Grothendieck Equivalence Classes of Fusion Categories Preserving the Group-theoretical Property
Advisor: Dr. Naor Movshovitz
Internship Location: Indiana University Bloomington
Onsite Mentor: Prof. Julia Plavnik, Charlotte Ann Griffin Associate Professor, Mathematics, Indiana University, Bloomington
My research will primarily involve formulating hypotheses and then proving or disproving them using formal mathematical techniques, and constructing examples and counter-examples. This will be based on study of existing research and background material in the area. My research may also involve investigating implications on adjacent fields. The outcome of the research will be mathematical claims formalized as theorems and proofs of the theorems, written up as a formal math report.
My final product will contain examples of groups for which the research question holds: any fusion category that is Grothendieck-equivalent to the group’s representations is group theoretical. Depending on my progress, it may also contain general classifications of such groups.
Project Blog
Week 0: Introduction
Hi everyone! I'm Rohan, and I'm excited to share my first senior project blog post with you. Here, I'll introduce my research question, its significance and background, and a brief conceptual overview of the relevant terms.
A Brief Conceptual Introduction
To state my research question, I must first discuss three key terms: fusion categories, Grothendieck equivalence, and representations of groups. Throughout this project, I might go into more detailed and rigorous explanations of these, depending on how accessible and enlightening they are.
Within quantum physics, anyons are a type of quasiparticle. I haven't studied them yet, but part of the reason they are interesting is due to their unique combination rules. Typically, information within quantum mechanics comes in discrete states -- for example, it often can only be an integer or half of an integer. However, combinations between anyons can also reach values in between these. Fusion categories are essentially a way to capture these relations. They consist of additive and multiplicative structures that correspond to the fusing and crossing of anyons.
Fusion categories also have other aspects that, in a way, make them more "conventional" structures mathematically. We say that its fusion rules is the information that tells us how it would make anyons interact. Then, two fusion categories are Grothendieck Equivalent if they have the same fusion rules. One way to think about Grothendieck Equivalence is that it checks if two fusion categories have the same function in the context of anyons.
Now, we pivot to group representations. A group is a set of elements endowed with some multiplication, which satisfies certain desirable properties such as associativity, having a multiplicative identity, and having multiplicative inverses. This is a very abstract concept. The set of elements can be anything -- they can be numbers, rotations of some shape, strings of letters, or even Taylor Swift's cats' names! The last one isn't so problematic because there is a notion of equivalence for groups as well; they are isomorphic if you can essentially relabel the elements of the first group to get the second group. However, the multiplication can also be anything as long as it fulfills the few conditions mentioned earlier.
Consequently, a group representation is a way to essentially change a group into an equivalent, concrete structure. For now, we can say that it renames the group elements to matrices, which are arrays of numbers, and the group's multiplication then just corresponds to the multiplication of the matrices, something concrete.
Background and My Research Question
Fusion categories are a relatively new concept; one of the founding works, [2], is from 2005. Since then, there has been progress towards classification of their structures, but even now very little is known. One specific class of fusion categories is one arising from a group's representation. We call these group theoretical fusion categories. Much more is known about group representations than about general fusion categories, so this property is useful for understanding the category's structure.
As mentioned earlier, Grothendieck Equivalence checks if two fusion categories have the same fusion rules. In [1], Naidu and Rowell posed the question of when Grothendieck Equivalence preserves the group theoretical property. Precisely, say that Rep(G) denotes the fusion category arising from representations of some group G. Then, they asked for which G is it true that all fusion categories C which are Grothendieck Equivalent to Rep(G) also group theoretical. They suggested that this may be true for all groups G and provided a few examples of groups that work. My goal in this project is to expand on their work, by finding other groups that work. A general classification would be ideal.
Over the next few weeks, I'll share more detailed definitions and conceptual explanations of fusion categories and Grothendieck equivalence. Stay tuned for these in my future posts!
References
[1] D. Naidu, E.C. Rowell. A finiteness property for braided fusion categories. Algebra Representation Theory 14, 837–855 (2011). https://doi.org/10.1007/s10468-010-9219-5.
[2] P. Etingof, D. Nikshych, V. Ostrik. On fusion categories. Annals of Mathematics, Volume 162, Issue 2, 581-642 (2005). https://doi.org/10.4007/annals.2005.162.581.
Week 1: An Introduction to Categories
Hello and welcome back to my blog! Today, I'll introduce categories, and specifically Abelian categories, in some detail.
Categories
I'll first define categories, the underlying concept of my project. For more details about the following definitions, the reader is pointed to [1].
A category has two main components: a collection of objects, and for each pair of objects, a collection of morphisms between them. This is essentially the most fundamental way to capture mathematical structures and maps between them. There are also two notions that must exist.
The first is the composition of morphisms. You may have encountered function composition before. If f: A –> B and g: B –> C are functions, then the composition gf does f first and does g on the result. For example, if f(x)=2x and g(x)=x^2+1, then gf(x)=(2x)^2+1=4x^2+1. We require that for any compatible morphisms f and g, their composition is also a morphism within the category.
The second is an identity morphism, denoted by 1, for each object. You can think of this as the function f(x)=x; formally, we say that for any other morphism g, g1=g and 1g=g whenever these morphisms are compatible for composition. We denote the identity morphism of an object A specifically by 1_A.
Before I describe some examples, I'll discuss the notation. If C is some category, we usually also say that C is the collection of objects. The objects themselves are denoted by capital, italicized letters, and for any objects A and B, Hom(A,B) denotes the class of morphisms from A to B.
Now, for some examples:
The category Set has all sets as objects. The morphisms are any functions between sets.
The one-object category N has the set of nonnegative integers as its only object. The morphisms are the identity, addition by one, addition by two, etc. Composition indeed exists: the composition of addition by m and addition by n is addition by m+n.
If you are familiar with linear algebra, the category Vect_k has all vector spaces over some field k as objects. The morphisms are linear transformations between these vector spaces.
Functors give a notion of maps between categories, as we'll see in later weeks. Then, Cat is the category of all "small" categories, with functors as its morphisms.
Universal Properties
Now that we've seen what categories are, I'll discuss universal constructions, a fundamental part of category theory. My description will be concise; if it seems confusing, it may make more sense once its used in practice for kernels and cokernels.
Essentially, for some property, the universal object is some object U along with a morphism u from U, such that for any (U',u') also satisfying the property, u' uniquely factors as a composition ul. It's hard to provide intuition for these universal constructions; for now, we can think of them as the "best" solutions.
Additive Categories
From here, the main reference point for definitions is [2].
Now, I'll start building up toward fusion categories. The first step is additive categories, which—as the name suggests—endow the category with addition. First, they have two other requirements:
• The collections of morphisms Hom(A,B) have an additive structure.
• For some object 0, Hom(0,0)=0. For example, in the category Set, 0 is the empty set. There are no functions between the empty set and itself, so Hom(0,0) is also the empty set. If you're familiar with linear algebra, a helpful exercise is to find the zero object of Vect_k.
The main requirement is the direct sum. For objects A and B, it necessitates the existence of an object C along with the following morphisms:
• p_1: C –> A,
• p_2: C –> B,
• i_1: A –> C,
• i_2: B –> C,
such that the compositions p_1i_1=1_A, p_2i_2=1_B, and i_1p_1+i_2p_2=1_C.
This may seem arbitrary at first, so I'll consider the example of vector spaces. Then, the direct sum essentially consists of ordered pairs (a,b), where a is a vector in A and b is a vector in B. The "projection" morphisms p_1 and p_2 send (a,b) to a and b, respectively. The "inclusion" morphisms i_1 and i_2 send a and b to (a,0) and (0,b), respectively. If you are curious, you can check for yourself that the three properties are satisfied.
We can see from this example that the direct sum essentially "contains" A and B.
Semisimple Categories
I wasn't planning to define semisimple categories yet, as they aren't too important until we ultimately reach fusion categories, but they follow very directly from the definition of additive categories. The definition comes in three parts:
• An object A is simple if it can not be expressed as the direct sum of B and C for nonzero objects B and C.
• An object A is semisimple if it is the direct sum of a finite number of simple objects.
• A category C is semisimple if all its objects are semisimple.
These properties might feel more natural if you think about the positive integers and multiplication. The zero object corresponds to 1, simple objects correspond to primes, and every positive integer is the product of finitely many primes.
Abelian Categories
Abelian categories expand on additive categories, ensuring that the morphisms act like linear transformations. They are based on kernels and cokernels.
For a morphism f: A –> B, the kernel is the universal object K with a morphism k: K –> A with the property that fk=0.
Similarly, the cokernel is the universal object C with a morphism c: B –> C with the property that cf=0.
If you've taken linear algebra, you probably learned that the kernel is the part of A which maps to 0. The universal construction reflects that: K essentially lies within A, and is the "best" object containing the part that maps to 0.
Finally, an Abelian category is an additive category where for each morphism f: A –> B with kernel (K,k) and cokernel (C,c), there exists an object I such that f splits into i: A –> I and j: I –> B. Furthermore, (I,i) is the cokernel of k and (I,j) is the kernel of c.
The intuition behind this definition is very complicated. Essentially, it ensures that some key ideas from linear algebra transfer over to our morphisms.
That's all for today. Make sure to tune in next week for an overview of the multiplicative aspect of categories! That description will likely emphasize intuition over the actual definitions, which get really bad really quickly.
