One of the things I find most fascinating is the interplay between mathematics and physics. One direction of this interplay is well-known i.e., how mathematics forms the basis of all physical concepts. In fact, this is now so ingrained in our minds that we naturally expect any new physical concept to have an appropriate mathematical description. Wigner calls this the “unreasonable effectiveness of mathematics” in his famous article. What is surprising, perhaps, is the other direction i.e., how physical intuition leads to new developments in mathematics. This is especially true with quantum physics and geometry.

Quantum field theories such as Yang-Mills and Chern-Simons theories provide a deep understanding of certain aspects of topology and geometry. Starting around in the 80s, these developments have led to the creation of the field of quantum topology and quantum algebra. Some of these results, which took mathematicians by surprise, were obtained from deep physical intuition. These developments are beautifully explained in a 2010 article by Atiyah, Dijkgraaf and Hitchin. For instance, understanding the space of the so called self-dual solutions of Yang-Mills theories leads to powerful invariants of four manifolds. Similarly, studying the holonomy of Wilson loop operators and their vacuum expectation values gives a new description of the Jones polynomial.

A basic and extremely important structure in mathematical physics and in the results mentioned above is that of a vector bundle. Several physical theories such as Yang-Mills theory and Chern-Simons theory can be cast in terms of vector bundles. Moreover, natural bundles on manifolds such as the tangent and cotangent bundles are useful in descriptions of general relativity and the phase space of classical mechanical systems (the latter, in fact, leads to the very useful concept of geometric quantization, which produces a quantum mechanical Hilbert space from the classical mechanical phase space). The concept of holonomy in a vector bundle, which measures the deviation produced in a standard vector when it is parallel transported around a loop, turns out to be the underlying mathematical description of the Aharonov-Bohm effect and the quantum field theoretic formulation of the Jones polynomial. These are just some examples of the use of vector bundles.

In this post, I want to discuss some aspects of vector bundles that are useful in the above applications. Aside from the definition and examples, I will talk about how finite dimensional vector bundles can be classified using homotopy.

**2. Definitions**

**2.1. Vector bundles**

A dimensional vector bundle over a -dimensional manifold can be specified as a triple , where is a projection map such that for every point on , the inverse image of is a finite dimensional vector space over some field. In addition, we need two other conditions.

**Local triviality.**This condition states that there exists an open cover of the manifold such that when the bundle is restricted to an open set from the cover, it is isomorphic to the trivial bundle i.e., there is a map (when is a dimensional real vector space).**Continuity of transition functions.**We also require that the maps defined above be homeomorphisms i.e., they are continuous maps of topological spaces. This means that on intersections of open sets of the cover, if we define the functions , we have that are homeomorphisms. The (with ) are called*transition*or*clutching*functions since they tell us how to identify the trivial bundles on different open sets of the cover. The group they generate is called the*structure*group.

It is easy to see that for a local trivialization , the transition functions also satisfy the cocycle condition i.e., is the identity. Since the transition functions are essentially invertible matrices, we can define the equivalence of two sets of transition functions and as the equivalence of the matrices representing them. In other words, and are equivalent if, on each open set , there are functions such that . The *structure group* of the vector bundle is the group in which the transition functions lie. Usually, this is . However, sometimes the structure group can be made into a subgroup of . Using the equivalence of transition functions, we can define the reduction of the structure group of a vector bundle. Suppose for a set of transition functions, we can find equivalent ones that lie in a subgroup of , then we say that the structure group is reduced (to that subgroup).

**2.2. Sections**

An important concept pertaining to a vector bundle is the notion of a section of a bundle. This notion and its properties naturally lead one to consider generalizations of vector bundles called *sheaves*. A section of a vector bundle is a map such that i.e., applied to any point takes it to a vector in the space over . Sections are important because they contain important information about the vector bundle. An example of a section is a vector field on a manifold, where the vector bundle is the tangent bundle of the manifold. Sections are useful to figure out if two vector bundles are isomorphic or not. For example, over the circle , one can construct two line bundles–the trivial bundle or the Möbius bundle defined below. Using the notion of the zero section (or rather, its complement), we can show that these two bundles are not isomorphic. The zero section is a map from to that simply assigns the zero vector to every point on the manifold (i.e., the zero vector in the vector space above that point). This map need not always be smooth. However, for the two vector bundles over , it is smooth. The complement of the zero section for the trivial bundle over is clearly not connected, where as in the Möbius bundle, it is connected. This means that they are not isomorphic as vector bundles. In fact, using the classification of vector bundles discussed later, we will see that these are the only possible non-isomorphic line bundles over .

