An algebraic model for rational $U(2)$-spectra

The abelian category $\mathcal{A}(G)$ takes the form of a category of sheaves of modules over the space $\mathfrak{X}_{G}=\mathrm{Sub}(G)/G$ of conjugacy classes of (closed) subgroups of $G$.

It is known from [3, 16] that the category of rational $SO(3)$-spectra splits as a product of 7 blocks. Of these, 5 are 0-dimensional (dominated by $SO(3),A_{5},\Sigma_{4},A_{4},D_{4}$) and 2 are 1-dimensional (dominated by the maximal torus ($SO(2)$) and its normalizer ($O(2)$)). The word ‘dimension’ applies both to the space of subgroups and to the injective dimension of the abelian model.

The overall conclusion of the present paper is that using the quotient map $U(2)\longrightarrow PU(2)\cong SO(3)$, the category of rational $U(2)$-spectra breaks up as the corresponding product of 7 blocks. The blocks of $U(2)$-spectra are one dimension larger than those in $SO(3)$, so that 5 of them are 1-dimensional and 2 of them are 2-dimensional. Each of the 7 blocks is dominated by the inverse image of the corresponding subgroup in $SO(3)$.

This conclusion is very clean, but the precise structure of the algebraic models is affected by the details of the group theory of $U(2)$.

This paper is the fourth in a series of 5 constructing an algebraic category $\mathcal{A}(SU(3))$ and showing it gives an algebraic model for rational $SU(3)$-spectra. This series gives a concrete illustrations of general results in small and accessible examples.

The first paper [7] describes the group theoretic data that feeds into the construction of an abelian category $\mathcal{A}(G)$ for a toral group $G$ and makes it explicit for toral subgroups of rank 2 connected groups.

The second paper [8] constructs algebraic models for all relevant 1-dimensional blocks, and the results are applied in the present paper to each of the five 1-dimensional blocks. The third paper [9] constructs algebraic models for blocks of rank 2 toral groups of mixed type, and the results are applied in the present paper to cover the block dominated by the normalizer of the maximal torus.

Finally, the paper [10] constructs $\mathcal{A}(SU(3))$ in 18 blocks and shows it is equivalent to the category of rational $SU(3)$-spectra. The most complicated parts of the model for $SU(3)$ are the blocks from $U(2)$, as described in the present paper.

This series is part of a more general programme. Future installments will consider blocks with Noetherian Balmer spectra [13] and those with no cotoral inclusions [11]. An account of the general nature of the models is in preparation [12], and the author hopes that this will be the basis of the proof that the category of rational $G$-spectra has an algebraic model in general.

In Section 2 we explain how the space of subgroups can be broken into blocks, giving a decomposition of the category of $G$-spectra into a product of pieces each dominated by a particular subgroup. This is illustrated for proper subgroups of $U(2)$.

In Section 3 we describe the models for 1-dimensional blocks as given in [8]. In Section 4 we describe the partition of subgroups of $U(2)$ into 7 blocks, of which 5 are 1-dimensional. In Section 5 we describe the model for each of the five 1-dimensional blocks. To describe the 2-dimensional blocks we need a systematic nomenclature for subgroups, which is introduced in Section 6. The 2-dimensional blocks of $G$ are the ones dominated by (i) the maximal torus and (ii) its normalizer. In earlier work, we have described the models for the corresponding blocks of subgroups of $H$, and in Section 7 we describe the fusion when passing from $H$-conjugacy to $G$-conjugacy. Finally, in Section 8 we give the model for the block dominated by the maximal torus and in Section 9 we give the model for the normalizer of the maximal torus.

The ambient group throughout this paper is $G=U(2)$, although we will usually write the full name. We write $\mathbb{T}$ for the maximal torus of diagonal matrices, $\mathbb{N}=N_{U(2)}(\mathbb{T})$ for its normalizer, $Z$ for the centre of scalar matrices, $\tilde{T}$ for the diagonal maximal torus of $SU(2)$ (consisting of matices $\mathrm{diag}(\lambda,\lambda^{-1})$). We write $W$ for the Weyl group (of order 2), and $\Lambda_{0}=H_{1}(\mathbb{T})=\mathbb{Z}W$ for the toral lattice.

