4,011 research outputs found
Elementary abelian subgroups: from algebraic groups to finite groups
We describe a new approach for classifying conjugacy classes of elementary abelian subgroups in simple algebraic groups over an algebraically closed field, and understanding the normaliser and centraliser structure of these. For toral subgroups, we give an effective classification algorithm. For non-toral elementary abelian subgroups, we focus on algebraic groups of exceptional type with a view to future applications, and in this case we provide tables explicitly describing the subgroups and their local structure. We then describe how to transfer results to the corresponding finite groups of Lie type using the Lang-Steinberg Theorem; this will be used in forthcoming work to complete the classification of elementary abelian p-subgroups for torsion primes p in finite groups of exceptional Lie type. Such classification results are important for determining the maximal p-local subgroups and p-radical subgroups, both of which play a crucial role in modular representation theory
Composition and Cobordism Maps
We study the relationship between the algebra of module homomorphisms under composition and 4-dimensional cobordisms in the context of bordered Heegaard Floer homology. In particular, we prove that composition of module homomorphisms of type- structures induces the pair of pants cobordism map on Heegaard Floer homology in the morphism spaces formulation of the latter, due to Lipshitz--Ozsv\'{a}th--Thurston. Along the way, we prove a gluing result for cornered 4-manifolds constructed from bordered Heegaard triples.
As applications, we present a new algorithm for computing arbitrary cobordism maps on Heegaard Floer homology and construct new nontrivial -deformations of Khovanov's arc algebras. Motivated by this last result and a K\"{u}nneth theorem for Heegaard Floer complexes of connected sums, we also prove the existence of a tensor product decomposition for arc algebras in characteristic 2 and show that there cannot be such a splitting over
Recognition of the symplectic simple group by the order and degree prime-power graph
Let be a finite group, the set of all irreducible character degrees of , and the set of all prime divisors of integers in . For a prime and a positive integer , let denote the -part of . The degree prime-power graph of is a graph whose vertex set is , where , and there is an edge between distinct numbers if divides some integer in . The authors have previously shown that some non-abelian simple groups can be uniquely determined by their orders and degree prime-power graphs. In this paper, the authors build on this work and demonstrate that the symplectic simple group can be uniquely identified by its order and degree prime-power graph
On the Global Topology of Moduli Spaces of Riemannian Metrics with Holonomy
We discuss aspects of the global topology of moduli spaces of hyperkähler metrics.
If the second Betti number is larger than , we show that each connected component of these moduli spaces is not contractible. Moreover, in certain cases, we show that the components are simply connected and determine the second rational homotopy group. By that, we prove that the rank of the second homotopy group is bounded from below by the number of orbits of MBM-classes in the integral cohomology. \\
An explicit description of the moduli space of these hyperkähler metrics in terms of Torelli theorems will be given. We also provide such a description for the moduli space of Einstein metrics on the Enriques manifold. For the Enriques manifold, we also give an example of a desingularization process similar to the Kummer construction of Ricci-flat metrics on a Kummer surface.\\
We will use these theorems to provide topological statements for moduli spaces of Ricci-flat and Einstein metrics in any dimension larger than . For a compact simply connected manifold we show that the moduli space of Ricci flat metrics on splits homeomorphically into a product of the moduli space of Ricci flat metrics on and the moduli of sectional curvature flat metrics on the torus
LIPIcs, Volume 251, ITCS 2023, Complete Volume
LIPIcs, Volume 251, ITCS 2023, Complete Volum
On Newton polytopes of Lagrangian augmentations
This note explores the use of Newton polytopes in the study of Lagrangian
fillings of Legendrian submanifolds. In particular, we show that Newton
polytopes associated to augmented values of Reeb chords can distinguish
infinitely many distinct Lagrangian fillings, both for Legendrian links and
higher-dimensional Legendrian spheres. The computations we perform work in
finite characteristic, which significantly simplifies arguments and also allows
us to show that there exist Legendrian links with infinitely many
non-orientable exact Lagrangian fillings.Comment: 26 page
On the complexity of isomorphism problems for tensors, groups, and polynomials IV: linear-length reductions and their applications
Many isomorphism problems for tensors, groups, algebras, and polynomials were
recently shown to be equivalent to one another under polynomial-time
reductions, prompting the introduction of the complexity class TI (Grochow &
Qiao, ITCS '21; SIAM J. Comp., '23). Using the tensorial viewpoint, Grochow &
Qiao (CCC '21) then gave moderately exponential-time search- and
counting-to-decision reductions for a class of -groups. A significant issue
was that the reductions usually incurred a quadratic increase in the length of
the tensors involved. When the tensors represent -groups, this corresponds
to an increase in the order of the group of the form ,
negating any asymptotic gains in the Cayley table model.