In the discussion above, we have been talking about real vector bundles, but all of the definitions and concepts can be extended to complex vector bundles (or more specifically holomorphic vector bundles). To obtain complex vector bundles, we need to replace (in the definition of vector bundles above) the base manifold and the fibers by their complex counterparts. If we make the maps and holomorphic (i.e., functions that satisfy the Cauchy-Riemann conditions), then we get holomorphic vector bundles.

**3. Examples**

- The trivial bundle , where the map is simply the restriction to the first factor in the direct product. It is easy to verify that it satisfies the two conditions in the definition of a vector bundle.
- The tangent (and co-tangent) bundles on a manifold are, in a sense, the “first” vector bundles. Arbitrary vector bundles are generalizations of these.
- Möbius line bundle is a quotient of the trivial bundle by the equivalence relation which identifies with , where . This is one of the vector bundles over discussed above.
- The Grassmannian is the space of dimensional subspaces of . In other words, any point on the Grassmannian manifold is a dimensional vector space . One can define a bundle called the tautological bundle by taking the fiber over any point in the Grassmannian as the vector space itself.
- Another important class of bundles are
*pullback*or*induced*bundles. Suppose there is a map between two manifolds , then one can pullback any vector bundle on to as follows. The fiber over any point is defined as , where is the projection map of the bundle over . - Using the standard constructions over vector spaces such as direct sums, tensor products, complements etc., one can define equivalent constructions of vector bundles over a base manifold by defining them fiber-wise.
- The direct sum or
*Whitney*sum of vector bundles can also be thought of as the pullback bundle of the diagonal embedding map taking to .

**4. Classification of bundles**

In order to understand the different types of vector bundles one can construct over a manifold, we need to know what it means to say that two bundles are the same i.e., we need the notion of isomorphism of vector bundles. A morphism of two vector bundles and is a morphism of the base spaces and a fiber-wise linear map between the total spaces. In other words, it is a pair of maps such that the following diagram commutes.

Here the vertical arrows are the projection maps of the respective vector bundles. The map is defined fiber-wise. When restricted to any fiber over a point in , it is a *linear* map to the space over the point in . Using this, we can define isomorphism of vector bundles as a morphism such that the composition of it and its inverse is the identity. Once we have a notion of isomorphism of bundles, we would like to know how to classify them up to equivalence.

It turns out that the Grassmannian and its tautological bundle defined above play an important role in the classification of finite dimensional vector bundles. First, notice that we can extend the definition of the Grassmannian above to subspaces of an infinite dimensional vector space i.e., define is the set of -dimensional subspaces of . Similarly, we have the complex Grassmannian .

The first step is to show that homotopic maps between two manifolds lead to isomorphic induced bundles. To be more concrete, suppose that and are two manifolds and let be a vector bundle over with a projection map . Let and be two maps from to that are homotopic. Each of these maps induces a pullback bundle (described above) from to . Let us denote these two bundles on as and . One can show that these two vector bundles over are isomorphic.

Using this, it can be shown that the equivalence classes of -dimensional vector bundles over any compact (or paracompact) topological space are in one-to-one correspondence with equivalence classes of maps (under homotopy) from to . Put another way, this means that every equivalence class of -dimensional vector bundles over a manifold can be obtained as a pullback bundle from the tautological bundle over via some equivalence class of homotopic maps. The space of equivalence classes of maps from to is denoted . So for any manifold , if we have a good understanding of the space , then we would also have an understanding of all possible vector bundles over .

One can apply this machinery to line bundles over and prove that there are only two isomorphism classes of complex line bundles. Using the classification theorem, we need to know the dimension of the space . First, note that is the infinite projective space ). So we need to find out the dimension of the space . This can be done using Eilenberg-McLane spaces and their properties.

Denoted , an Eilenberg-McLane space is one that has the homotopy group isomorphic to and others trivial. These spaces can be used to construct more complicated spaces with non-trivial homotopy groups of all orders. An important property of Eilenberg-McLane spaces is that i.e., the equivalence classes of maps from a manifold to an Eilenberg-McLane space is isomorphic to the singular cohomology group. It turns out that the infinite projective space is an Eilenberg-McLane space, . This means that . For the manifold , can be calculated to be equal to . Thus one can deduce that there are only two isomprphism classes of complex line bundles. In terms of characteristic classes, this just turns out to be the first Chern class and these line bundles are completely characterized by the first Chern class.

