Uniqueness of factorization for fusion-invariant representations

The $K$-theory of the classifying space of a finite group $G$ is determined by the Atiyah-Segal completion theorem [2], [3] which established it to be the completion of the representation ring of $G$ with respect to the augmentation ideal. This is generally different from the Grothendieck group of complex vector bundles over $BG$, which is described in [18].

These results were generalized recently to classifying spaces of $p$-local finite groups. Given a $p$-local finite group $(S,{\mathcal{F}},{\mathcal{L}})$, the Grothendieck group of complex vector bundles over $|{\mathcal{L}}|^{\wedge}_{p}$ was found in [8] to be isomorphic to the representation ring $R({\mathcal{F}})$, which can expressed as an inverse limit of representation rings over the orbit category of ${\mathcal{F}}$. On the other hand, the $K$-theory of $|{\mathcal{L}}|^{\wedge}_{p}$ is isomorphic to the completion of $R({\mathcal{F}})$ with respect to the augmentation ideal (see [4]).

The representation ring $R({\mathcal{F}})$ can also be regarded as the Grothendieck group associated to the monoid of equivalence classes of representations of $S$ which are ${\mathcal{F}}$–invariant. For instance, when ${\mathcal{F}}$ is the fusion system associated to a finite group $G$ with $p$-Sylow subgroup $S$, they are precisely the representations of $S$ whose characters are constant on $G$–conjugacy classes of elements of $S$. We begin the paper by giving a brief review of fusion-invariant representations and their main properties in Section 1. Even though we prove our results in the full generality of saturated fusion systems, our main examples come from finite groups, hence this paper should be of interest to people who are only interested in the representation theory of finite groups.

The group $R({\mathcal{F}})$ is free abelian and its rank is the number of ${\mathcal{F}}$–conjugacy classes of elements of $S$. However, there are examples in [12] and [23] where an ${\mathcal{F}}$–invariant representation does not decompose as sum of irreducible ${\mathcal{F}}$–invariant representations in a unique way. This represents a significant departure from the behaviour of the representation ring for finite groups, and this article aims to provide a better understanding of this phenomenon.

This lack of uniqueness is better understood from the point of view of the monoid $\operatorname{Rep}\nolimits({\mathcal{F}})$ of isomorphism classes of ${\mathcal{F}}$–invariant representations. Using the language of reduced commutative monoids, this is a submonoid of the free monoid $\operatorname{Rep}\nolimits(S)$, and these examples show that it is not necessarily free. Free monoids are also called factorial to stress the uniqueness of factorization as a sum of atoms. We find several criteria in Section 2 that guarantee the uniqueness of factorization, some of them characterizations in fact. The following theorem outlines the main results in this section.

The essential rank of any saturated fusion system over an abelian $p$-group $S$ is zero. In particular, factoriality is guaranteed when the order of $S$ is at most $p^{2}$. It is natural to wonder if this holds when the order is $p^{3}$. Saturated fusion systems over groups of order eight are few and easy to handle, and over groups of order $p^{3}$ with $p$ odd, there is a classification in [24]. Using a case-by-case analysis and the criteria from Section 2, we determine the factoriality of $\operatorname{Rep}\nolimits({\mathcal{F}})$ for all these possibilities. The following theorem corresponds to Theorem 3.2, which is the main result in Section 3.

This result can not be taken further, since the examples without uniqueness of factorization in [12] and [23] correspond to fusion systems over groups of order $3^{4}$ and $2^{4}$, respectively. They are in fact the fusion systems of $\Sigma_{9}$ and $PSL_{3}({\mathbb{F}}_{3})$ at the primes $3$ and $2$. One may wonder if weaker factorization properties hold in $\operatorname{Rep}\nolimits({\mathcal{F}})$ in general. One such property is half-factoriality, where we do not ask for uniqueness of factorization, but only for uniqueness of the length of factorization of each element. In Section 4, we show that $\operatorname{Rep}\nolimits({\mathcal{F}})$ is not half-factorial when ${\mathcal{F}}$ is the fusion system of $A_{6}$ at the prime $2$.