References
[1] S. MacLane. Categories for the Working Mathematician. Springer Science+Business Media New York (1971)
[2] P. Etingof, S. Gelaki, D. Nikshych, V. Ostrik. Tensor categories. American Mathematical Society (2015)
Week 2: Building Up to Fusion Categories
Hello and welcome back to my blog! Today I'll start with a brief introduction to linear algebra, and then I'll build up towards Fusion Categories.
Crash Course on Linear Algebra
In this quick introduction, I'll introduce fields and vector spaces, as well as other important concepts such as linear maps and bases.
A field essentially consists of scalars. Formally, a field is a set with two operations: addition and multiplication. These should satisfy some properties, such as commutativity, associativity, and distributivity. Furthermore, it has a 0 — an additive identity, and a 1 — a multiplicative identity. Lastly, it also has negatives and reciprocals (additive and multiplicative inverses). If this seems confusing, that's fine; whenever we encounter a field, we can usually think of it as the real numbers.
Then, a vector space is a set which also has two operations: addition, and multiplication by scalars from the field. It also has some properties, such as commutativity, associativity, and distributivity again, but their details aren't too important to us. For now, the most important part to remember is the addition and scalar multiplication.
I've alluded to linear maps (also called linear transformations) in previous posts; they are essentially functions that respect addition and scalar multiplication. In other words, for vectors u and v and a scalar c, we should have f(u+v)=f(u)+f(v) and f(cu)=cf(u).
Finally, given a set of vectors, the span is the set of all vectors that can be written as the sum of scalar multiples of them. For so-called "finite-dimensional" vector spaces, a basis is a minimal set of vectors which spans the entire space. Here, "minimal" means that with any vector removed, the basis would no longer span the space.
Monoidal Categories
Now, we pivot back to category theory. I've mentioned before that fusion categories are based on fusion rules, arising from two operations: addition and multiplication. Last week, we saw Abelian categories, which capture the additive structure. Now, we move to the multiplicative aspect.
While I find monoidal categories easier to intuitively grasp than Abelian categories, their definitions are far more technical. Accordingly, I'll be excluding much more detail here.
Firstly, functors give a notion of maps between categories. A functor from C to D maps C's objects to D's objects and C's morphisms to D's morphisms (between the corresponding objects). It also should respect composition of morphisms and the identity.
There are two additional notions that build on functors. One is the bifunctor, which is essentially a functor in two arguments, so it would be from C_1 x C_2 to D. This can further be extended to multifunctors. For our purposes, I'll just use functor as a blanket term including these. The other is natural isomorphism, which gives a notion of equivalence between functors.
For more detail about functors, multifunctors, or natural isomorphisms, an interested reader may refer to [2].
Finally, a monoidal category is a category C with one main new operation: a bifunctor ⊗ from C x C to C, giving a notion of multiplication. There's also a multiplicative identity object, 1. Then, we additionally require the following equivalences:
• C ⊗ 1 ≅ C,
• 1 ⊗ C ≅ C,
• C ⊗ (C ⊗ C) ≅ (C ⊗ C) ⊗ C.
These are unitality and associativity conditions. Finally, there are coherence conditions on these natural isomorphisms. I won't get into them as they aren't too important to this post, but detail about them can be found at [1].
Rigid Monoidal Categories
Linear algebra has a concept of dual spaces. Essentially, for a vector space V over a field k, the dual space V* is the space of linear maps from V to the base field k. These dual spaces are useful for a multitude of reasons. For example, they can extract information about the original space.
Rigid monoidal categories seek the same concept of a dual, although the definition may look different. For an object X, its left-dual is an object X* along with morphisms ev: X* ⊗ X –> 1 and coev: 1 –> X ⊗ X*. The necessary condition is essentially that if you do ev on one side and coev on the other, they cancel. Again, more detail can be found at [1].
There are a few ideas related to this which I find fascinating. First, in case the part about doing ev and coev on different sides doesn't make sense, there's actually a pretty nice interpretation with linear algebra. It can be shown that if X has a finite basis, then X* has a finite basis of the same size. Then, the concept of tensor calculus lets us encode ev and coev in matrices. Essentially, denote the basis elements of X as x_i and of X* as x*_j. Then entry (j,i) of the matrix corresponding to ev is ev(x*_j, x_i). With information stored in this way, the condition on ev and coev is simply that their associated matrices are inverses!
Additionally, ev and coev are short for "evaluation" and "coevaluation". To understand this, we can think about the case of vector spaces. There, an element of V* is a linear map f from V to k, which is the 1 object. As you may expect from the name, the evaluation map simply sends f x v to f(v).
Finally, I'll discuss a nice connection. The focus of my previous research project was Frobenius algebras. They endow vector spaces with multiple operations, notably the pairing and copairing. Interestingly, the relation between the pairing and copairing is exactly the relation between the evaluation and coevaluation. These are maps A x A –> 1 and 1 –> A x A, so A is its own dual. In fact, this notion of self-duality is exactly what inspired much early study of Frobenius algebras.
Anyway, I've defined left-duals but I haven't yet tied that back to rigid monoidal categories. Right-duals are an analogous concept but flipped, and a rigid monoidal category is a monoidal category where every object has a left-dual and a right-dual.
Conclusion
By now, we've seen most of the skeleton to defining fusion categories, although there are a few missing pieces. Next week, I'll briefly summarize the concepts we've seen so far and unite them to introduce fusion categories. Stay tuned for that!
References
[1] P. Etingof, S. Gelaki, D. Nikshych, V. Ostrik. Tensor categories. American Mathematical Society (2015).
[2] J. J. Rotman. An introduction to Homological Algebra. Universitext. Springer, New York, second edition (2009). https://doi.org/10.1007/b98977.
Week 3: Fusion Categories
Welcome back! As I mentioned last week, we are almost ready to define fusion categories. There is one last property to define. Yet again, these definitions are based on [1].
Finite Categories
I'll start by briefly discussing the main definition of finite categories. There is a more general notion of finite-length categories, based on two requirements.
First, recall that for a k-linear category, the morphisms between two objects form a vector space. We want this vector space to be finite-dimensional (i.e. it has a finite basis).
The "length" requirement is that for all objects, there is some finite sequence that iteratively "adds" simple objects, starting from 0, to reach it. However, this "adding" isn't based on the direct sum (the addition of Abelian categories). Instead, the simple objects should be the quotient of consecutive terms in the sequence. I won't discuss quotients of objects today because we won't actually use this definition, but it is more complicated than the rest of the operations.
For finite categories, there are two more requirements. There must be a finite number of simple objects (up to equivalence). The second requirement is that every object must have a "projective cover". This has a fairly technical definition, which I can explain if anyone is interested, but I'm not sure how to discuss it at an intuitive level. For that reason, I'll avoid using this definition of finite categories, for now...
Thankfully, there is another way to think about finite categories. It stems from rings and modules, analogues of fields and vector spaces. A ring is essentially a field without division, and the definition of a module over a ring is exactly the same as that of vector spaces over a field.
We'll use a slightly more specific case. An algebra is a ring which also forms a vector space over the field k. A module over an algebra is the same as a module over its underlying ring structure. Then, we can say that a finite category is a category equivalent to the category of finite-dimensional modules over some finite-dimensional algebra A.
Something I find interesting but haven't looked into yet is a connection to representations. We won't focus on representations of algebras, but they're exactly the same as modules over algebras. Then, a finite category might essentially be the category of representations of A. Given that we'll consider fusion categories that are the representations of some group, this might be notable.
Anyway, we're finally ready to define fusion categories!
Fusion Categories
A fusion category is a k-linear, Abelian, semisimple, rigid, monoidal, finite category with End(1) = k. These are concepts we've been seeing over the past two weeks, and I'll summarize what they are below.
• In a k-linear category, the class Hom(A,B) of morphisms between two objects is a vector space over k.
• In an Abelian category, there is an addition of objects (the direct sum), and morphisms have properties relating to kernels and cokernels that make them resemble linear algebraic structures.
• In a semisimple category, every object is the sum of finitely many simple objects (an analogue of primes).
• A monoidal category has multiplication of objects and a 1 object.
• A rigid monoidal category has a notion of duals, inspired by the same concept in linear algebra.
• Finite categories have been discussed above.
• End(1), which is notation for Hom(1,1), is a k-vector space due to the first property. It must have dimension 1.
Conclusion
We've finally arrived at the full definition of fusion categories. Some key examples of fusion categories include:
• The category of vector spaces,
• The category of graded vector spaces.
I probably won't prove that these are fusion categories as the proofs aren't very interesting.
In the coming weeks, there are a few more concepts to introduce. The first is the Grothendieck ring of fusion categories, which only captures its fusion rules. I also haven't yet discussed what representations of finite groups are. Stay tuned for these and more in future weeks!
[1] P. Etingof, S. Gelaki, D. Nikshych, V. Ostrik. Tensor categories. American Mathematical Society (2015)
Week 4: Grothendieck Rings
Hi everyone and welcome back to my blog! I've briefly discussed Grothendieck rings before; they capture the fusion rules of a fusion category, but not certain relations like associativity. Today, I'll formalize this notion. Not for the first time (but probably for the last) all definitions are due to [1].