]]>The reason for optimism is that in 2008 and then again in 2012, when the Republicans lost the general election, they did a lot of soul searching. In fact, they hired a few consultants to look into their failure and provide recommendations. That resulted in a 100 page report (you can see it here), which the press has since called the autopsy report. The report was a result of analysis and data collected from thousands of interviews. The report makes several recommendations, the most important of which is that the RNC should do a lot more to embrace women and minorities, especially hispanic voters. The Republicans should make them feel more welcome and not appear to be a party that wants them to disappear. The autopsy report was taken very seriously by many in the party. Ted Cruz and Marco Rubio organized their campaign strategies around it. Paul Ryan and even Newt Gingrich praised it as a model for the party in the coming years. It seemed at the time that the Republican party is moving away from far right tendencies or even away from social conservatism (that always seemed like a bit of a stretch). Of course, this was just a strategic move and not a major idealogical shift for the party. But still, this is not the direction that a *far right* party or a party mostly influenced by a far right ideology would take.

Trump, who derided the report from the beginning (even before he entered the race), did everything that flies in the face of the report. I don’t have to recount all the ways in which he did this. This, of course, made many in his party very uncomfortable and it seemed for a while that the party was tearing itself apart. But when he started getting more and more support and eventually won the primary, it seemed clear that the far right movement is not that small. But how big is it? It wasn’t clear and back then, I don’t think anyone really knew. That was also the time when people talked of a Clinton landslide. If Clinton had won (landslide or not), the Republican party would have been madder than ever at these Trump nationalists and it seemed to me that Trump and his supporters would have ended up starting their own party. That made a lot of sense because how could the US not have a far right movement when it is only growing everywhere else. That would also weaken the republican party and they would have to do more soul searching to reinvent themselves. I thought and even hoped at the time that this is the direction things would move.

But this unexpected and unthinkable election outcome has shown how flawed that thinking is. It has become abundantly clear that the far right movement is much stronger than anyone imagined. In fact, the US has had a far right party all along—the Republican party and it is not in the fringes like in western Europe. The far right is now the mainstream in the United States and it has all three branches of the government at its mercy.

]]>Since we are dealing with the orthogonal group (i.e., the fundamental representation is isomorphic to its conjugate), is the trivial representation of the diagonal action. So, it is easy to see that the contraction map commutes with the diagonal action. It turns out that there are no more elements that generate this algebra. This algebra is called the Brauer algebra. It is a diagram algebra in the sense that its basis elements can be represented using diagrams. Another interesting generalization is the action of the unitary group, where it action on as . This is equivalent to conjugation by using the Choi-Jamiolkowski isomorphism. It turns out that the centralizer algebra of this action of the unitary group is the so called walled Brauer algebra. This algebra is generated by the following elements: any permutation of the first tensor copies, any permutation of the last copies and contraction maps between pairs of tensor copies, where one of them is from the first half and the other from the second half. It is again easy to see that these elements (and the algebra generated by them) are in the centralizer. However, as always, proving that the centralizer is generated by these (and no more) elements is the hard part.

The generalization that I want to talk about next is the duality between the Hecke algebra (of type ) and the quantum group . Before we talk about the duality itself, I want to talk about each of these algebras.

**1. Hecke algebra**

A Hecke algebra is sometimes called a quantum generalization of the symmetric group although -deformation of , which is also used, is probably more appropriate. It is generated by elements (and their inverses) for , which satisfy the following relations.

which are the usual relations in the braid group and in addition, we have the relation

There are several types of Hecke algebras and the above one is usually called Iwahori-Hecke algebra. The irreducible representations of this algebra have been worked out by Wenzl. They bear a striking resemblance (perhaps not surprisingly) to the representation theory of . It turns out that the irreducible representations of can also be labeled by Young diagrams with boxes. When is not zero or a primitive root of unity, all the Young diagrams with boxes correspond to irreducible representations. For the case when is a root of unity, the irreducible representations belong to a restricted set. First, let us look at the case when is any complex number but not zero or a root of unity. In this case, it turns out that is isomorphic to the symmetric group algebra. The action of a generator inside an irreducible representation can be described as follows. Suppose is a standard Young tableau of shape and let be the tableau with and interchanged. Recall that a tableau can be identified with a path in the Bratelli diagram of the symmetric group. Let be the Manhattan distance between and in and let denote the quantum integer

Quantum numbers (or numbers) can be extended to operators, where is replaced by an operator . Using this notation, we have

if and are not in the same row or column. If they are in the same row or column, then one can put to be zero above. This action bears striking resemblance to the action of a transposition of the symmetric group inside an irreducible representation.

