Diagrammatic Categories which arise from Representation Graphs

The main result of this paper utilizes the representation graph of a group $G$, $R(V,G)$, and gives a general construction of a diagrammatic category $\mathbf{Dgrams}_{R(V,G)}$. The proof of the main theorem shows that, given explicit criteria, there is an equivalence of categories between a quotient category of $\mathbf{Dgrams}_{R(V,G)}$ and a full subcategory of $G-\textbf{mod}$ with objects being the tensor products of finitely many irreducible $G$-modules.

In this paper, we provide a framework in which we can develop diagrammatic categories from the data of a representation graph (or McKay quiver) for a group $G$. In general, there is a full and essentially surjective functor from this diagrammatic category onto a full monoidal subcategory of the category of all finite dimensional $G$-modules. Furthermore, we provide criteria which, when satisfied, results in an induced functor which is faithful and, thus, an equivalence of categories.

While this paper focuses on the relationship between our diagrammatic construction and $G$-modules over $\mathbb{C}$, the results of the main theorem extend to other contexts. For example, one can consider a fusion category. In a fusion category, the fusion rule can be described as a graph. Using this graph, we may define a diagrammatic category utilizing the construction in this paper which will be categorically equivalent to the fusion category with which we started.

To provide some concrete examples, we will focus on the following context for this paper. Felix Klein classified the finite subgroups of the special unitary group, $SU(2)$. There are two infinite families of finite subgroups along with 3 exceptional subgroups: the cyclic groups of order $n$, $\mathbf{C}_{n}$; the binary dihedral groups, $\mathbf{D}_{n}$, of order $4n$; the binary tetrahedral group $\mathbf{T}$; the binary octahedral group $\mathbf{O}$; and the binary icosahedral group $\mathbf{I}$. Around 1980, McKay made the observation that certain affine Dynkin diagrams and the representation graphs associated with these finite subgroups are identical [1]. We will focus on the cyclic groups and the binary tetrahedral group along with their representation graphs. See Section 2.1 for details.

As a motivating example, consider for $k\in\mathbb{Z}_{\geq 0}$, $\delta\in\mathbb{C}$, the diagrammatic Temperley–Lieb algebras, $TL_{k}(\delta)$, were developed by the authors of the same name in [2]. For $k\in\mathbb{Z}_{\geq 0}$, there are isomorphisms between the endomorphism algebra

of the natural module $V$ for $SU(2)$ and the Temperley–Lieb algebra $TL_{k}(2)$. In [3], Barnes, Benkart, and Halverson combined the work of McKay and Temperley–Lieb by describing the endomorphism algebras of the finite subgroups of $SU(2)$ and presenting diagrammatics for the $\mathbf{C}_{n}$ and $\mathbf{D}_{n}$ cases.

The study of endomorphism algebras like $Z_{k}(SU(2))$ can be generalized to more general homomorphism spaces. For example, we can study $\operatorname{Hom}_{SU(2)}\left(V^{\otimes k},V^{\otimes\ell}\right)$ for all $k,\ell\in\mathbb{Z}_{\geq 0}$. This gives us new tools, new perspective, and a richer understanding of the representation theory. With this generalization in mind, the diagrammatic Temperley–Lieb category was developed, see [4] and [5]. This category admits a fully faithful monoidal functor to the category whose objects are tensor products of $V$ and the morphisms are all $SU(2)$-linear maps. In particular, the Temperley–Lieb algebras appear as endomorphism algebras in the Temperley–Lieb category.

Surprisingly, entire categories can be easier to derive than individual endomorphism algebras. In particular, the Temperley–Lieb category has generating diagrams known as the cup, cap, and identity strand, and there is a diagrammatic basis for each space of homomorphisms, Hom${}_{SU(2)}(V^{\otimes k},V^{\otimes\ell})$, which can be described as all non-crossing diagrams with $k$ nodes on the bottom of the diagram and $\ell$ nodes on the top.

