Casson-Lin's Invariant and Floer Homology

1997 ◽  
Vol 06 (06) ◽  
pp. 851-877 ◽  
Author(s):  
Weiping Li
Keyword(s):  
Euler Number ◽  
Floer Homology ◽  
Natural Generalization ◽  
Conjugacy Classes ◽  
Irreducible Representations ◽  
Representation Space ◽  
Integral Homology

Casson defined an invariant which can be thought of as the number of conjugacy classes of irreducible representations of π1(Y) into SU(2) counted with signs, where Y is an oriented integral homology 3-sphere. Lin defined a similar invariant (the signature of a knot) for a braid representative of a knot in S3. In this paper, we give a natural generalization of Casson-Lin's invariant. Our invariant is the symplectic Floer homology for the representation space of π1(S3 \ K) into SU(2) with trace-zero along all meridians. The symplectic Floer homology of braids is a new invariant of knots and its Euler number is the negative of Casson-Lin's invariant.

EQUIVARIANT KNOT SIGNATURES AND FLOER HOMOLOGIES

2001 ◽  
Vol 10 (05) ◽  
pp. 687-701 ◽  
Author(s):  
WEIPING LI
Keyword(s):  
Trace Formula ◽  
Euler Number ◽  
Floer Homology ◽  
Point Of View ◽  
Representation Space ◽  
Symplectic Topology

In this paper, we give a description of the equivariant signature of knots from the symplectic topology point of view. For certain knots K in S3, we define a symplectic Floer homology for the representation space of the knot group π1 (S3\ K) into SU(2) with trace [Formula: see text] along all meridians (p is an odd prime and 0<k<p). The symplectic Floer homology of knots is a new invariant of knots and its Euler number is half of the equivariant signature of knots.

scholarly journals Lescop's invariant and gauge theory

2015 ◽  
Vol 24 (09) ◽  
pp. 1550050 ◽  
Author(s):  
Prayat Poudel
Keyword(s):  
Gauge Theory ◽  
Euler Characteristic ◽  
Betti Number ◽  
Floer Homology ◽  
Integral Homology ◽  
Similar Relationship

Taubes proved that the Casson invariant of an integral homology 3-sphere equals half the Euler characteristic of its instanton Floer homology. We extend this result to all closed oriented 3-manifolds with positive first Betti number by establishing a similar relationship between the Lescop invariant of the manifold and its instanton Floer homology. The proof uses surgery techniques.

scholarly journals ON RELATIVE COMPLETE REDUCIBILITY

2020 ◽  
Vol 71 (1) ◽  
pp. 321-334 ◽  
Author(s):  
Christopher Attenborough ◽  
Michael Bate ◽  
Maike Gruchot ◽  
Alastair Litterick ◽  
Gerhard Röhrle
Keyword(s):  
Lie Algebras ◽  
Algebraic Groups ◽  
Reductive Group ◽  
Natural Generalization ◽  
Conjugacy Classes ◽  
Closed Field ◽  
Complete Reducibility ◽  
Algebraic Description ◽  
Reductive Subgroup

Abstract Let $K$ be a reductive subgroup of a reductive group $G$ over an algebraically closed field $k$. The notion of relative complete reducibility, introduced in [M. Bate, B. Martin, G. Röhrle, R. Tange, Complete reducibility and conjugacy classes of tuples in algebraic groups and Lie algebras, Math. Z.269 (2011), no. 1, 809–832], gives a purely algebraic description of the closed $K$-orbits in $G^n$, where $K$ acts by simultaneous conjugation on $n$-tuples of elements from $G$. This extends work of Richardson and is also a natural generalization of Serre’s notion of $G$-complete reducibility. In this paper we revisit this idea, giving a characterization of relative $G$-complete reducibility, which directly generalizes equivalent formulations of $G$-complete reducibility. If the ambient group $G$ is a general linear group, this characterization yields representation-theoretic criteria. Along the way, we extend and generalize several results from [M. Bate, B. Martin, G. Röhrle, R. Tange, Complete reducibility and conjugacy classes of tuples in algebraic groups and Lie algebras, Math. Z.269 (2011), no. 1, 809–832].

scholarly journals Irreducible representations of the generalized symmetric group Bmn

1987 ◽  
Vol 29 (1) ◽  
pp. 1-6 ◽  
Author(s):  
M. Saeed-ul-Islam
Keyword(s):  
Symmetric Group ◽  
Wreath Products ◽  
Conjugacy Classes ◽  
Irreducible Representations ◽  
Linear Representations

This paper is devoted to the determining of the irreducible linear representations of the generalized symmetric group (elsewhere written as , Cm ≀ Sn or G(m, 1, n)) by considering the conjugacy classes of and then constructing the same number of inequivalent irreducible linear representations of . These have previously been determined by Kerber [2, Section 5] using Clifford's theory applied to wreath products.

scholarly journals INSTANTON FLOER HOMOLOGY FOR TWO-COMPONENT LINKS

2012 ◽  
Vol 21 (05) ◽  
pp. 1250054 ◽  
Author(s):  
ERIC HARPER ◽  
NIKOLAI SAVELIEV
Keyword(s):  
Euler Characteristic ◽  
Floer Homology ◽  
Integral Homology ◽  
Oriented Link ◽  
Linking Number ◽  
Two Component

For any oriented link of two components in an integral homology 3-sphere, we define an instanton Floer homology whose Euler characteristic is twice the linking number between the components of the link. We show that, for two-component links in the 3-sphere, this Floer homology does not vanish unless the link is split. We also relate our Floer homology to the Kronheimer–Mrowka instanton Floer homology for links.

scholarly journals Lifting Representations of Finite Reductive Groups I: Semisimple Conjugacy Classes