Since $Z\cap\tilde{T}$ is of order 2, we often need to consider central products, and if $A\subseteq Z,B\subseteq SU(2)$ we write $A\times_{2}B$ for the image of $A\times B$ in $U(2)=(Z\times SU(2))/C_{2}$ under the central quotient.

For any collection $\mathcal{V}$ of subgroups closed under conjugation we write $\mbox{$G$-{\bf spectra}}\langle\mathcal{V}\rangle$ for the category of rational $G$-spectra with geometric isotropy in $\mathcal{V}$. For various groups $G$ and collections $\mathcal{V}$ we will show that there is an abelian category $\mathcal{A}(G|\mathcal{V})$ and a Quillen equivalence

and in this case we say $\mathcal{A}(G|\mathcal{V})$ is an abelian model for $G$-spectra over $\mathcal{V}$.

The purpose of this note is to describe $\mathcal{A}(G)$ and to prove the conjecture when $G=U(2)$ and $\mathcal{V}$ consists of all subgroups.

Choosing a bi-invariant metric on $G$, we may consider the space $\mathrm{Sub}(G)$ as a metric space with the Hausdorff metric. We refer to the quotient topology on the space $\mathfrak{X}_{G}=\mathrm{Sub}(G)/G$ of conjugacy classes as the h-topology. In addition, the space $\mathfrak{X}_{G}$ also has the coarser Zariski topology, whose closed sets are h-closed sets that are also closed under passage to cotoral¹¹1$K$ is cotoral in $H$ if $K$ is normal in $H$ with $H/K$ a torus subgroups.

We write $\mathcal{F}G\subseteq\mathrm{Sub}(G)$ for the subspace of subgroups with finite Weyl group and $\Phi G\subseteq\mathrm{Sub}(G)/G$ for the corresponding space of conjugacy classes. T.tom Dieck showed that the rational mark homomorphism taking degrees of geometric fixed points induces an isomorphism

For a map $f:S^{0}\longrightarrow S^{0}$, the Borel-Hsiang-Quillen Localization Theorem says that if $K$ is cotoral in $H$ then the degree of $\Phi^{K}f$ is the same as that for $\Phi^{H}f$. Thus if $A$ is an h-clopen set of conjugacy classes closed under cotoral specialization there is an idempotent $e_{A}:S^{0}\longrightarrow S^{0}$ with support $A$.

The existence of idempotents shows that a partition

where $\mathcal{V}_{i}$ is clopen in the h-topology and closed under cotoral specilization gives rise to an equivalence

Similarly, the form of $\mathcal{A}(G|\mathcal{V})$ is that of a category of sheaves over $\mathcal{V}$, so that the decomposition of $\mathfrak{X}_{G}$ above gives

The decompositions of the category of $G$-spectra correspond to those of the algebraic model, so the model for $G$-spectra can be established by showing $\mbox{$G$-{\bf spectra}}\langle\mathcal{V}_{i}\rangle\simeq\mathcal{A}(G|\mathcal{V}_{i})$ for each $i$.

One may prove [12] that for any group $G$ there is a partition as above where each of the terms $\mathcal{V}_{i}$ has a dominant subgroup. We say that $\mathcal{V}$ has dominant subgroup $H$ if $H$ has finite Weyl group and controls $\mathcal{V}$ in the sense that there is a chosen neighbourhood $\mathcal{N}_{H}$ of $H$ in $\Phi H$ whose image $\overline{\mathcal{N}}_{H}$ in $\mathrm{Sub}(G)/G$ lies in $\Phi G$, and

is the cotoral down-closure of $\mathcal{N}_{H}$ up to $G$-conjugacy.

We note that the conjugacy class of $H$ is maximal in $\mathcal{V}$, but $H$ does not determine $\mathcal{V}$ since we also need to choose the neighbourhood $\mathcal{N}_{H}$. Nonetheless, in each case we will make a choice and write $\mathcal{V}^{G}_{H}$ for a chosen component dominated by $H$.