Now, moving onto the case when is a root of unity (say ), it turns out that the Hecke algebra is not semisimple. The irreducible representations of are still labeled by Young diagrams but with a restriction. One only considers the so-called Young diagrams, where . The Jones representations of the braid group are for . A Young diagram has at most parts and . Let the set of all box diagrams be . Now a standard tableau is one whose path in the Bratelli diagram only passes though other diagrams. It turns out that on the space of such paths, there is an irreducible representation of (denoted ). For a Young diagram, recall that the standard tableau of shape span the irreducible space of the symmetric group (or the Hecke algebra at non-zero or non roots of unity). For a root of unity, the irreducible space is a subspace of since we consider only standard tableau. In fact, it is an open problem (as far as I know) to obtain a closed form expression for this dimension (although recursive procedures to calculate it exist). The action of a generator in this irreducible space is given as follows ( and have the same meaning as before).

If and are in the same row, we get and if they are in the same column, . This description of the Hecke algebra has been worked on by Wenzl in this paper. Wocjan and Yard in this paper, use this to construct an efficient quantum algorithm to approximate the Jones and HOMFLYPT polynomials.

**2. Hopf algebras and quantum groups**

Now let us look at the algebras that make up the dual side. These are -deformations of as one might guess (since on the “primal” side, we have a deformation of the symmetric group). However, the way to -deform them is far from obvious. In order to describe this, let us look at Hopf algebras and quantum doubles of (compatible) Hopf algebras. It turns out that the quantum double construction gives us the appropriate -deformation of the universal enveloping algebra . Hopf algebras are the language in which one can describe the quantum double construction. This construction, proposed by Drinfeld, has the very important property of producing Hopf algebras that are braided. In other words, the doubled Hopf algebra comes equipped with so called universal matrices, which satisfy Yang-Baxter equations (more precisely their quantum versions). Kassel’s book has a nice description of Hopf algebras and quantum doubles and a lot of the discussion below is based on that.

A Hopf algebra is a vector space which has an (associative) algebra structure (i.e., with a multiplication map and a unit map defined in the usual way) and something called a (coassociative) coalgebra structure, which comes with comultiplication and a counit. Moreover, the algebra and coalgebra structures must be compatible in the sense that the coalgebra structure must be an algebra homomorphism (or vice versa). More precisely, a coalgebra is defined with respect to comultiplication and a counit . A comultiplication can be thought of as a description of how acts on the tensor product of two representations. For instance, for a finite group (which is also a Hopf algebra), , which corresponds to the diagonal action. For a general Hopf algebra, it can be a linear superposition of tensor product terms. The Sweedler notation is useful when dealing with the general case. In this notation, one denotes or sometimes written as . This stands for the sum . The point is that it gets very confusing to keep track of all the variables and indices if we introduce more than one for each . So in the Sweedler notation becomes

By coassociativity, this is equal to , which is

Since the above two equations are the same, we can drop the extra indices and write

In fact, sometimes the summation symbol itself is dropped. While there is no ambiguity in dropping the summation symbol because the indices are enough to remind us that there is a sum, it may be better to have the to keep things explicit.

The counit is a map from the coalgebra to the complex numbers i.e., . This can be thought of as being equivalent to the trivial representation. In other words, we have . The compatibility condition of the two structures implies that and i.e., they are algebra homomorphisms. We also need , where is the unit map of the algebra structure and is the identity of . In fact, so far we only have a bialgebra. In order make a Hopf algebra, one also needs an antipode with the property that . The antipode generalizes inverses in . In fact, in a group, is the usual inverse i.e., . A Hopf algebra is called commutative if the algebra structure is commutative i.e., and it is called cocommutative if the coalgebra structure is cocommutative i.e., , where is the operator that swaps the tensor copies.

