Real McKay Correspondence KR-Theory of Graded Kleinian Groups by

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Real McKay Correspondence: KR-Theory of Graded

Kleinian Groups

Jon Cheah

supervised by

Dmitriy Rumynin

MA4K9 Dissertation

Submitted to The University of Warwick

Mathematics Institute

April, 2022

arXiv:2210.03924v2 [math.RT] 28 May 2023

Contents

Contents ii

1 Introduction 1

2 Finite Subgroups 2

2.1 Finite Subgroups of O(3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2.2 Finite Subgroups of SU(2,C) .......................... 5

2.3 SpinorsandPinors................................ 9

3 K-Theory 12

3.1 VectorBundles .................................. 12

3.2 EquivariantK-Theory .............................. 13

3.3 AlgebraicK-theory................................ 14

4 McKay Correspondence as an Equivalence of K-Theories 15

5 Calculating Real Frobenius-Schur Indicators 19

5.1 ThePolyhedralGroups ............................. 20

5.2 TheAxialGroups ................................ 27

6 KR-Theory 30

6.1 Topological KR-Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

6.2 AlgebraicKR-Theory .............................. 31

7 McKay Correspondence on Real spaces 32

8 Bibliography 33

1 Introduction

This project considers the ﬁnite symmetry subgroups of the orthogonal group O(3) ⊂

GL(3,R) and how these can be embedded into one another. Of particular interest are

the index 2 containments, which were classiﬁed by Conway and Smith in [9]. The special

orthogonal group SO(3) ⊂SL(3,R) admits a double cover from the spinor group Spin(3) ∼

SU(2) ⊂SL(2,C), and lifting our subgroups up preserves our network of containments.

Those subgroups not contained in SO(3) are lifted to the pinor groups Pin±(3) of which

there are two choices. We then explore KR-theory as introduced by Atiyah [2] in 1966,

which is a variant of topological K-theory when dealing with a topological space equipped

with an involution. In the case of the index 2 containments G◁b

G, the quotient spaces

C2// G, can be equipped by an involution via the action of b

G/G. In 1983, Gonzalez-

Sprinberg and Verdier [15] showed how one can view the McKay correspondence from

[24] as an isomorphism between the G-equivariant K-theory KG(C2) and the K-theory of

C2//G, the minimal resolution of the singularity.

In section 2, we go over the ﬁnite subgroups of O(3) including explicit matrix generators

and construct a graph of the index 2 containments. These are then lifted by their double

covers to their respective spinor or pinor groups. This lifting preserves their network of

containments yielding an analogous graph. We also produce MAGMA code for each of

the subgroups of O(3) and Pin±(3) which can be found in [8].

In section 3, we take a brief diversion into topological K-theory, mainly citing [20] and [3].

We follow Atiyah and Segal [4] in constructing G-equivariant K-theory for use in section

4. In section 4, we give a brief expository overview of the McKay correspondence observed

by McKay in [24] and the statement in terms of K-theory from Gonzalez-Sprigberg and

Verdier [15], as well as an example in the explicit case of the binary dihedral group BD16.

In section 5, we use the previously constructed index 2 containments as C2graded sub-

groups, and calculate the Real and complex Frobenius-Schur indicators. Applying Dyson’s

classiﬁcation of antilinear block structures (see [28], [11], or [14]), we produce decorated

McKay graphs for each of the containments. Our MAGMA code for the calculation of the

indicators can also be found in [8].

In section 6, we build KR-theory as was introduced in [2] and apply it to the case of our

Kleinian singularities and C2-graded groups. This allows us to state the ﬁnal conjecture

in section 7, a form of the McKay Correspondence for KR-theory in the case of C2-graded

groups.

We use the following notation throughout. Finite groups G◁b

Ggive an index 2 con-

tainment. The symbol is used to represent the identity transformation, or the identity

matrix of appropriate dimension. The group C2is multiplicative and might be written as

{1,−1},{,− }, or {1,x}.

2 Finite Subgroups

2.1 Finite Subgroups of O(3)

We begin with a classical result.

Theorem 2.1. Every ﬁnite subgroup of GL(3,R)is conjugate to a ﬁnite subgroup of O(3).