First, when we have a generating set or a basis, it is often useful to talk about "coordinates" of an element. These are simply the coefficients of each basis element in its representation. For example, if a basis is 1, x, and x^2, then the coordinates of 476+48x+x^2 would be 476, 48, and 1.
Rings
Rings are essentially fields without division. To define them from scratch, they are a set with two operations: addition and multiplication, which are associative and distributive. Additionally, there are 1 and 0 elements such that 1 * a = a and 0 + a = a. Finally, there are negatives.
We'll only consider rings that have a finite generating set (or basis): all elements can be expressed as a finite sum (or difference) of generators.
Another concept that we won't use yet, but which certainly is relevant, is the semiring. It is a ring which doesn't necessarily have negatives. Its importance will be clearer when we finally see the Grothendieck ring.
Finally, fusion rings are rings with a few additional properties. The basis should contain 1. Also, the pairwise product of basis elements must only have nonnegative coordinates. There's another, more complicated property about swapping basis elements. I'll avoid this for now, and discuss it if it turns out to be crucial.
The Grothendieck Ring
One fusion ring is especially important: the Grothendieck ring of a fusion category. This is inherently based on suppressing equivalences of objects.
But what does it actually mean for two objects A and B to be equivalent? They are equivalent if there exists an isomorphism, a special kind of morphism, between them. There are a few ways to think about this. The main way is that it has an inverse. Alternatively, it uniquely corresponds elements of A to elements of B.
If you've taken linear algebra, you might remember that all n-dimensional vector spaces over some field k are equivalent. Therefore, we refer to all of them with a blanket term, k^n. This is essentially an equivalence class.
The idea behind equivalence classes is that when we have many objects that are equivalent, we group them into the same class. (It should be noted that equivalence is transitive: if A and B are equivalent and B and C are equivalent, then A and C are equivalent.) Then, we pick one representative from this class.
We can apply this logic to the simple objects of a fusion category, breaking them into equivalence classes and picking representatives. Then, the Grothendieck ring is the ring generated by these. Addition is the direct sum, and multiplication is the tensor product.
While the fusion category has a direct sum, it doesn't have a "direct difference." The parts of the Grothendieck ring that correspond to actual objects are exactly the ones which are generated with addition. In other words, they're the ones with nonnegative coordinates. This is why I mentioned semirings: the Grothendieck semiring exactly captures the fusion category's equivalence classes. However, this usually doesn't add too much value.
Furthermore, this "nonnegative coordinates" seems to echo the definition of fusion rings. In fact, since the pairwise products are by construction objects, they must have nonnegative coordinates too. The condition I excluded for fusion rings also is satisfied because fusion categories are rigid.
Finally, we say that two fusion categories are Grothendieck equivalent if their Grothendieck rings are isomorphic.
Frobenius Perron Dimension
My research question is fundamentally about properties preserved by Grothendieck equivalence. The most notable such data is the Frobenius Perron dimension, or FPdim for short. In fact, it's defined over fusion rings. But to understand it, we first have to define eigenvalues. For a matrix A, an eigenvalue λ is a scalar such that for some vector v, Av=λv.
If you remember the matrix interpretation of duals (from my second post), the key idea is similar. Let's index the basis elements as x_1, ..., x_n. For each x_k, we can construct a "multiplication matrix" of it. Specifically, entry (i,j) of the matrix is the j-th coordinate of x_k * x_i.
Due to the second property of fusion rings, all entries of the matrix are nonnegative. Then, a theorem by Frobenius and Perron shows that its eigenvalue λ with the largest magnitude is real and nonnegative. Then, we set FPdim(x_k) to λ.
This only defines the Frobenius Perron dimension on basis elements. However, we can extend this by additivity. For example, we can define FPdim(x_1+4x_2) = FPdim(x_1)+4FPdim(x_2). This defines FPdim on the entire fusion ring.
For a fusion category, FPdim of an object is FPdim of whichever element in the Grothendieck ring represents its equivalence class. We can also define FPdim of the entire category, which is the sum of the squares of FPdim of the Grothendieck ring elements. Regardless, we will primarily use FPdim of objects.
Conclusion
We've finally covered all of the fusion category-related prerequisites for this project. In the future, we'll encounter more group theory and what it means for a category to be group theoretical. We might also see concrete applications of the Frobenius Perron dimension to solve certain cases of my research question. Stay tuned for this!
References
[1] P. Etingof, S. Gelaki, D. Nikshych, V. Ostrik. Tensor categories. American Mathematical Society (2015)
Week 5: Group Theoretical Fusion Categories
Hi everyone and welcome back to my blog! So far, I've been discussing fusion categories. Today, I'll get to the other main components of my research question: group representations and the group theoretical property. This post will be a bit shorter, because some important details are extremely technical and I haven't yet grasped them well enough to share fully.
Group Representations
I've vaguely touched on the concept of group representations a few times. A group is essentially a multiplicative structure: it is a set with an associative multiplication operation, an identity 1 (such that 1 times anything is itself), and multiplicative inverses.
Then, a representation is an assignment of each group element to a matrix, such that the group multiplication is mapped to the standard concept of matrix multiplication.
Here, matrices are ways to encode linear transformations, a key concept in linear algebra. There is technically a finite-dimensional requirement, but we will only consider finite-dimensional representations anyway.
In prior weeks, I shortly discussed the direct sum of vector spaces and their matrices. For vectors, it essentially just keeps the components separate and writes them next to each other. The direct sum of matrices works in a pretty similar way: write the matrices such that their bottom-right and top-left corners touch, and fill the rest with zeroes.
Meanwhile, the tensor product of vector spaces is a crucial concept in linear algebra, but it's probably too technical for this blog at the moment. The important part is that direct sums and tensor products of representation are also representations! This lets us think about the category of representations in a similar nature as fusion categories. Representations aren't exactly fusion categories, but the Grothendieck ring concept still applies. Specifically, this ring would be generated by all irreducible representations.
The Group-theoretical Property
I intended to make this the main focus of today's post, but the definition is extremely technical and there isn't an easy intuitive perspective I can discuss either. My own intuitive understanding is based on something I haven't discussed yet: braided monoidal categories. Recall that monoidal categories are essentially categories with multiplication. Braids are a sort of twisting map that reorders this multiplication. Most of my prior work is with symmetric monoidal categories, where the braids only swap the order. In the general case, they can do much more. One notable use is to add quantum natures to systems.
Meanwhile, graded vector spaces are ways to decompose vector spaces based on a group, such that the tensor product is based on group multiplication. This definition is more accessible, so I might share more about it in future weeks. These also have braidings, given by cocycles – another term that is too technical for now. An intuitive perspective is that these capture "twisted" representations.
The Drinfeld Center of a fusion category is a way to find its underlying braided monoidal category structure. Then, a fusion category is group-theoretical if its Drinfeld Center is equivalent to some braided, graded cocycle.
Conclusion
Now, we have a vague sense of what it means for a fusion category to be group-theoretical! This is a very complicated concept, and it might make more sense as we work through a specific example next week. Hopefully this can also illuminate topics from previous weeks, such as Grothendieck equivalence. Stay tuned!
Week 6: Representation Rings
Hello and welcome back to my blog! Last week, I briefly touched on group representations. I omitted the definition of tensor products, as it's too technical for this blog. However, I've realized that at least the multiplication tables of tensor products are essential. Throughout this week, I've also realized that these tables are surprisingly interesting to compute, so I'll focus on them today. I'll first review and introduce some notions from linear algebra and group theory.
Linear Algebra Concepts
Recall that matrices are arrays of numbers that can be interpreted as "linear maps" of vector spaces.
Furthermore, eigenvalues of a matrix A are scalars λ such that for some nonzero vector v, the output Av is equal to the scalar multiple λv. This essentially means that the matrix A has the same effect as scalar multiplication for any nonzero vector. In fact, another notion arises from this: the multiplicity of an eigenvalue is the dimension of the space of valid vectors v.
I've discussed eigenvalues before, in the context of calculating Frobenius-Perron dimension. Here, their use is much simpler and more intuitive: the trace of a matrix is the sum of its eigenvalues, with multiplicity.
One notable property about the trace is that it is preserved by conjugation. Conjugation of some matrix A by some other matrix B outputs the product BAB⁻¹. Through direct computation, one can show that this has the same trace as A. (If you want to try to prove this yourself, it is significantly easier to instead define the trace as the sum of the diagonal entries. Due to this result, the definitions are equivalent.)
Finally, if a matrix A has an eigenvalue λ, then any power Aⁿ has an eigenvalue λⁿ. You can prove this by directly showing that if v is the eigenvector corresponding to λ, then Aⁿv = λⁿv. I recommend this as an exercise if you're interested.
Group Theory Basics
As a reminder, a group is essentially a multiplicative structure which can capture a wide variety of mathematical concepts. Groups have the exact same notion of conjugation: the conjugation of some element g by h is hgh⁻¹. We can also divide a group into its conjugacy classes: two elements g and i are "equivalent," and therefore in the same class, if i is the conjugate of g by some h.
Additionally, while groups often represent concrete mathematical ideas, in representation theory it's usually easier to consider group presentations. These are ways to express the group in terms of elements that generate it, as well as relations between these generators.
That may be confusing, so let's consider an example. The group Dₙ, also called the dihedral group, is the group of symmetries of a regular n-gon. These symmetries can all be expressed as some combination of rotations and reflections. Let r be a counter-clockwise rotation by 360/n degrees. Then the rest of the rotations can be expressed as r², r³, etc. We also pick some axis of symmetry (arbitrarily), and denote reflection across it by s.