Once we have a Hopf algebra, we can impose extra structure on it that makes it more interesting. One important class of Hopf algebras are called braided or quasitriangular Hopf algebras. These are interesting because they give rise to solutions of the Yang-Baxter equations and hence to representations of the braid group. A braided Hopf algebra has a so called matrix which is an invertible element of satisfying the following equations.

which says that is cocommutative up to conjugation by . In addition, we need

In the above, if , then and similarly for the others. It is easy to show that a braided Hopf algebra satisfies the quantum Yang-Baxter equations, which are given by

**3. Quantum doubles**

Now, I would like to describe a construction by Drinfeld that gives systematic solutions to Yang-Baxter equations. It also gave one of the first examples of Hopf algebras that are neither commutative nor co-commutative. The construction takes a Hopf algebra and its dual and combines them in a fashion that looks like a semidirect product on groups (actually it is closer to another operation on groups called bicrossed product). The result is a Hopf algebra that is braided and hence has matrices that give rise to braid group representations.

Suppose that is a Hopf algebra. Recall that and are the multiplication and unit maps of the algebra structure and and are the comultiplication and counit maps. Now, one can construct its dual and this turns out to be the Hopf algebra . Technically, the dual is not of , but of the so called opposite Hopf algebra . In other words, the multiplication in is , which takes to rather than . It’s not hard to see that for this dual, the algebra and coalgebra structures are exchanged – with a small change: the comultiplication of the coalgebra structure comes from the opposite algebra. On the tensor product of these two Hopf algebras denoted , one can obtain a Hopf algebra structure as follows. First, as a vector space, it is . It is easy to describe the coalgebra structure, so let us do that first. The comultiplication map is simply a tensor product of the individual comultiplication maps (and then a rearranging of the tensor copies). In other words,

where swaps the second and third tensor copies. The counit is also the tensor product of the individual counits i.e., , which is . One needs to work harder for the algebra structure. In order to define algebra structure, a construction known as bicrossed products of algebras is exploited. Bicrossed products can be defined on a pair of algebras that are compatible. This compatibility is the main reason we need the dual to be rather than . Another way to look at the product might be useful. We think of as consisting of creation and annihilation operators. The creation operators can be thought of as belonging to and the annihilation operators to . Then, defining the product means defining the normal order. So in order to define what we mean by , we have to define how to normal order the expression. Rewriting the expression in more shorthand notation as , we need to normal order it i.e., commute past . Using bicrossed products, this turns out to be

where and is the natural embedding of and into . Typically, in a normal ordering, one has annihilation operators to the right. This can also be done by modifying the above formulae appropriately.

Once we have the quantum double, one can obtain the universal matrices since is braided. The expression for the matrices turn out to be

where is some basis of and is the dual basis. Here even though we have considered the matrix in what can be thought of as the fundamental representation of the doubled Hopf algebra, one can extend this definition of the matrix to any pair of representations of the double. In other words, if and are two finite dimensional representation of , then on the tensor product of the two, the matrix will be

where the s are the identity on the respective spaces.

**4. The quantum group and its representations**

It turns out that one can construct -deformations of Lie algebras as quantum doubles of the Borel subalgebras. More precisely, let (resp. ) be the Borel subalgebra consisting of the Cartan subalgebra and the set of positive roots (resp. negative roots). One can show that the universal enveloping algebras and the (and their quantum versions) are Hopf algebras dual to each other. The quantum double of (the quantum versions of) these two is denoted . Let us now take a closer look at the structure of the quantum group .

is defined to be the following algebra generated by and .

and

A couple of remarks:

- While one cannot simply substitute in the above equations to obtain , it turns out that there is an equivalent formulation of the above algebra where one can substitute and obtain the usual universal enveloping algebra as .
- Sometimes is written as to obtain a definition that looks closer to the defining relations of .

The Hopf algebra structure on is given by

and the group like elements

The map is zero for and and for and . The antipode is defined as

The matrix can be obtained from the fact that it is a quantum double (i.e., using the Eq.~(14)) and turns out to be

where and . It is easy to see that the matrices generate the Hecke algebra. Identifying with , we can see that it satisfies the definition of the Hecke algebra (the Yang-Baxter equation turns into the braid group condition). Another consequence of the braiding is that the matrices and hence the Hecke algebra commutes with the action of the quantum group. This can be seen from the equation , which essentially expresses the fact that the action of the quantum group commutes with .

The irreducible representations of for a root of unity are different from the case when is a root of unity. First, let us look at the simpler case of when is not a root of unity. In this case, the irreducible representations can be labeled by the highest weight , which in turn depends on an integer as , where . Given an , one can construct an dimensional irreducible module with highest weight . If is the highest weight vector, then we have and . Defining , we obtain a basis of the finite dimensional module. The action of the generators on this space is as follows.