In this paper, we explore diagrammatic categories which expand the set of generating objects which correspond to all of the simple $\mathbf{C}_{n}$-modules and provide explicit relations giving a diagrammatic description of the monoidal full subcategory generated by the irreducible $\mathbf{C}_{n}$-modules, $\mathbf{C}_{n}$-mod${}_{\text{irr}}$. In order to give a more general constructions, in section 4, we utilize the representation graph of a group $G$, $R(V,G)$, and give a general construction of a diagrammatic category $\mathbf{Dgrams}_{R(V,G)}$. The proof of the main theorem shows that, given explicit criteria, there is an equivalence of categories between a quotient category of $\mathbf{Dgrams}_{R(V,G)}$ and $G$-mod${}_{\text{irr}}$. In the final section, we give a few final remarks regarding generalization to directed graphs and give a few examples which show that these results apply outside of the context of $SU(2)$ and its finite subgroups.

We shall close out the introduction with a discussion of certain directions in which this work might extend. For the constructions in this paper, the functor to the category of $G$-modules can be thought of as a functor to the category of $\mathbb{C}$-vector spaces once we forget the $G$-action. In other words, we have a representation of each of these diagrammatic categories. Just as one group can have many representations, one category can have many interesting representations. This is an active area of research.

For example, Sam and Snowden explore the representation theory of the Brauer category in [6]. They specifically mention that much of the theory they develop could be transferable to other categories, like the Temperley–Lieb categories and its variants. In particular, we expect it applies to the categories introduced in this paper.

Similarly, Brundan and Vargas give a concrete diagrammatic definition of the affine partition category, and use it to study the representation theory of the partition category [7]. It is with these two papers in mind that we may ask the following questions.

In addition to the above questions, there are other directions one might consider exploring. The hands-on combinatorial nature of this area makes it easy to compute interesting examples and special cases. Another direction could be to explore how these categories react to changes in certain parameters. For example, the Temperley–Lieb category, when not considering the connection to $SU(2)$, can be defined with a parameter $\delta\in\mathbb{C}$ where $\delta=2$ is the Temperley–Lieb category for $SU(2)$. What would introducing such a parameter to these diagrammatic categories change about the combinatorics or representation theory? For example, one might explore how these categories decategorify.

Let us explore some new families of categories. We will again use the convention that the empty diagram is the morphism from $(0)$ to $(0)$ which represents multiplication by $1$ where $(0)=\bar{0}$ is the identity object.

Given a diagram $d$, we will denote $s_{d}$ as the number of split diagrams used in the construction of $d$ and $m_{d}$ as the number of merge diagrams used in $d$.

The above lemmas will be helpful in proving Theorem 3.7. Now let us explore a particular category from representation theory.

We denote $\mathbf{C}_{n}\textbf{-mod}_{\text{irr}}$ as the full $\mathbb{C}$-linear monoidal subcategory of $\mathbf{C}_{n}\textbf{-mod}$ where the generating objects are the irreducible $\mathbf{C}_{n}$-modules $\mathbf{C}_{n}^{(a)}$ where $a\in\mathbb{Z}/n\mathbb{Z}$. As all irreducible $\mathbf{C}_{n}$-modules are $1$-dimensional, and

then by Schur’s Lemma, Hom${}_{\mathbf{C}_{n}}(M,N)$ is either $0$-dimensional or $1$-dimensional. If it is $1$-dimensional, then $M\cong\mathbf{C}_{n}^{(a)}\cong N$ for some $a\in\mathbb{Z}/n\mathbb{Z}$. We pick bases for these irreducible $\mathbf{C}_{n}$-modules and let $v_{a}$ denote our chosen basis vector of $\mathbf{C}_{n}^{(a)}$.

We can choose $\mathbf{C}_{n}$-module homomorphisms,

where $c\equiv a+b\mod n$,

where $c\equiv a+b\mod n$, and

is the identity map. Notice that $m_{a,b}^{c}\circ s_{c}^{a,b}=id_{c}$ for all $a,b,c\in\mathbb{Z}/n\mathbb{Z}$.