There are two clear ways in which we can get back to where we started. Rotating n times returns us to the original polygon, as does reflecting twice. Therefore, rⁿ and s² should be the identity symmetry, which is just not changing anything. We can find one more relation. Since reflecting flips the orientation of rotations, if we rotate once, reflect, and then rotate again, the two rotations cancel each other out. Therefore, rsr=s.
These are the only three relations, so they fully determine the group.
A brief clarification
When I say that r and s generate the group, you might think I mean that every element in the group is of the form rᵃsᵇ for some a and b. After all, that's exactly what it would mean in a vector space. However, generate just means that the elements can be expressed as any multiplication of r's and s's. Vector addition is commutative, which is why we rearrange them into the conventional format. However, group multiplication usually isn't commutative, so we can't reorder it.
Regardless, our relations are enough. For example, if we have an element sr, by the relations we can rewrite it as rⁿ⁻¹s.
Some dihedral group calculations
First, the dihedral group has exactly 2n elements, which are of the form rᵃsᵇ where a can be anything at least 0 but less than n, and b is either 0 or 1.
By playing around with the group relations for some time, we can also derive that the number of separate conjugacy classes is (n+3)/2 if n is odd, and n/2+3 if n is even. These calculations aren't too interesting and aren't in the spirit of what I want to discuss today, so I'll exclude them.
Another fundamental group theory concept is the commutator subgroup. It is basically the set of ghg⁻¹h⁻¹ across all pairs of group elements g and h. We'll see later why this is important; for now, by a quicker but still not too interesting calculation, it consists of the even powers of r. For odd n, this includes all rotations, so it has size n. For even n, it only includes half of them, so it has size n/2.
Character Theory
Last week, I discussed how to take the direct sum of two representations — by taking the direct sum of the matrices each group element is sent to. Just as with simplicity of objects in fusion categories, a representation is irreducible if it can't be expressed as the sum of two representations.
Before we transition to character theory, there are two useful facts about irreducible representations that don't rely on characters.
• The number of irreducible representations equals the number of conjugacy classes of the group.
• The squares of the dimensions of the irreducible representations add to the size of the group.
Character theory, at least as we'll see it in this project, is about the traces of irreducible representations. In particular, the character of a representation is a map that assigns each element to a complex number: the trace of the matrix from the representation.
Because matrix conjugation preserves trace, group elements in the same conjugacy class correspond to the same complex number. For this reason, instead of saying a character is a function from the group to the complex numbers, we often say it is a function from the conjugacy classes.
There is a specialized dot product between characters. Let z* denote the complex conjugate of some number z. Then, the dot product of characters χ₁ and χ₂ is the average of χ₁(g)*χ₂(g) across all group elements g.
Character tables: some facts
Character tables record the characters of each irreducible representation. Below, I'll list many important and interesting facts about them, with which we can compute a few examples. I have actually proved all of these as exercises and they're quite intriguing, but today's post is already quite long — if you would like to see them or if I have space, I'll try to include them in later posts.
• The dot product of an irreducible character with itself is 1.
• The dot product of two different irreducible characters is zero. (These conditions state that the rows are orthonormal.)
• Define a similar dot product of columns: for two group elements g and h, it is the sum over all irreducible characters χᵢ of χᵢ(g)*χᵢ(h). If g and h are in the same conjugacy class, this is the size of the group divided by the size of the shared conjugacy class.
• If g and h are in different conjugacy classes above, then the dot product is zero. (This gives an orthogonality condition on the columns as well).
• For a character χ and element g, χ(g)*=χ(g⁻¹).
• For a character χ of dimension n and element g such that gᵐ is the identity, χ(g) decomposes as the sum of n complex numbers of the form z, where zᵐ=1.
Computing the D_3 table
Using these facts, we can compute the character table of many small-sized groups. Let's do the dihedral group D₃ as an example. It has three conjugacy classes: the identity {1}, as well as {r, r²} and {s, rs, r²s}.
Then there are three irreducible representations. The squares of their dimensions sum to 6, so they must have dimensions 1, 1, and 2 in some order.
Every group has the trivial representation: just assign each element to 1. This character also sends each conjugacy class to 1.
There is one more 1-dimensional representation. Suppose it sends r to some number z. Then it sends r² to z². Because r and r² are in the same conjugacy class, they have the same character. Then z=z², so it's 1. All 1-dimensional representations also send the group identity to 1.
Now, let's look at the dot product of this new representation with the trivial one. If the last conjugacy class is sent to z, then the dot product is 1/6 (1+1+1+z+z+z). Since this must be zero, z=-1.
The third representation still must send the group identity to the matrix identity. In dimension 2, this has trace 2 as well. Let the third character send the other conjugacy classes to a and b, respectively. By taking the dot product with each of the one-dimensional representations, we get that 2+2a+3b=0 and 2+2a-3b=0. This is a system of equations, and solving, we find that a=-1 and b=0.
Thus, the character table has rows {1,1,1}, {1,1,-1}, and {2,-1,0}.
A general statement about dihedral groups
In fact, irreducible representations of any dihedral group have dimension one or two.
To prove this, we need another result: the number of dimension one irreducible representations equals the size of the group divided by the size of the commutator subgroup. Referring back to the calculations section, there are 2 1-dimensional irreducible representations if n is odd, and 4 if it is even.
Also using the number of conjugacy classes from the calculations section, there are (n-1)/2 representations left for odd n, and n/2-1 left for even n. The squares of their dimensions are at least four, and they must add to 2n-2 or 2n-4 respectively, as that's the size of the group minus the contribution from the 1-dimensional representatoins.
This can only happen if all the dimensions are exactly two, which achieves the desired result.
Computing multiplication tables
This post has been quite long, so we seem to be far from the original goal: to compute multiplication tables of the tensor product without actually discussing the tensor product. However, this goal follows quite directly from character tables. I learned about this really clean connection from Section 4.9 of [1], which is also a great source for character theory.
Let Vᵢ denote the representation corresponding to χᵢ. Because these irreducible representations generate the rest, we know that the tensor product Vᵢ ⊗ Vⱼ is a linear combination of them. The characters tell us what the coefficients are. In particular, the coefficient of Vₖ is (χᵢχⱼ)•χₖ. Here, the first part is normal multiplication (i.e. multiply the outputs), and then we take a dot product.
I'll omit the actual calculation here because it's just direct arithmetic, but this allows us to exactly grasp what the representation ring's structure is — at least, when the groups are small enough that the character tables are actually feasible to compute by hand (or with programmed assistance). However, as we'll soon see, even if we know exactly what the ring looks like, determining if a fusion category is group-theoretical is an extremely technical and difficult task. Stay tuned as I try to navigate this across the next few weeks!
References
[1] Pavel Etingof, Slava Gerovitch, Oleg Golberg, Sebastian Hensel, Tiankai Liu, Alex Schwender, Dmitry Vaintrob, Elena Yudovina. Introduction to Representation Theory. Student Mathematical Library (2011)
Week 7: part 1: Solving a Specific Case
Welcome back to my blog! In [3], Naidu and Rowell posed the question I aim to answer in this project. They also gave a few examples of groups which work. The main case they addressed was of the group D(2n), the group of symmetries of a regular polygon with 2n sides. Today, I'll discuss the theory behind their approach.
In this post, I'll establish the main approach which they used and derive the relevant properties of the dihedral group. In part 2 of this week, I'll show these ideas in action, using them to reach the result.
An alternative formulation of the group-theoretical property
In a prior post, I gave a very vague description of the group-theoretic property, written in terms of complicated category theory notions such as the Drinfeld center. However, there's a much cleaner way to think about it (given in [2]), which is also far easier to use in practice.
This is based on the concept of modules, which I've discussed a few weeks ago. Essentially, over a ring R, a module M is a group along with some multiplication R x M –> M. Modules in the categorical sense are very similar: modules M over an object A are primarily characterized by a morphism from M ⊗ A to M.
This is technically a right-module, while a left-module arises from a morphism from A ⊗ M to M. A bimodule comes with both of these structures, as well as some compatibility conditions. Then, for some object A in our fusion category C, we want the category of A-bimodules to be "pointed".
A fusion category is pointed if all its simple objects are invertible. I haven't yet defined invertible, but as you may expect, it essentially means that the object has a multiplicative inverse. Formally, the tensor product of the object and its dual should be isomorphic to the unit object. A more detailed definition can be found in [1].
Simplifying the conditions
We can think about this in terms of dimensions, specifically the Frobenius-Perron dimension. The object and its dual have the same dimension, which we can call n. Then their tensor product has dimension n². This should match the unit object's dimension, which is 1. Then any invertible object must have Frobenius-Perron dimension 1. It turns out, by a standard result, that this condition is also sufficient.
Recall that the Frobenius-Perron dimension of the entire category is the sum of the squares of FPdim(C) over all simple objects C (up to isomorphism). This value is the same between C and the bimodule category.
In particular, when C is Grothendieck equivalent to some representation ring Rep(G), they share a Frobenius Perron dimension. One of the results I mentioned last week implies that FPdim(C) is the size of the group G. Accordingly, we can aim to find a number of simple bimodule structures equal to the number of elements of G.
Some important results
Now, I'll discuss some key ideas needed for the proof.