It is not necessary to appeal to the general result since we will give an explicit decomposition for $U(2)$. We may tabulate the dominant subgroups $H$ and their components as follows. Any element normalizing $H$ normalizes the identity component, so $H$ is determined by $H_{e}$ and the finite subgroup $H_{d}=H/H_{e}$ of $W_{G}(H_{e})$. Accordingly we will index $H$ by the pair $(H_{e},F)$ where $F$ is a finite subgroup of $W_{G}(H_{e})$. In the rest of the section we will give more details of $\mathfrak{X}_{G}$ for proper subgroups $G$ of $U(2)$, and from Section 4 onwards we give more details on $G=U(2)$ itself.

In the following table, $T$ is the maximal torus of $SO(3)$ (a circle), $\mathbb{T}$ is the maximal torus of $U(2)$ (diagonal matrices, a 2-torus) and $Z$ is the centre of $U(2)$ (the scalar matrices, a circle). The entry ‘Discrete’ means that the geometric isotropy space is discrete.

The closed subgroups of $SO(2)$ are the finite cyclic subgroups and $SO(2)$ itself, so that with the h-topology we have

the one point compactification of the discrete space $\mathcal{C}=\{C_{n}\;|\;n\geq 1\}$ of cyclic subgroups. We note that $\mathcal{C}^{\#}$ has dominant subgoup $SO(2)$, so

The closed subroups of $O(2)$ are the closed subgroups of $SO(2)$ together with a single conjugacy class of dihedral subgroups of order $2n$ for each $n\geq 1$ and $O(2)$ itself. Thus, with the h-topology we have

where $\mathcal{C}$ is as before and

We note that $\mathcal{C}^{\#}$ has dominant subgroup $SO(2)$ and $\mathcal{D}^{\#}$ has dominant subgroup $O(2)$ so we have

The conjugacy classes of subgroups of $SO(3)$ admit a partition into seven pieces

as follows. In the five cases $H\in\{SO(3),A_{5},\Sigma_{4},A_{4},D_{4}\}$ the space $\mathcal{V}^{SO(3)}_{H}=\{(H)\}$ is a singleton. If $H=SO(2)$ we have the space of cyclic subgroups

Finally, when $H=O(2)$, we have the space of dihedral subgroups

Note that this starts with $D_{6}$: in $SO(3)$ the dihedral group $D_{2}$ is conjugate to the cyclic subgroup $C_{2}$ of $SO(2)$, and $D_{4}$ has been treated as exceptional because its Weyl group is bigger than the others.

All seven sets are clopen in the Hausdorff metric topology and closed under cotoral specialization. This therefore gives a partition of the Zariski space $\mathfrak{X}_{U(2)}$ into seven Zariski clopen pieces.

From [14] we see that $G$-spectra with singleton geometric isotropy has a simple model:

where

This gives a model for $G$-spectra with geometric isotropy in $\mathcal{V}$ for any discrete space $\mathcal{V}$. This describes the model in the cases marked ‘discrete’ in the table in Subsection 2.D.

We describe the construction of the model $\mathcal{A}(G|\mathcal{V})$ in the simplest cases, typified by $G=SO(2)$ or $O(2)$. We suppose that there is a countable set $\mathcal{K}$ with $\mathcal{V}\cong\mathcal{K}^{\#}$; since we are thinking of $\mathcal{V}^{G}_{H}$ we write $H$ for the compactifying point. The set $\mathcal{K}$ is countable, and we will say that something holds ‘for almost all $k$’ if it holds for all subgroups cotoral in $H$ and for all but finitely many of the rest.

In general, to specify the model $\mathcal{A}(\mathcal{K},\mathcal{R},\mathcal{W})$ we need additional data:

• •

  a system $\mathcal{R}$ of rings (i.e., rings $\mathcal{R}(H)$, $\mathcal{R}(k)$ for $k\in\mathcal{K}$. (In general we will need ring maps $\mathcal{R}(H)\longrightarrow\mathcal{R}(k)$ for almost all $k$, but in our case $\mathcal{R}(H)=\mathbb{Q}$, so this is not additional structure)).