In this paper, we present a new kind of tensor gadget that allows us to
replace those quadratic-length reductions with linear-length ones, yielding the
following consequences:
1. Combined with the recent breakthrough -time
isomorphism-test for -groups of class 2 and exponent (Sun, STOC '23),
our reductions extend this runtime to -groups of class and exponent
where .
2. Our reductions show that Sun's algorithm solves several TI-complete
problems over , such as isomorphism problems for cubic forms, algebras,
and tensors, in time .
3. Polynomial-time search- and counting-to-decision reduction for testing
isomorphism of -groups of class and exponent in the Cayley table
model. This answers questions of Arvind and T\'oran (Bull. EATCS, 2005) for
this group class, thought to be one of the hardest cases of Group Isomorphism.
4. If Graph Isomorphism is in P, then testing equivalence of cubic forms and
testing isomorphism of algebra over a finite field can both be solved in
time , improving from the brute-force upper bound
Redundant relators in cyclic presentations of groups
A cyclic presentation of a group is a presentation with an equal number of generators and relators that admits a particular cyclic symmetry. We characterise the orientable, non-orientable, and redundant cyclic presentations and obtain concise refinements of these presentations. We show that the Tits alternative holds for the class of groups defined by redundant cyclic presentations and show that if the number of generators of the cyclic presentation is greater than two, then the corresponding group is large. Generalising and extending earlier results of the authors, we describe the star graphs of orientable and non-orientable cyclic presentations and classify the cyclic presentations whose star graph components are pairwise isomorphic incidence graphs of generalised polygons, thus classifying the so-called (m,k,ν) -special cyclic presentations
Commensurations of and its Torelli subgroup
For , the abstract commensurators of both and
its Torelli subgroup are isomorphic to
itself.Comment: 29 pages, 5 figure
Locality and Exceptional Points in Pseudo-Hermitian Physics
Pseudo-Hermitian operators generalize the concept of Hermiticity. Included in this class of operators are the quasi-Hermitian operators, which define a generalization of quantum theory with real-valued measurement outcomes and unitary time evolution. This thesis is devoted to the study of locality in quasi-Hermitian theory, the symmetries and conserved quantities associated with non-Hermitian operators, and the perturbative features of pseudo-Hermitian matrices.
An implicit assumption of the tensor product model of locality is that the inner product factorizes with the tensor product. Quasi-Hermitian quantum theory generalizes the tensor product model by modifying the Born rule via a metric operator with nontrivial Schmidt rank. Local observable algebras and expectation values are examined in chapter 5. Observable algebras of two one-dimensional fermionic quasi-Hermitian chains are explicitly constructed. Notably, there can be spatial subsystems with no nontrivial observables. Despite devising a new framework for local quantum theory, I show that expectation values of local quasi-Hermitian observables can be equivalently computed as expectation values of Hermitian observables. Thus, quasi-Hermitian theories do not increase the values of nonlocal games set by Hermitian theories. Furthermore, Bell's inequality violations in quasi-Hermitian theories never exceed the Tsirelson bound of Hermitian quantum theory.
A perturbative feature present in pseudo-Hermitian curves which has no Hermitian counterpart is the exceptional point, a branch point in the set of eigenvalues. An original finding presented in section 2.6.3 is a correspondence between cusp singularities of algebraic curves and higher-order exceptional points. Eigensystems of one-dimensional lattice models admit closed-form expressions that can be used to explore the new features of non-Hermitian physics. One-dimensional lattice models with a pair of non Hermitian defect potentials with balanced gain and loss, Δ±iγ, are investigated in chapter 3. Conserved quantities and positive-definite metric operators are examined. When the defects are nearest neighbour, the entire spectrum simultaneously becomes complex when γ increases beyond a second-order exceptional point. When the defects are at the edges of the chain and the hopping amplitudes are 2-periodic, as in the Su-Schrieffer-Heeger chain, the PT-phase transition is dictated by the topological phase
of the system. In the thermodynamic limit, PT-symmetry spontaneously breaks in the topologically non-trivial phase due to the presence of edge states.
Chiral symmetry and representation theory are utilized in chapter 4 to derive large classes of pseudo-Hermitian operators with closed-form intertwining operators. These intertwining operators include positive-definite metric operators in the quasi-Hermitian case. The PT-phase transition is explicitly determined in a special case
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