Even though the example of $A_{6}$ is completely described mathematically, we found this example thanks to algorithms that we programmed and implemented in GAP [11]. Namely, for a fusion system ${\mathcal{F}}$ associated to a finite group $G$ with $p$-Sylow subgroup $S$, we built functions which can determine the number of ${\mathcal{F}}$–conjugacy classes, the complexification of the representation ring, and the number of irreducible ${\mathcal{F}}$–invariant representations which are subrepresentations of the regular representations. In some cases, we noticed that these pieces of information can be used to determine all the irreducible ${\mathcal{F}}$–invariant representations. Apart from the examples from [12] and [23], we found new examples of fusion systems without uniqueness of factorization into irreducible fusion-invariant representations. Namely, they are the fusion systems of the groups:

• •

  $\Sigma_{6}$, $\Sigma_{8}$, $PSL_{2}({\mathbb{F}}_{17})$, $PSU_{3}({\mathbb{F}}_{5})$, $M_{10}$, $PSL_{3}({\mathbb{F}}_{5})$, $PSL_{2}({\mathbb{F}}_{31})$, $PSU_{3}({\mathbb{F}}_{9})$ and $GL_{3}({\mathbb{F}}_{3})$ at the prime $2$.

• •

  $A_{9}$ and $PSL_{3}({\mathbb{F}}_{19})$ at the prime $3$.

Apart from the non-factorial examples above, we found that factoriality holds for $A_{8}$ at the prime $2$ and for $PSp_{4}({\mathbb{F}}_{3})$ and $PSL_{4}({\mathbb{F}}_{7})$ at the prime $3$. Even though we have characterizations of the factoriality of $\operatorname{Rep}\nolimits({\mathcal{F}})$, they are not completely satisfactory because they require determining the number of irreducible ${\mathcal{F}}$–invariant representations or studying the inclusion of $R({\mathcal{F}})$ into $R(S)$ to find a particular basis, but we do not have a direct criterion in terms of ${\mathcal{F}}$.

There is another interesting aspect of the irreducible ${\mathcal{F}}$–invariant representations which we could not settle completely in this paper. In Section 2 we show that if $\operatorname{Rep}\nolimits({\mathcal{F}})$ is factorial, these representations must be subrepresentations of the regular representation of $S$. We conjecture that this is always the case, even when $\operatorname{Rep}\nolimits({\mathcal{F}})$ is not factorial, and indeed it holds for all the examples where we managed to determine all the irreducible $G$–invariant representations.

In this section we provide an introduction to fusion-invariant representations. Note that all the representations in this paper are finite-dimensional unitary representations and we are only interested in representations up to equivalence. We begin with a particular case which covers most of the examples found in this paper. However, the main results apply to the general definition.

The category of subgroups of $S$ with $G$–subconjugations as morphisms appears as the indexing category in the Theorem of Cartan-Eilenberg on the $p$-local cohomology of finite groups (see for instance Theorem III.10.3 in [6]). This theorem can be improved to a homotopical decomposition of $BG^{\wedge}_{p}$ and C. Broto, R. Levi and B. Oliver [5] created a suitable language and techniques to study these decompositions and other spaces with similar properties. In this paper the authors define the notion of saturated fusion system over a finite $p$-group $S$. This is a category whose objects are the subgroups of $S$ and whose morphisms are group homomorphisms which are required to satisfy some axioms.

The interested reader can see Definitions 1.1 and 1.2 in [5] for the formal definition, but the intuitive idea is that these morphisms behave like conjugations by elements of a bigger group. In fact, the main source of examples of saturated fusion systems comes from finite groups. If $G$ is a finite group and $S$ is a $p$-Sylow subgroup of $G$, there is a saturated fusion system called ${\mathcal{F}}_{S}(G)$ whose morphism sets are given by

But there are also exotic fusion systems, meaning that they are not of this form. At this moment, there are many constructions of exotic fusion systems in the literature, although we will not encounter them in this paper. We direct the interested reader to [19] and [24] for some examples. Next we extend Definition 1.1 to this general setting.

In the particular case of ${\mathcal{F}}_{S}(G)$, this corresponds to $G$–invariant representations. When ${\mathcal{F}}$ is clear, we also call them fusion-invariant representations. These representations have been studied in [4], [8], [12] and [23], but also in [7], [22], [27] and [28] in a more general context.

In order to determine whether a representation is fusion-invariant, we can use character theory and certain families of subgroups which generate the fusion system in a certain sense. For the following lemma, we say that two elements $x$, $y$ of $S$ are ${\mathcal{F}}$–conjugate if there exists a morphism $f$ in ${\mathcal{F}}$ such that $f(x)=y$. The definitions of centric, radical and essential subgroups can be found in Definition 1.6 of [5], Definition A.9 of [5] and in the second paragraph of page 2 in [16], respectively.