• •

  a component structure $\mathcal{W}$ (i.e., finite groups $\mathcal{W}_{H}$ and $\mathcal{W}_{k}$ for $k\in\mathcal{K}$ together with group homomorphisms $\mathcal{W}_{k}\longrightarrow\mathcal{W}_{H}$ for almost all $k$.)

• •

  the group $\mathcal{W}_{H}$ acts on $\mathcal{R}(H)$ and the group $\mathcal{W}_{k}$ acts on $\mathcal{R}(k)$.

In our case $\mathcal{R}(k)$ and $\mathcal{W}_{k}$ will be independent of $k$ and $\mathcal{R}(H)=\mathbb{Q}$. We simplify the notation accordingly, writing $\mathcal{A}(\mathcal{K},R,\mathcal{W})$ where $\mathcal{R}(k)=R$ for all $k$. When $\mathcal{W}$ is trivial in the sense that $\mathcal{W}_{k}=\mathcal{W}_{H}=1$, we simply write $\mathcal{A}(\mathcal{K},R)$ for the model, and if $\mathcal{W}$ is constant at $W$ we will see that $\mathcal{A}(\mathcal{K},R,\mathcal{W})=\mathcal{A}(\mathcal{K},R)[W]$

There are two subcases with slightly different characters.

For Type 1, we take $R=\mathbb{Q}[c]$ where $c$ is of degree $-2$. We then take $\mathcal{O}_{\mathcal{K}}=\prod_{k\in\mathcal{K}}\mathbb{Q}[c]$, together with the multiplicatively closed set

The standard model $\mathcal{A}(\mathcal{K},\mathbb{Q}[c],\mathcal{W})$ has objects $\beta:N\longrightarrow\mathcal{E}^{-1}\mathcal{O}_{\mathcal{K}}\otimes V$ where $N$ is a $\mathcal{O}_{\mathcal{K}}$-module and $V$ is a graded rational vector space, and the map $\beta$ is inverting the multiplicatively closed set $\mathcal{E}$. The idempotent summand $e_{k}N$ has an action of $\mathcal{W}_{k}$, the vector space $V$ has an action of $\mathcal{W}_{H}$ and the composite $e_{k}N\longrightarrow N\longrightarrow\mathcal{E}^{-1}\mathcal{O}_{\mathcal{K}}\otimes V\longrightarrow\mathbb{Q}[c,c^{-1}]\otimes V$ is $\mathcal{W}_{k}$-equivariant.

In Type 0, we take $\mathcal{A}(\mathcal{K},\mathbb{Q},W)$ to be sheaves $\mathcal{F}$ of $\mathbb{Q}$-modules over $\mathcal{K}^{\#}$. We then take $\mathcal{O}_{\mathcal{K}}=\prod_{k\in\mathcal{K}}\mathbb{Q}$, together with the multiplicatively closed set

The stalk $\mathcal{F}_{k}$ has an action of $\mathcal{W}_{k}$, the stalk $\mathcal{F}_{H}$ has an action of $\mathcal{W}_{H}$ and the horizontal spreading map $\mathcal{F}_{H}\longrightarrow\mathcal{I}^{-1}\prod_{k}\mathcal{F}_{k}$ is $\mathcal{W}_{k}$-equivariant.

The space of subgroups of $U(2)$ mapping to $H$ isolated subgroups has a simple form.

For Parts (ii) and (iii), we suppose first that $H=SO(3)$, so that $p^{-1}(H)=U(2)$. If $K$ is of rank 2 then $K=U(2)$. Otherwise $K$ is of rank 1, and since it maps on to $SO(3)$ it is not a circle. The identity component of $K$ cannot be $SO(3)$ since $SO(3)$ does not have a faithful 2 dimensional representation, so $K_{e}\cong SU(2)$, and there is a single conjugacy class of subgroups of this type so we may suppose $K_{e}=SU(2)$, and by Part (i) $K=SU(2)\times_{C_{2}}C_{2s}$. Part (iii) is the same as Part (ii) when $H=SO(3)$.

