Searcharxiv⌕ Search

arXiv subjects

Chi-Kwong Fok

Publications and source records attributed to Chi-Kwong Fok.

13 recordsLinked to original sources
arXiv2026

Equivariant formality and representation theory

Let $G$ be a compact connected Lie group and $K$ its connected Lie subgroup. Using the $K$-theoretic version of equivariant formality developed in [F2] which involves analysis of vector bundles over $G/K$, we find two characterizations of equivariant formality of the isotropy action of $K$ on $G/K$. The first one equates equivariant formality with a "smoothness" condition of the restriction map of the representation ring of $G$ to that of $K$. The second characterization asserts that equivariant formality amounts to the equivariant $K$-theoretic index pairing of two certain special $K$-theory classes being nontrivial in some sense. By applying these characterizations, we are able to give a representation theoretic criterion for equivariant formality of the isotropy action by a circle subgroup, as well as an invariant theory criterion for the case where $K$ is a torus of dimension one less than the rank of $G$, culminating in a complete classification of equivariant formality where $G$ is further assumed to be simple.

math.AT↗
arXiv2024

Cohomology and $K$-theory rings of the space of commuting elements in $SU(2)$

In this paper, we compute explicitly both the $K$-theory and integral cohomology rings of the space of commuting elements in $SU(2)$ via the $K$-theory of its desingularization. We also briefly discuss the different behavior of its cohomology with complex and $\mathbb{Z}_2$ coefficients in the context of representation stability and FI-modules.

math.AT↗
arXiv2024

Equivariant formality of istropy actions

Let $G$ be a compact connected Lie group and $K$ a connected Lie subgroup. In this paper, we collect an assortment of results on equivariant formality of the isotropy action of $K$ on $G/K$. If the isotropy action of $K$ on $G/K$ is equivariantly formal, then $G/K$ is formal in the sense of rational homotopy theory. This enables us to strengthen a theorem of Shiga--Takahashi to a characterization of equivariant formality in this case. Using a K-theoretic analogue of equivariant formality introduced and shown by the second-named author to be equivalent to equivariant formality in the usual sense, we provide a representation-theoretic characterization for equivariant formality of the isotropy action and give a new, uniform proof of equivariant formality for some classes of homogeneous spaces for which it was previously known.

math.AT↗
arXiv2023

A stroll in equivariant $K$-theory

Equivariant $K$-theory is a generalized equivariant cohomology theory which is a hybrid of the $K$-theory of a topological space and the representation theory of the group acting on it. In this article, we review the basics of equivariant $K$-theory and focus on the localization theorem and formula due to Atiyah and Segal, which have become important tools in equivariant topology nowadays. We then discuss the application of equivariant $K$-theory to equivariant formality, and briefly mention some recent developments.

math.KT↗
arXiv2021

Quantum multiplication through equivariant Schubert calculus

In this note, we rederive quantum Pieri's formula and the rim hook algorithm in quantum Schubert calculus by studying multiplication in the equivariant cohomology ring of Grassmannians with respect to equivariant Schubert classes which are characteristic classes. We also extend this idea in studying equivariant quantum Schubert calculus, and obtain the equivariant quantum Giambelli's and Pieri's formulae in terms of characteristic classes, with the former formula shown to be free of quantum deformation.

math.AT↗
arXiv2021

The ring structure of twisted equivariant $KK$-theory for noncompact Lie groups

Let $G$ be a connected semisimple Lie group with its maximal compact subgroup $K$ being simply-connected. We show that the twisted equivariant $KK$-theory $KK^{\bullet}_{G}(G/K, τ_G^G)$ of $G$ has a ring structure induced from the renowned ring structure of the twisted equivariant $K$-theory $K^{\bullet}_{K}(K, τ_K^K)$ of a maximal compact subgroup $K$. We give a geometric description of representatives in $KK^{\bullet}_{G}(G/K, τ_G^G)$ in terms of equivalence classes of certain equivariant correspondences and obtain an optimal set of generators of this ring. We also establish various properties of this ring under some additional hypotheses on $G$ and give an application to the quantization of $q$-Hamiltonian $G$-spaces in an appendix. We also suggest conjectures regarding the relation to positive energy representations of $LG$ that are induced from certain unitary representations of $G$ in the noncompact case.

math.KT↗
arXiv2019

Equivariant formality in $K$-theory

In this note we present an analogue of equivariant formality in $K$-theory and show that it is equivalent to equivariant formality \emph{à la} Goresky-Kottwitz-MacPherson. We also apply this analogue to give alternative proofs of equivariant formality of conjugation action on compact Lie groups, left translation action on generalized flag manifolds, and compact Lie group actions with maximal rank isotropy subgroups.

math.AT↗
arXiv2017

Equivariant twisted Real $K$-theory of compact Lie groups

Let $G$ be a compact, connected, and simply-connected Lie group viewed as a $G$-space via the conjugation action. The Freed-Hopkins-Teleman Theorem (FHT) asserts a canonical link between the equivariant twisted $K$-homology of $G$ and its Verlinde algebra. In this paper we give a generalization of FHT in the presence of a Real structure of $G$. Along the way we develop preliminary materials necessary for this generalization, which are of independent interest in their own right. These include the definitions of Real Dixmier-Douady bundles, the Real third cohomology group which is shown to classify the former, and Real $\text{Spin}^c$ structures.

math.KT↗
arXiv2016

Adams operations on classical compact Lie groups

Let $G$ be $U(n)$, $SU(n)$, $Sp(n)$ or $Spin(n)$. In this short note we give explicit general formulas for Adams operations on $K^*(G)$, and eigenvectors of Adams operations on $K^*(U(n))$.

math.AT↗
arXiv2015

Picard group of isotropic realizations of twisted Poisson manifolds

Let $B$ be a twisted Poisson manifold with a fixed tropical affine structure given by a period bundle $P$. In this paper, we study the classification of almost symplectically complete isotropic realizations (ASCIRs) over $B$ in the spirit of \cite{DD}. We construct a product among ASCIRs in analogy with tensor product of line bundles, thereby introducing the notion of the Picard group of $B$. We give descriptions of the Picard group in terms of exact sequences involving certain sheaf cohomology groups, and find that the `Néron-Severi group' is isomorphic to $H^2(B, \underline{P})$. An example of an ASCIR over a certain open subset of a compact Lie group is discussed.

math.SG↗
arXiv2015

$KR$-theory of compact Lie groups with group anti-involutions

Let $G$ be a compact, connected, and simply-connected Lie group, equipped with an anti-involution $a_G$ which is the composition of a Lie group involutive automorphism $σ_G$ and the group inversion. We view $(G, a_G)$ as a Real $(G, σ_G)$-space via the conjugation action. In this note, we exploit the notion of Real equivariant formality discussed in \cite{Fo} to compute the ring structure of the equivariant $KR$-theory of $G$. In particular, we show that when $G$ does not have Real representations of complex type, the equivariant $KR$-theory is the ring of Grothendieck differentials of the coefficient ring of equivariant $KR$-theory over the coefficient ring of ordinary $KR$-theory, thereby generalizing a result of Brylinski-Zhang's (\cite{BZ}) for the complex $K$-theory case.

math.KT↗
arXiv2014

The Real K-Theory of Compact Lie Groups

Let $G$ be a compact, connected, and simply-connected Lie group, equipped with a Lie group involution $σ_G$ and viewed as a $G$-space with the conjugation action. In this paper, we present a description of the ring structure of the (equivariant) $KR$-theory of $(G, σ_G)$ by drawing on previous results on the module structure of the $KR$-theory and the ring structure of the equivariant $K$-theory.

math.KT↗