In the case of ${\mathcal{F}}_{S}(G)$, the last condition states that a representation $\rho$ is $G$–invariant if and only $\chi_{\rho}(x)=\chi_{\rho}(y)$ for any pair of $G$–conjugate elements of $S$. We now enumerate some immediate properties of ${\mathcal{F}}$–invariant representations.

• •

  If $\alpha$ is equivalent to $\beta$ and $\alpha$ is ${\mathcal{F}}$–invariant, so is $\beta$.

• •

  If $\rho$ and $\alpha$ are ${\mathcal{F}}$–invariant, so are $\rho\oplus\alpha$ and $\rho\otimes\alpha$.

• •

  The trivial representation and the reduced regular representation are ${\mathcal{F}}$–invariant for any ${\mathcal{F}}$.

• •

  If $\rho$ and $\rho\oplus\alpha$ are ${\mathcal{F}}$–invariant, so is $\alpha$.

• •

  If $n\rho$ is ${\mathcal{F}}$–invariant for some $n\neq 0$, so is $\rho$.

• •

  If $\rho\oplus\alpha$ is isomorphic to $\rho\oplus\beta$, then $\alpha$ is isomorphic to $\beta$.

It should be clear that any ${\mathcal{F}}$–invariant representation can be decomposed as a direct sum of irreducible ${\mathcal{F}}$–invariant representations. Note that an irreducible ${\mathcal{F}}$–invariant representation is not necessarily irreducible if we regard it as a representation of $S$. The following example illustrates this.

The isomorphism classes of ${\mathcal{F}}$–invariant representations of $S$ form a semiring with the operations of direct sum and tensor product. We denote it by $\operatorname{Rep}\nolimits({\mathcal{F}})$ and when ${\mathcal{F}}={\mathcal{F}}_{S}(G)$, we denote it by $\operatorname{Rep}\nolimits_{G}(S)$ instead. By definition, this is a subsemiring of the semiring $\operatorname{Rep}\nolimits(S)$. For the purpose of studying the uniqueness of decomposition as a sum of irreducibles, we can ignore their multiplicative structures and regard them simply as commutative monoids.

The theory of ${\mathcal{F}}$–invariant representations has some similarities with the theory of supercharacters and superclasses [10]. For instance, ${\mathcal{F}}$ determines a partition of $S$ given by ${\mathcal{F}}$–conjugacy classes, the set $\{1\}$ is one of the classes and characters of ${\mathcal{F}}$–invariant representations are constant on each of these $G$–conjugacy classes. We can appreciate the difference in the case of ${\mathcal{F}}_{D_{8}}(\Sigma_{4})$. The group $D_{8}$ is partitioned into four $\Sigma_{4}$–conjugacy classes and Example 4.5.4 in [12] shows that any representation of $D_{8}$ which is $\Sigma_{4}$–invariant is generated by $1$, $X+Z$, $Y+Z$ and $XY$, where $X$, $Y$ and $XY$ are the one-dimensional irreducible representations of $D_{8}$ and $Z$ is the two-dimensional irreducible representation. If this partition of $D_{8}$ were a part of a supercharacter theory, there would exist a supercharacter which would contain $2Z$ as a subrepresentation. But then the partition of the set of irreducible representations of $D_{8}$ would have less than four elements.

On the other hand, it is easier to produce a supercharacter theory over a finite $p$-group $S$ which does not come from a saturated fusion system over $S$. The two-part partition of $S$ where all the nontrivial elements are in the same class always corresponds to a supercharacter theory. But if $S$ is a $p$-group whose nontrivial elements have at least two different orders, the subsets forming this partition can not be ${\mathcal{F}}$–conjugacy classes.

In this section we recall some terminology and results from the theory of reduced commutative monoids in order to study uniqueness of factorization in the monoids of fusion-invariant representations introduced in the previous section. We follow the terminology from [13]. Since our monoids are commutative, we will denote by $0$ the identity element. In the rest of the paper, ${\mathcal{F}}$ will denote a saturated fusion system over a $p$-group $S$.

The commutative monoid $\operatorname{Rep}\nolimits({\mathcal{F}})$ is reduced because it is a submonoid of $\operatorname{Rep}\nolimits(S)$, which is reduced.

The monoid $\operatorname{Rep}\nolimits({\mathcal{F}})$ is atomic and its atoms are the irreducible ${\mathcal{F}}$–invariant representations of $S$.