Now suppose $H$ is finite. Thus we suppose $K$ is a full subgroup of the toral group $p^{-1}(H)$.

(ii) Since $H^{2}(H;Z)=H^{3}(H;\mathbb{Z})$ is of order 2, there is a subgroup $K$ with $C_{n}=K\cap Z$ if and only if $n$ is even. By [7, Lemma 3.3], the $p^{-1}H$-conjugacy classes of such subgroups are in bijection to $H^{1}(H;Z/C_{s})\cong\mathrm{Hom}(H,T)=\mathrm{Hom}(H^{ab},T)$, and hence in bijection to the abelianization $H^{ab}$. This gives the numbers $a(H)$.

(iii) Since $Z$ is the centre of $U(2)$, any further $U(2)$ conjugacy comes through $PU(2)=SO(3)$. The normalizers of the subgroups $H$ are well known and recorded in Lemma 5.3 below. They are non-trivial only for $A_{4}$ and $D_{4}$, and in both cases they act transitively on the non-canonical conjugacy classes. This gives the numbers $b(H)$. ∎

For the four finite totally isolated subgroups $H$ of $SO(3)$, there is an off the shelf model $\mathcal{A}(\tilde{H}\times_{C_{2}}Z\;|\;\mathrm{full})$ for full subgroups of the toral group $\tilde{H}\times_{C_{2}}Z$ [8]. This can then be adapted to the block $\mathcal{V}^{U(2)}_{H}$ of $U(2)$ by taking account of fusion. As described in Section 3, off-the-shelf models for a space of the form $\mathcal{K}^{\#}$ uses additional structure coming from Weyl groups.

The group theory is well known.

Since the conjugacy classes of $H$ are totally isolated we have the model

for $SO(3)$-spectra with geometric isotropy $H$.

Inflating to $U(2)$, since $H$ is finite the group $p^{-1}(H)=\tilde{H}\times_{C_{2}}Z$ is 1-dimensional, so we have the off-the-shelf model of the form $\mathcal{A}(\mathcal{K},R,W)$ [8].

The very easiest case is when $H=SO(3)$. The space $\mathcal{V}^{U(2)}_{SO(3)}$ is 1-dimensional with a single generic point, we could argue that the methods of [8] apply (despite the fact that the height 0 subgroups are not finite). The set of height 0 subgroups is

The compactifying point is $U(2)$ itself. The Weyl groups of subgroups in $\mathcal{K}$ are all copies of the circle, and the Weyl group of $U(2)$ is trivial; all of these are connected so the component structure is trivial. This is a classic Type 1 example.

However it is much simpler to point out that $\mathcal{V}^{U(2)}_{SO(3)}$ consists of the subgroups containing the normal subgroup $SU(2)$. We may then use the traditional passage to geometric fixed points.

The required model then follows from [15].

We have already dealt with $H=SO(3)$, but since the statements in this section apply to it we will include that in the statements, just giving the proofs for the remaining four toral cases.

To prepare for the statement about models we describe the analogous situation for a finite group $G$. If we have $K\subseteq H\subseteq G$ and if all $G$-conjugates of $K$ lies inside $H$ then the $G$-conjugacy class of $K$ breaks up into a number of different $H$-conjugacy classes

where $K_{i}=K^{g_{i}}$. Now Given a representation $V$ of $W_{G}(K)$, we may view it as a $G$-equivariant bundle $\mathcal{V}$ over $(K)_{G}$ fixed by $K$. Over $K^{g}$ we have the representation $(g^{-1})^{*}V$ of $K^{g}$ fixed by $K^{g}$. Thus $V$ restricts to $((g_{i}^{-1})^{*}V)_{i}$. It is then natural to factorize restriction from $G$ to $H$ as follows (where the superscript indicates that in the fibre over $K$, the group $K$ fixes the bundle pointwise (the bundle is ‘Weyl’, in the sense of Barnes)).

With this preparation, the statement in $U(2)$ should seem natural.