Dihedral group characters
Last week, I discussed the conjugacy classes of the Dihedral group D(n). For even n, which we are focusing on today, there are n/2+3 conjugacy classes. Two of these classes encompass the symmetries which aren't orientation-preserving, i.e. the ones that involve reflection.
As we saw last week, for even n, D(n) has four irreducible representations of dimension 1. We can exactly determine two of them. One of them is the identity representation 1, which just sends each element to 1. The other is the sign representation, which we call X which sends the orientation-preserving elements to 1 and the rest to -1. Since only two classes aren't orientation-preserving, only two classes are sent to 1.
For any other character χ, we must have χ•1=0 and χ•X=0. Then χ•(1-X)=0.
Here, 1-X is zero for all but the two reflection classes, where it's the same value (two). Consequently, the corresponding values of the character must be negatives of each other.
Let one of the remaining 1-dimensional representations be Y. Then Z=XY is also a valid representation, and exactly matches Y except at those two classes, where they are opposite.
Then, for any character χ separate from these four, we also have χ•(Y-Z)=0, which tells us that the values of the character for the two conjugacy classes must be the same.
But if they are the same and also negatives, they must both be zero. As a result, for any character χ of dimension greater than one, χX=χ.
Dihedral group representations
So far, we have four one-dimensional representations: 1 and X, and Y and Z. As I proved last week, the rest of our irreducible representations are two-dimensional.
Consider some two-dimensional irreducible representation V with character χ. As discussed last week, the coefficient of some other representation V' (with character χ') in V ⊗ X is χX•χ' = χ•χ'. One of the key results from last week is that this is 1 if χ=χ', and 0 otherwise. Therefore, V ⊗ X is (equivalent to) V!
This may not seem too important yet, but it is central to Naidu and Rowell's methods. Stay tuned for part 2 of this post, where we'll explore how they used this property!
References
[1] P. Etingof, S. Gelaki, D. Nikshych, V. Ostrik. Tensor categories. American Mathematical Society (2015)
[2] P. Etingof, D. Nikshych, V. Ostrik. On fusion categories. Annals of Mathematics (2005)
[3] D. Naidu, E. Rowell. A finiteness property for braided fusion categories. Algebras and Representation Theory (2011)
Week 7: part 2: Solving a Specific Case
Welcome back! This is a continuation of my Week 7 part 1 post. There, I discussed the general approach which Naidu and Rowell took in [2], as well as some helpful computations. Here, I'll synthesize that with some other key ideas to demonstrate their proof.
A few technical results
As I mentioned in the first part of this post, they approached the problem by considering a more friendly definition of the group-theoretical property: that for some object A, the category of A-bimodules is pointed.
However, it is not given that this category is even a fusion category. A result mentioned in [1] addresses this.
Fact 1
In a fusion category C, consider a simple object V such that for some natural number n, Vⁿ=1. Let A=V⊕V²⊕...⊕Vⁿ. Then the A-bimodule category is a fusion category.
Once we know that the bimodule category is indeed a fusion category, we want to focus on its simple objects. To do this, we refer to another result from [1].
Fact 2
Consider a simple A-bimodule M which has a nontrivial morphism into some simple object X in C, such that V⊗X=X. Then M is equivalent to X as objects in C, and M is invertible in the category of A-bimodules.
Another helpful property derived from the proof of Fact 2 is that there are exactly n² bimodule structures on each such simple object X.
These facts are quite technical and we'll have to accept them as true for now. However, as I aim to generalize in later posts, we'll see more detail about why these are true.
The proof
The key idea is that most simple objects X are within the jurisdiction of Fact 2. As I showed in the first part of this post, all two-dimensional representations X indeed have V⊗X=X. There are n/2-1 of these, and by Fact 2, they each contribute 4 unique bimodule structures, each of which is invertible.
In total, these contribute 4•(n/2-1)=2n-4 to the Frobenius-Perron dimension of the A-bimodule category, which should match the number of elements in the dihedral group. As discussed when we introduced this group, it has 2n elements. Therefore, the remaining contribution must be 4.
There are two possibilities:
• We have one more bimodule structure, which has Frobenius-Perron dimension 2 in the bimodule category.
• We have four more bimodule structures, each of which are invertible.
Any object forms a bimodule over itself, so A gives one more bimodule structure. To conclude, they found that Y⊕Z, the direct sum of the other two one-dimensional irreducible representations, also gives bimodule structures. Then there can't only be one more structure, so we must be in the second case.
Some takeaways
The key property of dihedral groups which Naidu and Rowell exploited is the fact I derived in part 1, that every irreducible representation either is one-dimensional or has V⊗X=X. In particular, their approach largely hinges on most bimodule structures corresponding exactly to the original simple objects in C (the irreducible representations).
I haven't thought too precisely about this yet, but I would expect that any irreducible object in the bimodule category must have the same dimension as our choice of A. If this is true, Naidu and Rowell's approach will be quite difficult to employ. Most groups don't have this really elegant property that all irreducible representations are either 1 or 2-dimensional, so we would need to search for much more complicated possibilities.
References
[1] P. Etingof, D. Nikshych, V. Ostrik. On fusion categories. Annals of Mathematics (2005)
[2] D. Naidu, E. Rowell. A finiteness property for braided fusion categories. Algebras and Representation Theory (2011)
Week 8: Families of Groups
Welcome back! In [1], Naidu and Rowell found a few classes of groups which satisfy our research question. Their main result about this question is the dihedral group case, and we saw their proof in last week's post. Today, I'll discuss some other groups they considered, as well as common constructions which we can try to use.
Abelian groups
A group is Abelian if its multiplication is commutative. Naidu and Rowell pointed out that any finite Abelian group works. This is based on one key property: all irreducible representations of a finite Abelian group are one-dimensional.
One way we can see this is that since G is Abelian, all multiplications ghg⁻¹h⁻¹ are just the identity. Then the commutator subgroup has size 1. As mentioned a few weeks ago, the number of dimension 1 representations is the number of elements divided by the commutator size, so it is simply |G|. We can also note that each element is in its own conjugacy class, by the same reason, so there are |G| conjugacy classes. I also mentioned a few weeks ago that the total number of irreducible representations is the number of conjugacy classes. Therefore, since there are |G| irreducible representations, of which |G| are one-dimensional, we conclude.
Since all irreducible representations are one-dimensional, they are all invertible. Therefore, the original fusion category is itself pointed (which means that all simple objects are invertible).
The importance of Abelian groups is there unique property that all irreducible representations are one-dimensional. No other group has this, and the closest we can get is if all irreducible representations are one-dimensional or two-dimensional.
As we saw last week, this is a crucial property of dihedral groups. A few other important groups share this property, and might be interesting to consider.
Direct products
I've discussed direct products of vector spaces before (although we usually call them direct sums in this context). Essentially, they are ordered pairs of a vector from each space. These components are entirely independent, so this construction replicates the structures of the two original groups.
Due to this, it seems natural that the direct product of any working groups would also work. Indeed, Naidu and Rowell noted this property.
Many groups have representation rings that are just really difficult to work with. By taking a direct product with a better-behaved group, it might be possible to obtain another group that is easier to tackle.
Semidirect products
Semidirect products are similar to direct products, but they change the structure through conjugation. To define them, we need a few more group theory ideas.
Some more group theory
For a group G, a subgroup H is any subset of G which is closed under multiplication (i.e. the product of any two elements in H is still in H). In particular, a normal subgroup N is closed under conjugation: for any element g in G and n in N, gng⁻¹ is still in N.
Recall that homomorphisms are maps which preserve multiplication, and an automorphism is a homomorphism from a group to itself. An important fact is that the automorphisms themselves form a group, under composition. We call this the automorphism group Aut(G).
Semidirect products are constructed based on a homomorphism φ from a group H to an automorphism group Aut(N). Here, φ(h) is itself a map, so for clarity we denote this by φₕ.
The construction
Then, the semidirect product has the same underlying set as the direct product N x H. It's based on the same idea of ordered pairs. However, this time the multiplication is not independent. Instead, the product of (n,h) and (n',h') is (nφₕ(n'), hh').
The semidirect product recovers the direct product in the case when the homomorphism φ is "trivial": it maps everything to the identity.
The working case
Naidu and Rowell proved that one specific case of the semidirect product works. To discuss this, we'll need some more group theory.
Recall that "modulo n" refers to the remainder when something is divided by n. Then, the cyclic group, denoted Zₙ, is the group of integers modulo n, up to addition. In other words, the group is based on normal addition, except adding 1 to itself n times returns us to zero.
The multiplicative group, denoted Zₙˣ, is also based on remainders modulo n. However, this group now consists of integers that are relatively prime to n, and the operation is now multiplication.
This explanation was a bit rushed because for now, these aren't too important to understand. Both of them have a very simple structure and are even Abelian. However, the semidirect product adds a much more complicated structure. Regardless, Naidu and Rowell noted that one restrictive condition works: the semidirect product of a multiplicative group and cyclic group of the same size, which additionally is a prime power.
Conclusion
Today, we got a deeper look at some of the other groups Naidu and Rowell considered, and we saw some interesting directions to pursue. In the coming weeks, we'll get a more concrete picture of these new directions.