It turns out that this module is irreducible, any simple finite dimensional module is generated by a highest weight vector and any two finite dimensional modules with the same highest weight are isomorphic. In this sense, the structure of finite dimensional simple modules parallels that of Lie algebras.

The case when is a root of unity is more complicated since one can have nilpotent elements. First define to be the order of (smallest integer such that ). Then define if is odd and if it is even. The source of the complications comes from the fact that and, in fact, for any multiple of . In spite of this, it turns out that simple modules of dimension smaller than are the same as for the generic case and that there are no simple modules of dimension greater than . For dimension equal to , there turn out to be two kinds of simple modules. These are described in Kassel’s book.

**5. Ribbon Hopf algebras**

There is a special kind of structure one can impose on a Hopf algebra that makes it a ribbon Hopf algebra. The quantum groups discussed above happen to be ribbon Hopf algebras. Exploiting this structure makes it easy to calculate irreducible representations of centralizer algebras of quantum groups. This viewpoint is nicely developed in this paper by Leduc and Ram. A ribbon Hopf algebra is a braided Hopf algebra with a ribbon element i.e., an invertible element in the center of the algebra. Formally, a ribbon Hopf algebra is braided (i.e., has matrices) and has an element such that

and

In the above equations the element is

where and come from the matrix i.e.,

For a quantum group , where is a simple Lie algebra, the ribbon element is , where is the so called Weyl vector (half the sum of positive roots).

**6. Quantum Schur-Weyl duality**

It turns out that there is a Schur-Weyl duality between the quantum group and the Hecke algebra of type on the fold tensor product space , where is the dimensional representation of . However, when is a root of unity, the actions of and turns out to be non-semisimple even though they are full commutants of each other. But first, let us look at the more straightforward case (or generic case) when is not a root of unity. In general, this duality extends to the quantum group corresponding to a simple Lie algebra , where is the fundamental representation of . The full centralizer of the tensor product action of on copies of the fundamental representation of is either an Iwahori-Hecke algebra or the Birman-Wenzl-Murakami (BMW) algebra depending on whether is of type , , or . Type gives the Iwahori-Hecke algebra and other types give the BMW algebra.

**6.1. Generic case**

In this case, it is easy to show that any finite dimensional module is semisimple. It also turns out that is semisimple (although it follows from the facts that action is semisimple and that is the full centralizer of that action). This means that we can decompose the tensor product space into irreducible spaces corresponding to the two actions and these spaces can be labeled by Young diagrams just as in classical Schur-Weyl duality.

where and are irreducible spaces of and respectively. This duality can be extended to and . The proof of this in the generic case is not so hard. There are several ways to prove it. One uses semisimplicity of the two algebras. One can also use the fact that the Hecke algebra is isomorphic to the symmetric group algebra in this case, then use the fact that their dimensions are . We have seen that the Hecke algebra is generated by the universal matrices (composed by the tensor swap). This means that actions of the quantum group and the Hecke algebra are already in the centralizer of each other.

**6.2. Roots of unity**

Things are not so simple in this case. First, the algebras (Hecke and the quantum group) are not semisimple. Although they are still full commutants of each other, it is not very satisfying to have nonsemisimple algebras. Moreover, this is the case that seems interesting from the point of view of link invariants and topological quantum computing. This is because the link invariants at roots of unity are more powerful and include Jones and HOMFLYPT polynomials. From the point of view of topological quantum computing, the representations of the braid group that one obtains at the roots of unity are dense in the unitary group and hence are `universal’ in the sense that one can express any unitary as a sequence of braids to arbitrary precision. One can appeal to the Solovay-Kitaev theorem to show that the number of crossings is polynomial in the number of strands.

An interesting question is whether there are quotients of these algebras (say and ) that are full centralizers of each other and are semisimple. For instance, could be the following quotient of . Recall that we defined earlier. Let be the direct sum of , where runs over all diagrams. The action of restricted to this space is preserved because this space consists of simple modules. Because of non-semisimplicity, the action of on the complement of this space need not be preserved. In this paper, Goodman and Wenzl analyze the Littlewood-Richardson (LR) coefficients of this quotient and find that there is an interesting duality called level-rank duality (which one finds in the fusion rules of certain conformal field theories). These LR coefficients themselves occur in certain WZW models. In fact, the irreducible representations of can be used to express the Jones polynomial and the one variable HOMFLYPT polynomial of a the trace closure of any braid. Suppose is a braid on strands, then the Jones polynomial of the trace closure of the braid at a root of unity is given by

where is the sum of the exponents of the generators in the word for (recall that any element can be expressed as a word in and ). Here we denote by the element of the Hecke algebra corresponding to the braid (by a slight abuse of notation). The quantity has the following expression

where recall that is the representation of the Hecke algebra inside a Young diagram and Tr is the usual trace.