This is also called a unique factorization monoid. It is easy to see that a reduced commutative monoid is factorial if and only if it is free, that is, isomorphic to a monoid of the form

for some set $X$. For a finite group $G$, the monoid $\operatorname{Rep}\nolimits(G)$ is factorial because any representation of $G$ can be decomposed as a direct sum of irreducible representations in a unique way up to reordering. The atoms of $\operatorname{Rep}\nolimits_{\Sigma_{3}}(S)$ are the representations $\rho_{0}$ and $\rho_{1}+\rho_{2}$ described in Example 1.5 and this monoid is factorial since

implies $m=m^{\prime}$ and $n=n^{\prime}$.

In order to describe a non-factorial example, we introduce the Grothendieck group of $\operatorname{Rep}\nolimits({\mathcal{F}})$, which is denoted by $R({\mathcal{F}})$ and called the representation ring of ${\mathcal{F}}$. We also use the notation $R_{G}(S)$ when ${\mathcal{F}}={\mathcal{F}}_{S}(G)$. In general, $R({\mathcal{F}})$ is a free abelian group whose rank is the number of ${\mathcal{F}}$–conjugacy classes of elements of $S$ (see Corollary 2.2 in [4]). However, the monoid $\operatorname{Rep}\nolimits({\mathcal{F}})$ is not free in general as the following example from [12] shows.

Another known example of a non-factorial monoid of fusion-invariant representations, shown in Example A.2 of [23], is $\operatorname{Rep}\nolimits_{PGL_{3}({\mathbb{F}}_{3})}(SD_{16})$.

The following result is well known in the theory of non-unique factorizations, but we include the proof of this particular case for completeness.

In particular, the proof shows that if $M$ is an atomic monoid such that $K(M)$ is free abelian of finite rank, the number of atoms in these atomic monoids is greater or equal than the rank of $K(M)$.

It is illustrative to see an example where these numbers differ.

Given a representation $\rho$ of $S$, let us denote by $\operatorname{Irr}\nolimits(\rho)$ the set of its irreducible subrepresentations.

We say that two virtual representations of $S$ are disjoint if their decompositions as integral linear combinations of irreducible representations of $S$ do not share any irreducible representation. More generally, given a finite number of virtual representations of $S$, we say that they are disjoint if any two of them are disjoint.

Proposition 2.13 is not a characterization of the factoriality of $\operatorname{Rep}\nolimits({\mathcal{F}})$ as the following example shows.

However, Proposition 2.13 is useful in two extreme cases. The following terminology is taken from the Introduction of [17].

This second extreme case corresponds to the case when every morphism is a restriction of an automorphism of $S$.

If $S$ is abelian, the essential rank of ${\mathcal{F}}$ is automatically zero and $p$-groups of order at most $p^{2}$ are abelian, so it is natural to next treat the case when $|S|=p^{3}$. We will do this in the next section.

The argument in the previous remark shows that any irreducible${\mathcal{F}}$–invariant representation is a subrepresentation of the regular representation when $\operatorname{Rep}\nolimits({\mathcal{F}})$ is factorial. Apart from the examples from [12] and [23], we enumerate in the appendix several examples of saturated fusion systems where $\operatorname{Rep}\nolimits({\mathcal{F}})$ is not factorial, found using the software GAP. We have checked that many of them still satisfy that any irreducible ${\mathcal{F}}$–invariant representation is a subrepresentation of the regular representation (we did not include these details in the paper). Hence it is reasonable to pose the following conjecture.

In this section we determine that the monoids of fusion-invariant representations for saturated fusion systems over $p$-groups of order $p^{3}$ are all factorial.

We begin with $p=2$. Let $S$ be a group of order eight and ${\mathcal{F}}$ a saturated fusion system over $S$. If $S$ is abelian, then the essential rank of ${\mathcal{F}}$ is zero, hence $\operatorname{Rep}\nolimits({\mathcal{F}})$ is factorial by Lemma 2.17. It is well known that, up to isomorphism, there are three saturated fusion systems over $D_{8}$, the fusion systems of $D_{8}$, $S_{4}$ and $A_{6}$, and two over $Q_{8}$, corresponding to the groups $Q_{8}$ and $SL_{2}({\mathbb{F}}_{3})$. For both $D_{8}$ and $Q_{8}$, let us denote by $X$, $Y$ and $XY$ the one-dimensional irreducible representations and by $Z$ the two-dimensional irreducible representation. The monoid of fusion-invariant representations is factorial in all these cases:

• •

  When the group is $D_{8}$ or $Q_{8}$, because it is the monoid of representations of a group.