References
[1] D. Naidu, E. Rowell. A finiteness property for braided fusion categories. Algebras and Representation Theory (2011)
Week 9: The Dicyclic Groups
Welcome back! Two weeks ago, I explained the approach that Naidu and Rowell took in [1] to prove that the dihedral groups work. As I noted last week, a key reason their proof worked is that all irreducible representations of dihedral groups are either one-dimensional or two-dimensional. This is a very rare property which makes them much easier to work with. However, this property is not unique to them. Another important class of groups, the dicyclic groups, share it.
What are the dicyclic groups?
Dicyclic groups are actually very similar in nature to dihedral groups. This parallel is easier to see from the first of two definitions that I will share.
Dicyclic groups from group presentations
This definition arises from group presentations, which express a group in terms of their generating elements and relations of these generators. I discussed this before in the context of the dihedral group, and I'll briefly review that now. The dihedral group is generated by a single rotation and by a reflection, which we can call a and b. If the polygon has n sides, then rotating n times should leave no overall change. Similarly, reflecting twice does nothing. We express these conditions as aⁿ=1 and b²=1. Finally, since reflecting changes the orientation of rotations, bab=a⁻¹.
If we work with a polygon that has 2n sides, then the relations will be a²ⁿ=1, b²=1, and bab=a⁻¹.
The presentation of a dicyclic group is almost identical: it is generated by two elements a and b, where a²ⁿ=1, b²=aⁿ, and b⁻¹ab=a⁻¹.
The conjugacy classes of the dicyclic groups
The computation of the conjugacy classes and commutator subgroup of the dicyclic group are quite routine. The conjugacy classes are {1}, {aⁿ}, the pairs of inverses {aᵏ, a⁻ᵏ}, and the classes of {aᵏb} for even and odd k.
These form a total of n+3 conjugacy classes. We can also compute that the commutator subgroup has size 4. Then by the same argument as for the dihedral groups, there are four one-dimensional irreducible representations, and the rest are two-dimensional.
Like for the dihedral groups, one of these one-dimensional irreducible representations is just the trivial representation, sending everything to 1. Another is the sign representation, which sends all rotations to 1 and all reflections to -1. Proceeding by the same arguments, we can derive the same key property: that if X is the sign representation and V is any two-dimensional irreducible representation, then X⊗V=V.
Quaternion groups
Quaternion algebras are a crucial concept used in theoretical physics to better capture three-dimensional structures. They are generated from the real numbers by three terms: i, j, and k. This is an extension of the complex numbers, which are generated by i. The quaternion group captures how these generators interact. Specifically, they are given by the relations i²=j²=k²=-1 and ijk=-1.
The full list of elements is 1, -1, i, -i, j, -j, k, and -k. If we compute the conjugacy classes, we find that they are {1}, {-1}, {i, -i}, {j, -j}, and {k, -k}. This resembles the dicyclic conjugacy classes. If we look more closely, we find that the quaternion group is exactly the dicyclic group with n=2.
This doesn't help us too much with the general dicyclic groups, but it does show their importance: they are considered generalizations of the quaternion groups.
In fact, we can define dicyclic groups from quaternion algebras: a is a 2n-th root of unity in the usual complex number sense, and b is j. To verify that this is equivalent, we only need to check that the group presentation relations hold.
Other related groups
Before we decide on which other groups we should consider, it's important to again emphasize why the dicyclic groups are important. Like the dihedral groups, they have a sign representation which is almost always 1, but is -1 for just two conjugacy classes. In general, any one-dimensional representation which is 1 for most conjugacy classes would be very helpful, because by similar arguments, we can try to show that it shares the X⊗V=V property for most V.
In fact, there's a structural way to interpret this, which I didn't mention earlier because it's more technical. Recall that the cyclic group Cₙ is the group of integers modulo n (i.e. adding 1 to itself n times returns us to 0). Then, the dicyclic group is an extension of the cyclic group with 2n elements by the cyclic group with 2 elements.
I won't formally discuss extensions as they are beyond the scope of this blog, but if you're interested, they relate to exact sequences.
This also gives us a more intuitive way to think about sign representations. In particular, they focus on the C₂ part of the group. Specifically, it assigns 1 if the component is 0, and -1 if the component is 1.
With this perspective, group extensions (and semidirect product constructions in general) may produce "sign representations". Accordingly, I'll focus on these next.
References
[1] D. Naidu, E. Rowell. A finiteness property for braided fusion categories. Algebras and Representation Theory (2011)
Week 10: Bimodules
Welcome back to my blog! In part 1 of my Week 8 post, I mentioned an alternative formulation of the group theoretical property: that for some algebra A, the category of A-bimodules is pointed. To further explore the research question, we must better understand how to work with bimodules.
Modules and bimodules
Recall that a ring is a structure with addition and multiplication. Then, as I discussed in previous weeks, a left module is the analog of a vector space for rings. To be precise, it's an additive group along with multiplication r•m that returns another module element. This multiplication needs to distribute with addition: r•(m+n)=r•m+r•n and (r+s)•m=r•m+s•m. (Here, r and s are ring elements, while m and n are module elements). It must also be compatible with multiplication: (rs)•m=r•(s•m).
This is the standard definition of modules, but it isn't sufficient for our purposes. In category theory, we usually don't have any elements to work with, so we must formulate this in terms of morphisms. Multiplication is a morphism from R x M to M with the compatibility and distributivity conditions which I mentioned earlier. These conditions make the morphism bilinear, so it uniquely factors through the tensor product R⊗M. The phrase "uniquely factors" is a fundamental idea in category theory, which I can explain if anyone is interested. For now, I'll just say that it lets us instead work with morphisms from R⊗M to M, i.e. morphisms in Hom(R⊗M,M)
A right module is analogous, but multiplication is done on the other side, so now it uses morphisms in Hom(M⊗R,M). Then, bimodules unite these concepts. They are an additive group M along with the two multiplications such that r•(m•s)=(r•m)•s. To phrase this in categorical language, we can take two paths from R⊗M⊗R to M: we either do left multiplication or right multiplication first, and we do the other second. Then, these paths should result in the same morphisms. I'm essentially describing a commutative diagram, if you're interested in the formulation with paths.
Semisimple algebras
By default, we can't have modules over objects from a fusion category. The R from prior discussion must be a ring, so it needs to have multiplication. However, multiplication does not come along with the fusion category structure. Therefore, we need to construct a new multiplicative structure.
There are two basic requirements for the multiplication: it must be associative, and it must have a multiplicative identity. Both of these conditions are usually phrased in terms of elements, but can also be written categorically. For example, associativity says that the two paths from A⊗A⊗A to A (which multiply on one side and then multiply the results) are the same. However, the details of this aren't too important for our purposes, as most multiplications we construct will have this property.
By adding multiplication, we have turned our object into an algebra, and we can now consider modules over it. There is one more key property: if the algebra is semisimple. We say that an element a annihilates a module if a•m=0 for all m in the module. Then, the algebra is semisimple if no element other than 0 annihilates all left modules.
Alternatively, an algebra is semisimple if (and only if) it is the direct product of simple algebras, i.e. algebras which have no proper ideals. This definition provides less clarity about why it's an important property, but because we already know how the objects decompose under the direct sum, this view sheds more light on our specific case.
Conclusion
Bimodules were the last missing piece needed to attack the research question. Now, I should have all the necessary theory to consider any group, although some might be extremely computation-expensive while others might just not be possible. Stay tuned to see what will work!
Week 11: Semidirect Products of Cyclic Groups
Welcome back! I've found many working groups, most of which are the semidirect product of two cyclic groups. Today, I'll discuss what this means and how we can check if they work.
The Cyclic Group Cn
I haven't discussed cyclic groups in detail yet. They are essentially the group of rotations of a regular polygon with n sides. There are two other ways to think about this. One is that they are the groups of integers under addition, with everything taken as a remainder when divided by n. The other is that it is generated by one element a with an = e (where e is the identity).
Semidirect Products G ⋊ H
I've discussed the semidirect product before, so I'll quickly review that now. An automorphism is a homomorphism from G to itself, i.e. a map φ: G -> G where φ(gg') = φ(g)φ(g). Let's consider a homomorphism from H to the automorphism group of G. This is a collection of maps φh: G -> G where φhh' equals the composition φh • φh'.
Then, the semidirect product is the set of ordered pairs (g,h) where g and h are elements of G and H, respectively. Multiplication is given by (g,h)•(g',h') = (gφh(g'), hh').
Possible φ for Cyclic Groups
Let's now consider the case when G is the cyclic group Cn, and H is the cyclic group Cm.
Automorphisms of cyclic groups are relatively easy to determine. Any homomorphism is determined from how it acts on generators, and cyclic groups only have one generator. Then the automorphism is uniquely defined by φ(a). As long as this is an order n element, we get a valid automorphism. Then these are given by φ(a) = ak, where k is relatively prime to n.
Let b be the generator of H. Then φb(a) = ak for some k. Ideally, k is not 1 -- otherwise, this is a trivial homomorphism and just gives us the normal direct product.
Now, because bm is the identity, φbm(a) = a. Because it's a homomorphism, this also equals a(k^m). Then, km should leave a remainder of 1 when divided by n.
Because we are working with modular arithmetic, we first focus on when n is a prime. A classic result from number theory is that if m and n-1 are relatively prime, then 1 is the only possibility with n | km-1. Therefore, we can only consider primes n such that n-1 shares a factor with m.
What Do We Want?