The Schur functions are given by the following formula.

where is the usual hook length of the box with coordinates . The above formula for the HOMFLYPT polynomial suggests that the Schur functions are proportional to the dimensions of irreducible representations of the commutant of the quotient . The following are some very interesting questions that I don’t know the answers to.

- What quotient of is the commutant of ? The dimensions of its irreducible representations should be proportional to . How does this quotient relate to the WZW models mentioned above?
- What is the space on which acts? Presumably, this space does not have a tensor product structure. Of course, one way to specify it is as the space of all paths on the Bratelli diagram, but it would be interesting to know how to project on to this space.
- How can we write in terms of generators and relations? Since it is a quotient of , there must extra relations that satisfy.

**1. Statement of Schur-Weyl duality**

Schur-Weyl duality (in one form) says that the diagonal action of the general linear group commutes with the action of the symmetric group on the tensor product of finite dimensional vector spaces. Consider the -fold tensor product of a finite dimensional vector space of dimension . We have the diagonal action of the general linear group , which applies an invertible operator on each tensor copy. For this action of the general linear group, we are interested in the centralizer algebra i.e., . It is easy to see that the set of operators that permute the tensor copies are in this algebra and the question is – are there more? Schur-Weyl duality is the statement that there are no more operators and so, this centralizer algebra is isomorphic to the symmetric group algebra. We will actually consider the action of the so-called universal enveloping algebra of the Lie algebra instead of the general linear group. Schur-Weyl duality can then be stated as

Theorem 1The image of and the image of in are centralizers of each other.

Before we prove this theorem, let us recall some facts about universal enveloping algebras of Lie algebras.

**2. Universal enveloping algebras**

A universal enveloping algebra of is a unital associative algebra such that its Lie algebra is . In order to motivate this, first recall that a Lie algebra is not associative. In fact, the Jacobi identity expresses this non-associativity. But, why do we care about having an *associative* algebra? The main reason is that we are usually interested in representations of Lie algebras rather than the algebras themselves. This means that if is a representation of the Lie algebra, then one would like to say that , where is the multiplication in the Lie algebra. But this is not the case with Lie algebra representations. We would like to have an associative algebra, so that we have this property for its representations.

Now, from any associative algebra , one can construct a Lie algebra by simply defining the Lie bracket to be the commutator i.e., . It is easy to check that this satisfies the axioms of a Lie algebra. So, a natural question is the converse, that is, given a Lie algebra , what is the associative algebra such that ? We do not know if such an object exists and even if it exists, it may not be unique. For some problems, existence can often be shown by a direct construction, but uniqueness can become problematic. In order to solve the uniqueness problem, we require that the object we are searching for, satisfy a *universal property*. This universal property says that if we find any other algebra whose Lie algebra is also , then there exists a *unique* algebra isomorphism from to such that . This guarantees that the algebra we find is the only algebra (up to a unique isomorphism) whose Lie algebra is . This also ensures that the representations of the Lie algebra (which are essentially homomorphisms from to an associative algebra) are in one to one correspondence with the representations of . This also means that Schur-Weyl duality, which we will prove for the action of , is true for the action of (and also the Lie group of ). Now for the formal definition of .

Definition 2Given a Lie algebra , a unital associative algebra (with a Lie algebra homomorphism ) is said to be a universal enveloping algebra of if it satisfies the following universal property: for any unital associative algebra and a Lie algebra homomorphism , there exists a unique algebra homomorphism such that .