• •

  For $\Sigma_{4}$, we proved it in Example 2.14.

• •

  For $A_{6}$, there are three $A_{6}$–conjugacy classes of elements of $D_{8}$. It is easy to check that the irreducible $A_{6}$–invariant representations are $1$, $X+XY+Z$ and $Y+Z$.

• •

  For $SL_{2}({\mathbb{F}}_{3})$, there are three $SL_{2}({\mathbb{F}}_{3})$–conjugacy classes of elements of $Q_{8}$. It is easy to check that the irreducible $SL_{2}({\mathbb{F}}_{3})$–invariant representations are $1$, $X+Y+XY$ and $Z$.

For the rest of the section we assume that $p$ is an odd prime. We first consider the extraspecial group $S=p_{+}^{1+2}$ with presentation

Recall that saturated fusion systems over $p_{+}^{1+2}$ are determined by the automorphism group of $p_{+}^{1+2}$, the elementary abelian groups of rank $2$ which are centric radical and their automorphisms groups (see Corollary 4.5 in [24]). Therefore a representation $\rho$ of $S$ is ${\mathcal{F}}$–invariant if and only if the following two conditions are satisfied:

1. (1)

   $f^{*}\rho\cong\rho$ for all $f\in\operatorname{Aut}\nolimits_{{\mathcal{F}}}(S)$.

2. (2)

   $h^{*}\rho|_{V}\cong\rho|_{V}$ for all $h\in\operatorname{Aut}\nolimits_{{\mathcal{F}}}(V)$ for each ${\mathcal{F}}$–centric radical subgroup $V$ which is elementary abelian of rank $2$.

The first condition states that $\rho$ is a fixed point of the action of $\operatorname{Aut}\nolimits_{{\mathcal{F}}}(S)$ on $\operatorname{Rep}\nolimits(S)$. As in the proof of Lemma 2.17, the fixed points form a free monoid, a basis given by adding over the orbits of the irreducible representations of $S$ under the action of $\operatorname{Aut}\nolimits_{{\mathcal{F}}}(S)$.

By Theorem 5.5.4 in [14], the group $S$ has $p^{2}$ irreducible one-dimensional representations and $p-1$ irreducible representations of dimension $p$. The one-dimensional representations come from the quotient

and are denoted by $x^{i}y^{j}$, just like the corresponding representations of ${\mathbb{Z}}/p\times{\mathbb{Z}}/p$. Let $\omega$ denote a primitive $p^{2}$-root of unity. The remaining irreducible representations $\phi_{j}$ for $j=1,\ldots,p-1$ are described by their values on $a$ and $b$. Namely,

and $\phi_{j}(b)$ is the linear transformation that sends $e_{1}$ to $e_{p}$ and $e_{n}$ to $e_{n-1}$ if $n\geq 2$. Here $\{e_{1},\ldots,e_{p}\}$ denotes the standard basis of ${\mathbb{C}}^{p}$. Note that $\phi_{j}(c)=w^{j}I$.

To determine the orbits of the irreducible representations of $S$ under the action of $\operatorname{Aut}\nolimits_{{\mathcal{F}}}(S)$, we can use the following lemma.

Hence we will start determining the monoid $\operatorname{Rep}\nolimits(S)^{\operatorname{Out}\nolimits_{{\mathcal{F}}}(S)}$ and then impose the conditions given by the elementary abelian subgroups. The elementary abelian subgroups of rank two of $S$ are the subgroups $V_{i}$ generated by $c$ and $ab^{i}$ for $i=0,\ldots,p-1$, and the subgroup $V_{p}$ generated by $c$ and $b$. These subgroups have a natural action of $GL_{2}({\mathbb{F}}_{p})$ by identifying each of them with ${\mathbb{F}}_{p}^{2}$ in such a way that the generating elements mentioned above form the standard basis. Note that the irreducible representations of an elementary abelian group $V$ of rank two can be identified with elements of $\operatorname{Hom}\nolimits_{{\mathbb{F}}_{p}}(V,{\mathbb{F}}_{p})$ and the action of $\operatorname{Aut}\nolimits(V)\cong GL_{2}({\mathbb{F}}_{p})$ under this identification is the dual action, which corresponds to multiplication by the transposed matrix.

We now go through the all the possibilities in Tables 1.1 and 1.2 from [24] to determine the factoriality of the monoid of fusion-invariant representations in each case.