By now, we have a sufficient understanding of what the semidirect product of cyclic groups is. But why did I think that they're especially nice to consider?
In my Week 8 post, I mentioned that a key characteristic of the dicyclic group is its "sign representation". In particular, it's generated by two elements a and b, and we can take a representation that only considers the component of b. Among the conjugacy classes, very few actually have a component of b. In fact, only two of the n-3 conjugacy classes differ from the trivial representation here. This was key to the argument I discussed for the dicyclic groups, as well as the argument Naidu and Rowell used for dihedral groups.
The reason we can take this sign representation is because if we imagine splitting the generators a and b into two components, as in the definition of the semidirect product, then the multiplication of the second components is identical to that of a cyclic group. This property is shared by semidirect products of cyclic groups in general.
Therefore, we will have some one-dimensional representations. These assign all group elements (g, e) to 1.
To elaborate on the properties I want, it's probably easiest to demonstrate some working examples.
The Groups I've Found
Before I show the groups, I'll quickly re-explain how to determine the multiplication rules of the irreducible representations. Let χi be the character corresponding to Vi. Then, Vi ⊗ Vj = Σ (χiχj • χk) Vk.
In particular, Vi ⊗ Vj = Vj exactly when for all conjugacy classes (represented by some g), χi(g)χj(g) = χj(g). This is true when either χi(g) = 1 or χj(g) = 0.
C13 ⋊ C3
Now, let's look at the character table of C13 ⋊ C3. The homomorphism is φb(a) = a9 (note that 93-1 = 728 = 13•56). All character tables are taken from [1].
That may look complicated, especially because the bottom-right 4x4 square of entries is quite messy.
Firstly, ζ denotes the roots of unity, so ζ3 is a complex number with ζ33 = 1, and similar for ζ13. The terms involving the latter are not easy to work with. However, if we look more closely at the one-dimensional representations, they indeed have the property I mentioned earlier: in every column, either they give 1, or all the other characters give zero.
Let V denote the the representation corresponding to the second row. Then for any of the three-dimensional representations, V ⊗ X = X. Take the algebra A = 1 ⊕ V ⊕ V2. By following the same argument as in the dicyclic case, in the category of A-bimodules, the four three-dimensional representations each are irreducible objects with nine unique structures each. The other simple objects combine for a Frobenius-Perron dimension of 3, and since the category is integral, they must all be invertible too.
C11 ⋊ C5
Let's look at another example. Now, the homomorphism is φb(a) = a3.
Again, in each column, either the one-dimensional representations give 1 or everything else gives 0. Therefore, if A is the direct sum of the one-dimensional irreducible representations, then in the category of A-bimodules, the final two representations are simple objects, with 25 structures each. Then, the remaining simple objects account for a Frobenius-Perron dimension of five.
It is now possible that we have a two-dimensional simple object, so we need to do some more work here. However, the key property of these character tables has reduced the problem into a much simpler one, and one which we can directly tackle.
Some other character tables
For now, I'm focusing on the semidirect product of two cyclic groups. However, I've encountered many other examples of character tables which fit my desired archetype.
One is the character table of C3⋊D3:
Recall that D3 is a dihedral group, which we know also has simple conjugacy classes and a sign representation.
Another example is C32⋊C4:
I'll quickly discuss what the first group in the product is, because while it isn't related to today's blog, it's a very logical next direction. The exponent denotes the direct product, so C32 is the group of ordered pairs (a,b), where each component has independent multiplication by the C3 rules. To describe it with presentations, it's generated by elements a and b with a3 = b3 = 1 and ab = ba.
The second condition makes the group commutative. In fact, the Fundamental Theorem of Abelian Groups says that any finite, commutative group is the direct product of some cyclic groups. The Chinese Remainder Theorem, a basic number theory result, lets us strengthen that: each of the cyclic groups has an order that is a prime power.
Direct products are structurally really simple -- they just separately have the structures of their two components. Therefore, Abelian groups in general are a reasonable next step to take.
Anyway, that's for the future. For now, we just want to understand the structure of semidirect products of Abelian groups.
Conjugacy classes of Cn⋊Cm
I'll first give an exact characterization of the structure of the group. The simplest way to do this is with group presentations. However, we must first address a number theoretical question.
The choice of k
For convenience, we denote the generators of each of the cyclic groups by 1. Suppose our homomorphism is given by φ1(1) = 1k. (Here, since the operation is addition, 1k is the result of adding 1 to itself k times, which is just k.)
As before, km-1 must be a multiple of n. Since we're now considering multiplication modulo n, we restrict our attention to when n is a prime. Then, it's guaranteed that we can choose some k other than 1 such that km-1 is a multiple of n.
However, this choice might not always be ideal. For example, if m=4 and k=n-1, then k2-1 is a multiple of n, so we don't get as much freedom as we might want.
Thankfully, a standard result from number theory gives a satisfying answer. Let m' = gcd(m, n-1), which we assumed is greater than 1. Then, we can pick a k such that m' is the smallest exponent where km'-1 is a multiple of n. Furthermore, all other such exponents are multiples of m'.
This result has a fun proof, but I'll omit it for the sake of space. If you're interested, please feel free to ask! For now, one more assumption simplifies our work a lot: we assume that m is a factor of n-1. Then, m' is just m, and we pick some k as discussed above.
The group presentation
Let a = (1,0) and b = (0,1). Then a and b generate the semidirect product, and an = bm = 1. Moreover, based on the multiplication rules of a semidirect product, we have ab-1 = (1,-1) and b(ab-1) = (φ1(1), 0) = (k, 0) = ak.
Now we have our three rules. I'll rewrite these together for clarity: an = bm = 1 and bab-1 = ak. It's often helpful to rewrite the last one as ba=akb.
An important property that I've hinted at before, which is why we can consider the sign representation, is that the power of b is "invariant." This isn't a rigorous term, but essentially, to get the power of b in any multiplication, we can just ignore all the a's. We'll see why exactly this is more important once we are ready to characterize some of the representations.
Multiplication rules
We now have characterized the group, but it's usually helpful to derive some more multiplication rules to simplify calculations.
First, we can compute ba2 = (ba)a = (akb)a = ak(ba) = ak(akb) = a2kb. We can inductively generalize this to bac = ackb. (If you're interested, this can be a quick exercise.)
Now, we have b2ac = b(bac) = b(ackb) = (back)b = (ackkb)b = ackkb2. Again, we can inductively generalize this to bdac = ack^dbd.
Let's quickly check that this is sensible. We know that bm=e, so bmac should equal acbm. Indeed, the ratio between these is ac(k^m-1), and as I showed earlier, k^m-1 is a multiple of n, so ac(k^m-1) = e.
Describing the conjugacy classes (part 1)
Again, we'll focus on the case when n is a prime, and this implies that n-1 shares a common factor with m.
For some element g, the members of its conjugacy class can be expressed as hgh-1 for some h. Let g=acbd, and let h=aebf. Then h-1=b-fa-e.
First, we have hg = aebfacbd = ae+ck^fbf+d. Then, hgh-1 = ae+ck^fbf+db-fa-e = ae+ck^fbda-e = ack^f+e-ek^dbd.
As I've mentioned before, we can observe that any two elements in the same conjugacy class have the same exponent of b. Picking some arbitrary f, the exponent of a is ckf-(kd-1)e. Another common fact in number theory is that any equation of this form, where kd-1 shares no factors with n, achieves all possible residues modulo n. (The main idea is that by Bezout's Lemma, kd-1 has a multiplicative inverse modulo n). Anyway, because we reach all possible residues modulo n, any two terms acbd and ac'bd are in the same conjugacy class (when d is nonzero). In other words, the only condition is that they have the same, nonzero exponent of b.
The d=0 case is a bit different, because for some ac, the elements in its conjugacy class are of the form ack^f. To fully describe these classes, I'll introduce another idea from group theory.
Cosets
For a group G, a subgroup is any subset H which is "closed under multiplication". Specifically, for any h and h' in H, their product h•h' should also be in H.
For an element g, let gH denote the set of elements that can be expressed as gh for some h in H. The main idea is that if we write out gH for each g, any two overlapping cosets are the same. Additionally, each coset has the same size, which is the size of H. The proof of this is quite elegant, but my post is already slightly long so I'll omit it.
Therefore, if there are x unique cosets, then since these exactly partition the full group G, we have x|H|=|G|.
Describing the conjugacy classes (part 2)
When we think about cosets, the main group G is the multiplicative group modulo n. To be precise, it contains of the elements 1, 2, ..., n-1, and its operation is multiplication, followed by taking the remainder when divided by n. The subgroup H is generated by k, so it contains the powers of k. Since km=1, and since m is the smallest such exponent, H as m elements. The conjugacy classes of elements ac are exactly described by cosets, and because G has n-1 elements, there are (n-1)/m of these.
In summary, we have one conjugacy class for each of the m-1 nonzero exponents of b. We have (n-1)/m conjugacy classes for powers of a. We also have one final class, which only consists of the identity (which we can just call 1).
The one-dimensional representations
We've already done most of the work to describe the one-dimensional representations.
Recall that the commutators are elements of the form ghg-1h-1. If we generate these as h=acbd and g=aebf, then we already computed that ghg-1 = ack^f+e-ek^dbd. Since h-1=b-da-c, we have ghg-1h-1 = ac(k^f-1)-e(k^d-1). By similar logic as before, if we arbitrarily pick c, f, and some nonzero d, then by varying e, we achieve all powers of a. Then there are n commutators (the powers of a, since an=1).