2014 ◽  
Vol 66 (6) ◽  
pp. 1201-1224 ◽  
Author(s):  
Jeffrey D. Adler ◽  
Joshua M. Lansky
Keyword(s):  
Fixed Point ◽  
Finite Group ◽  
Fixed Points ◽  
Point Group ◽  
Reductive Group ◽  
Conjugacy Classes ◽  
Irreducible Representations ◽  
Reductive Groups ◽  
A Finite Group ◽  
The Relationship

AbstractSuppose that is a connected reductive group defined over a field k, and ┌ is a finite group acting via k-automorphisms of satisfying a certain quasi-semisimplicity condition. Then the identity component of the group of -fixed points in is reductive. We axiomatize the main features of the relationship between this fixed-point group and the pair (,┌) and consider any group G satisfying the axioms. If both and G are k-quasisplit, then we can consider their duals *and G*. We show the existence of and give an explicit formula for a natural map from the set of semisimple stable conjugacy classes in G*(k) to the analogous set for *(k). If k is finite, then our groups are automatically quasisplit, and our result specializes to give a map of semisimple conjugacy classes. Since such classes parametrize packets of irreducible representations of G(k) and (k), one obtains a mapping of such packets.

scholarly journals QUASI-ALTERNATING MONTESINOS LINKS

2009 ◽  
Vol 18 (10) ◽  
pp. 1459-1469 ◽  
Author(s):  
TAMARA WIDMER
Keyword(s):  
Open Problem ◽  
Floer Homology ◽  
Natural Generalization ◽  
Khovanov Homology ◽  
Knot Floer Homology ◽  
Interesting Open Problem ◽  
Alternating Links

The aim of this article is to detect new classes of quasi-alternating links. Quasi-alternating links are a natural generalization of alternating links. Their knot Floer and Khovanov homology are particularly easy to compute. Since knot Floer homology detects the genus of a knot as well as whether a knot is fibered, characterization of quasi-alternating links becomes an interesting open problem. We show that there exist classes of non-alternating Montesinos links, which are quasi-alternating.

scholarly journals DUAL SPACE AND HYPERDIMENSION OF COMPACT HYPERGROUPS

2016 ◽  
Vol 59 (2) ◽  
pp. 421-435 ◽  
Author(s):  
MAHMOOD ALAGHMANDAN ◽  
MASSOUD AMINI
Keyword(s):  
Representation Theory ◽  
Finite Groups ◽  
Dual Space ◽  
Conjugacy Classes ◽  
Irreducible Representations ◽  
Compact Groups ◽  
Dual Spaces ◽  
Heisenberg Inequality

AbstractWe characterize dual spaces and compute hyperdimensions of irreducible representations for two classes of compact hypergroups namely conjugacy classes of compact groups and compact hypergroups constructed by joining compact and finite hypergroups. Also, studying the representation theory of finite hypergroups, we highlight some interesting differences and similarities between the representation theories of finite hypergroups and finite groups. Finally, we compute the Heisenberg inequality for compact hypergroups.

On the restriction of cuspidal representations to unipotent elements

2002 ◽  
Vol 132 (1) ◽  
pp. 35-56 ◽  
Author(s):  
DIPENDRA PRASAD ◽  
NILABH SANAT
Keyword(s):  
Conjugacy Class ◽  
Maximal Torus ◽  
Reductive Group ◽  
Conjugacy Classes ◽  
Irreducible Representations ◽  
Cuspidal Representation ◽  
Linear Character ◽  
Maximal Tori ◽  
Complex Linear ◽  
Cuspidal Representations

Let G be a connected split reductive group defined over a finite field [ ]q, and G([ ]q) the group of [ ]q-rational points of G. For each maximal torus T of G defined over [ ]q and a complex linear character θ of T([ ]q), let RGT(θ) be the generalized representation of G([ ]q) defined in [DL]. It can be seen that the conjugacy classes in the Weyl group W of G are in one-to-one correspondence with the conjugacy classes of maximal tori defined over [ ]q in G ([C1, 3·3·3]). Let c be the Coxeter conjugacy class of W, and let Tc be the corresponding maximal torus. Then by [DL] we know that πθ = (−1)nRGTc(θ) (where n is the semisimple rank of G and θ is a character in ‘general position’) is an irreducible cuspidal representation of G([ ]q). The results of this paper generalize the pattern about the dimensions of cuspidal representations of GL(n, [ ]q) as an alternating sum of the dimensions of certain irreducible representations of GL(n, [ ]q) appearing in the space of functions on the flag variety of GL(n, [ ]q) as shown in the table below.

The number of set-orbits of a solvable permutation group

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Yinling Gao ◽  
Yong Yang
Keyword(s):  
Lower Bound ◽  
Permutation Group ◽  
Solvable Group ◽  
Solvable Groups ◽  
Conjugacy Classes ◽  
Irreducible Representations ◽  
Finite Permutation Group ◽  
Explicit Bound ◽  
Power Set ◽  
Explicit Lower Bound

Abstract A permutation group 𝐺 acting on a set Ω induces a permutation action on the power set P ⁢ ( Ω ) \mathscr{P}(\Omega) (the set of all subsets of Ω). Let 𝐺 be a finite permutation group of degree 𝑛, and let s ⁢ ( G ) s(G) denote the number of orbits of 𝐺 on P ⁢ ( Ω ) \mathscr{P}(\Omega) . In this paper, we give the explicit lower bound of log 2 ⁡ s ⁢ ( G ) / log 2 ⁡ | G | \log_{2}s(G)/{\log_{2}\lvert G\rvert} over all solvable groups 𝐺. As applications, we first give an explicit bound of a result of Keller for estimating the number of conjugacy classes, and then we combine it with the McKay conjecture to estimate the number of p ′ p^{\prime} -degree irreducible representations of a solvable group.

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