cancer Research Group guidelines using immunohistochemistry and in situ hybridization. TC and IC PD-L1 protein levels were positively correlated with patient survival. TC PD-L1 expression was negatively correlated with tumor grade. TC and IC PD-L1 expression was associated with improved prognosis in Epstein-Barr virus negative (EBV-), microsatellite instability (MSI) rather than microsatellite stability (MSS) subgroup GC patients. PD-L1 protein expression in TCs and ICs can be used as a prognostic indicator for GC patients, particularly in the EBV-, MSI, and MSS subgroups.