As I mentioned in my Week 6 post, the number of one-dimensional representations is the size of the group divided by the number of commutators. The former is mn, so there are m one-dimensional representations.
When describing a representation, we only need to define how it acts on the generators. As I've mentioned before, these resemble the "sign representations" in that they entirely neglect components of a. To formalize this, if we call the representation ψ, then ψ(a)=1.
We still need the other two relations from the presentation: ψ(b)m=1 and ψ(b)ψ(a)ψ(b)-1=ψ(a)k. The latter is automatic, because we can ignore the a terms and the rest cancel. The first tells us that ψ(b) can be any complex m-th root of unity.
There are m such roots of unity, so there are m one-dimensional representations that neglect a. In fact, this matches the total number of one-dimensional representations, so we've fully characterized them!
The other representations
First, let's find the dimensions of these representations! As I discussed in my Week 6 post, the sum of the squares of each irreducible representation's dimension is the size of the group, which is mn. Also, the number of irreducible representations is the number of conjugacy classes, which is m+(n-1)/m.
We already have m one-dimensional representations, so there are (n-1)/m remaining representations, and the sum of the squares of their dimensions is m(n-1).
You may immediately guess that all remaining representations therefore have dimension m. This does hold in all relevant character tables I found online! However, it's not so easy to show, and we don't need it just yet. It still seems like a fun exercise, so I might try it later. A possibly important idea is that the conjugacy classes given by cosets are basically indistinguishable, so we can try to "clone" a working representation.
While I haven't yet figured out how to prove this, I did succeed in proving the main condition.
Checking the key property
The original reason I noticed these groups are nice to consider is because of the property I mentioned earlier: that in each column, either the one-dimensional characters all give 1, or everything else gives 0. Here, I'll prove that our semidirect product satisfies this property.
Let the one-dimensional characters be 1, χ, χ2, ..., χm-1. The representatives of the relevant conjugacy classes are just powers of b, so if χ(b)=ζ, then χc(bd)=ζcd.
The critical idea is that the sum Σ ζ-idχi is m for bd, and zero for everything else. This is a fairly standard computation with roots of unity; again, I'll omit it due to this post already being just a bit too long.
Given this fact, because χ' • χi = 0 for any multi-dimensional character χ' and integer i between 1 and m-1, by linearity χ' • Σ ζ-idχi = 0. Therefore, χ'(bi) = 0, as desired.
Conclusion/Summary
That was a really long post, so I'll take some more time to summarize it. The main update is that I've found a few new groups that answer my research question. I discussed why these work, and I also looked quite deeply at the general case. Specifically, I described the structure of the semidirect product of Cn and Cm, where n is a prime and m is a factor of n-1. I classified the conjugacy classes, and determined a few key properties of the character tables.
This isn't yet enough to fully prove the general case (although it does suffice when m=3). However, the remaining work seems quite similar to proving that Abelian groups satisfy the research question, so I expect that these will all work.
Finally, there are a few ways in which we can further generalize this case. I'm not sure if they'll succumb to a similar argument, but at least we can determine the conjugacy class composition. I didn't include them in this post because it's already long enough, but they might be promising future directions.
References
https://people.maths.bris.ac.uk/~matyd/GroupNames/characters.html
Week 11: 5: Semidirect Products of Cyclic Groups (continued)
Welcome back! In my previous post, I laid most of the groundwork to attack one specific case of these semidirect products: Cp⋊Cn, where p is a prime that gives a remainder of 1 when divided by n. Today, my goal is to finish this proof. To do this, I'll need to take a new approach.
Solving the Cp⋊Cn Case
A new approach
First, I'll discuss the change I'm making to my approach. Initially, I was following Naidu and Rowell's arguments: to show that the category of bimodules is pointed, we find enough simple objects to guarantee that all simple objects are invertible. This essentially "counts" total Frobenius-Perron dimension.
However, a complexity in this approach is that we need to be able to exactly count how many bimodule structures each object has. To get around this, we can instead model our approach after the work in [1]. Specifically, we chase the loftier goal of directly classifying simple bimodules. I expect that this is usually the harder approach, and that may even be true in this case; regardless, I usually like more theoretical arguments so I'm more comfortable with this approach.
The first step is to classify simple bimodules which are made of higher-dimensional representations. First, we need to understand more about their characters.
Higher-dimensional representations in the character table
In my previous post, I showed the following:
For some χ, the one-dimensional representations are given by the powers of χ. Furthermore, for all higher-dimensional representations V, χ⊗V = V.
I also explained why I'd expect that all other representations have dimension n. I didn't actually prove that, because I hadn't yet figured it out and I didn't think it was important. As I've worked more on this problem, however, I've realized that it is necessary. Fortunately, I've also found a beautiful proof. To understand this, we first need an idea from linear algebra.
Eigenvalues of similar matrices
Two matrices A and B are called similar if we can write B = PAP-1 for some invertible matrix P. Then, a key fact is that any two similar matrices have the same set of eigenvalues. In fact, these matrices are each similar to their Jordan Canonical Form. I won't discuss this in too much detail, but it's a matrix which has their eigenvalues along its diagonal.
This fact is important in its own right, but another corollary we'll use relates to traces. Recall that the trace of a matrix is the sum of its diagonal elements, and that any two similar matrices have the same trace. The trace of a Jordan Canonical Form is, by construction, the sum of its eigenvalues. Therefore, in general, the trace of a matrix is the sum of its eigenvalues.
If we use this fact along with the argument I'll soon make, we can even fully write the character table. However, I don't actually need the entire table, so all I'll use for now is the fact that two similar matrices have the same eigenvalues.
Eigenvalues of the a terms
Let the representation map a to the matrix A. Then it maps ap to Ap. Because ap is the group identity, Ap must be the identity matrix.
First, I'll briefly discuss why A has an eigenvalue other than 1. If A's eigenvalues were all 1, then A would be the identity matrix itself. Then, the resulting representation would just be a representation of Cn, trivially extended to our semidirect product. Because Cn is a commutative group, its irreducible representations are only one-dimensional, so the representation we're considering can't be irreducible.
Now, let λ be an eigenvalue of A other than 1. Then λp is an eigenvalue of the identity matrix, so λp = 1. Therefore, λ is a p-th root of unity.
We also know that λ2 is an eigenvalue of A2, and generally, λk is an eigenvalue of Ak. Therefore, each power of λ (other than 1) is the eigenvalue of some power of A (other than the identity matrix).
The elegant finish
Because λ is a p-th root of unity, these p-1 powers of it are all distinct. Furthermore, they are split among the conjugacy classes of the powers of a. As I showed last post, there are (p-1)/n such classes.
Each class has one set of eigenvalues, and the number of eigenvalues is the representation's dimension. Therefore, for (p-1)/n classes to total at least p-1 eigenvalues, the representation must have dimension at least n.
Finally, another calculation I discussed last post is that among the remaining (p-1)/n representations, the sum of the squares of their dimensions is (p-1)•n. Combining this with the fact that each dimension is at least n, we can conclude that each dimension is exactly n, as desired.
Classifying the simple bimodules
Let A be the unique semisimple algebra structure on 1⊕χ⊕...⊕χn-1.
Let M be any simple bimodule which contains a copy of some higher-dimensional representation V. By Lemma 6.2 of [1], M must exactly be V.
Now, suppose that some simple bimodule M is composed only of the χi representations. Note that χiM is also a simple bimodule for all exponents i.
Now, let's look at the bimodule M⊕χM⊕χ2M⊕...⊕χn-1M. We just showed one way to decompose it in terms of simple bimodules. Now, let's find another.
By the distributive property, we can factor this as A⊗M. Now, recall that M is composed of powers of χ. Let's apply the distributive property again for this decomposition. The key idea is that for any exponent i, A⊗χi is still A. Therefore, A⊗M is just some tensor power of A.
In a semisimple category, the decomposition of an object into the sum of simple objects must be unique (just as how prime factorizations are unique). Therefore, we must have M=A.
The finish
Now, we just need to show that these simple objects are invertible in the bimodule category. We do this by looking at their Frobenius-Perron dimension.
Notice that the simple bimodules are all either higher-dimensional representations, or A itself. Importantly, the first thing we proved today is that all higher-dimensional representations have dimension n. Therefore, all the simple bimodules have the same dimension in our original fusion category, and this matches the dimension of A.
When we switch from our original fusion category to the category of bimodules, the Frobenius-Perron dimensions are proportional. Since A must have dimension 1 in its bimodule category, so must everything else. Therefore, all simple objects have Frobenius-Perron dimension 1 in the bimodule category, so they are invertible. Consequently, this category is pointed, so our original category is group-theoretical.
Conclusion
Today, I finished the proof that I started in the previous post. This is a quite general result, and it seems pretty important. Last week, I mentioned the case where we replace Cp with any Abelian group, or where we replace n with any positive integer that shares a factor with p-1. I've worked more on each of these cases, and they don't seem like something I can tackle at the moment. I have made some progress on the case where we replace p with any prime power, but this is still quite messy and I don't expect it to work out cleanly.
[1] P. Etingof, S. Gelaki, V. Ostrik. Classification of fusion categories of dimension pq, International Mathematics Research Notices (2004)