Proof. Let G⊂GL(3,R) be ﬁnite, and ⟨·,·⟩ be the usual inner product on R3. We

construct ⟨·,·⟩G:R3×R3→Rby,

⟨u, v⟩G:= 1

|G|X

g∈G

⟨g·u, g ·v⟩.

This is well deﬁned as Gis ﬁnite, and we note that ⟨·,·⟩Ginherits symmetry and bilinearity

from ⟨·,·⟩. Furthermore, for any u∈R3\ {0}

⟨u, u⟩G=1

|G|X

g∈G

⟨g·u, g ·u⟩>0

and for any h∈G,

⟨h·u, h ·v⟩G=1

|G|X

g∈G

⟨gh ·u, gh ·v⟩=1

|G|X

gh∈G

⟨gh ·u, h ·v⟩=⟨u, v⟩G,

so ⟨·,·⟩Gis positive deﬁnite and G-invariant. Thus up to conjugation (or a change of

coordinates in R3), Gis a subgroup of O(3).

The ﬁnite subgroups of O(3) are well studied and can be found in [9] or [12]. Up to

conjugacy, there are 14 ﬁnite subgroups of O(3), and we list these in Table 1. These are

be split into 7 inﬁnite families which each leave an axis of R3invariant, and 7 sporadic

groups which only leave the origin invariant. The latter manifest themselves as symmetry

groups of polyhedra centred on the origin, whereas the former are the symmetry groups

of various prisms/antiprisms [9]. Following the work of these authors, we have reproduced

these groups in Magma code with explicit generators.

The groups Cn, D2n, T12, O24, I60 have all of their elements with determinant equal to 1,

and are precisely the orientation preserving groups contained in the special orthogonal

group SO(3).

To use Conway and Smith’s terminology, the four groups CC2n,CD4n,DD4nand T O24

are called hybrid groups. Given an index 2 containment of groups H◁G⊂O(3), we let

the set HG consist of all h∈H, and −gfor all g∈G\H, and make this into a group by

the standard composition of orthogonal transformations.

Axial groups Polyhedral groups

Cyclic Cn=⟨A⟩Tetrahedral T12 =I2, Y 

Diplo-cyclic 2C2n=⟨A, − ⟩ Diplo-tetrahedral 2T24 =I2, Y, −

Cyclo-cyclic CC2n=⟨−A′⟩Octahedral O24 =⟨I, Y ⟩

Dihedral D2n=⟨A, B⟩Tetra-octahedral T O24 =⟨−I, Y ⟩

Cyclo-dihedral CD2n=⟨A, −B⟩Diplo-octahedral 2O48 =⟨I, Y, − ⟩

Dihedro-dihedral DD4n=⟨−A′, B⟩Icosahedral I60 =⟨X, Z⟩

Diplo-Dihedral 2D8n=⟨A, B, − ⟩ Diplo-icosahedral 2I120 =⟨X, Z, − ⟩

Table 1: The ﬁnite subgroups of O(3) and their matrix generators. The axial groups

are generated by rotation matrices A=cos 2π

n−sin 2π

sin 2π

ncos 2π

0 0 1 ,A′=cos π

n−sin π

sin π

2ncos π

0 0 1 , and

B=−1 0 0

0 1 0

0 0 −1. The polyhedral groups are generated by rotation matrices Y=001

100

010,

I=1 0 0

0 0 1

0−1 0 ,X=1

√5−√500

0−1 2

0 2 1 , and Z=cos 2π

5−sin 2π

sin 2π

5cos 2π

0 0 1 .

The groups with labels of the form 2Gare the direct sums of a group G⊂SO(3) and

the inversion group {± } where is the identity transformation. The inversion −is

central in O(3). Conway and Smith call these diploid groups and denote these by ±G.

We avoid this for the sake of cleaner notation when lift up to their covering groups. The

diplo- groups 2Gshould not be confused with the binary groups BG to be introduced in

the next section.

The index 2 containments are given by Conway and Smith in [9], and we recreate their

graph of containments in Figures 1 and 2.

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RealMcKayCorrespondence:KR-TheoryofGradedKleinianGroupsbyJonCheahsupervisedbyDmitriyRumyninMA4K9DissertationSubmittedtoTheUniversityofWarwickMathematicsInstituteApril,2022ContentsContentsii1Introduction12FiniteSubgroups22.1FiniteSubgroupsofO(3)............................22.2FiniteSubgroupsofSU(2,C)...

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