Here we will construct the universal enveloping algebra, but we will not prove the universal property. This is proved, for instance, in chapter 3 of Knapp’s book. From any vector space (such as ), one can construct an associative algebra called the tensor algebra , which satisfies the universal property with respect to . The tensor algebra is defined as

where is the fold tensor product of with itself. So, for instance, is the ground field and is . The tensor algebra is a graded algebra which means that it has a filtration such that and is a direct sum of the . A filtered algebra is more general; it satisfies the first condition and instead of the second condition, we would have i.e., need not be the direct sum of . The tensor algebra has two interesting algebras as its quotients. The symmetric algebra (which we will encounter again later in the proof of Schur-Weyl duality) and the exterior algebra are quotients of the tensor algebra by the ideals and respectively (for ). It is easy to see that the symmetric and alternating algebras are graded with and . It turns out that one can construct the universal enveloping algebra of as a quotient of as well. To include the Lie structure into this algebra, we impose the Lie bracket by modding out by the ideal generated by for . So, the universal enveloping algebra is . Since is not generated by homogeneous elements as in symmetric and alternating algebras, the quotient is filtered but not graded. The filtration is given by . For any filtered algebra, one can construct the so-called *associated graded algebra*. In the case of , the associated graded algebra turns out to be isomorphic to the symmetric algebra . This is part of the content of the Poincaré-Birkhoff-Witt (PBW) theorem. The PBW theorem also gives a basis for in terms of an ordered basis for . To construct the associated graded algebra of , we define the grading as . This can be shown to be isomorphic to the grading of the symmetric algebra. Using the PBW theorem, one can show that is generated by the elements , where and is an ordered basis for . Now, we are ready to prove Schur-Weyl duality.

**3. Proof of Schur-Weyl duality**

As mentioned earlier, we will prove this duality between and . But this can be extended to the one between and as well as the one between and .

*Proof:* Denote the image of in by , the image of by and by . Now, observe that from Maschke’s theorem, it follows that is semisimple. This means that one can decompose into irreducible invariant spaces of in the following way.

where are (spaces corresponding to) irreducible representations of and , i.e., the set of maps embedding into or the multiplicity spaces. Applying Schur’s lemma, we see that one can identify with the direct sum . In particular, this means that is semisimple.

The next step is to observe that since we are dealing with the commutator of the symmetric group, we have that is the symmetric space . Now, let us look at the image of the universal enveloping algebra on . The image is generated by elements of the form

where lies in . It is clear that this element lies in and that all elements that lie in are not in this image. If we show that the kernel does not contain any more elements of , then we have that the image is , which is isomorphic to by the PBW theorem and we would be done. In order to show that there are no more elements in the kernel, we can examine the elements in . These are generated by elements of the type , where . It can be seen that these are not in the kernel.

**4. Irreducible representations of Lie algebras**

For any simple Lie algebra , there exist a set of operators in the algebra such that (this is sometimes confusingly stated as commutation of and . The bracket here is multiplication in the Lie algebra and not commutation. But if one considers a matrix representation of , then one could think of it as commutation. Perhaps, a better way is to think of the is as commuting operators in the adjoint action as we will see below). The maximal such set is called the Cartan subalgebra , whose dimension is called the rank of the Lie algebra. This plays a central role in the structure of a Lie algebra. Since is a vector space, there also exists an action of the algebra on itself via the Lie bracket. This action is called the adjoint representation and can be written as , for any in the Lie algebra thought of as states. Now, in terms of the adjoint representation, we can see that the commute with each other (we need to use the Jacobi identity to see this). This means that they are simultaneously diagonalizable. Moreover, it can be seen that all the are in the null space of each of the . If we call the eigenvectors of the rest of the eigenspaces (where ranges over some set), we have that for some eigenvalue . It turns out that these eigenspaces are one-dimensional (if they are not, then it means that we have not chosen a *maximal* set of commuting operators for ). It is easy to show that are eigenvectors with eigenvalues and the has eigenvalue . If we define as , then and form an subalgebra inside this Lie algebra. This is useful to determine the structure of finite dimensional representations of . A last thing to note here before we move on to the dual space is that there exists a symmetric, bilinear form on the Lie algebra, called the Killing form, defined as (here is the dual Coxeter number of , but let’s not worry about that now).

[It is interesting that the existence of SICs or a complete set of MUBs can be related to existence of certain structures in Lie algebras. Boykin et al., show that the number of Cartan subalgebras of which are pairwise orthogonal with respect to the Killing form corresponds to the number of mutually unbiased bases in (if the Cartan subalgebra is closed under Hermitian conjugation). Appleby et al., showed that the existence problem of SIC-POVMs can be related to the existence of a particular basis of .]

One can construct the dual space of as the space of linear superpositions of maps , where . This dual space is called the root space and the are called roots. It turns out that the root space can be split into positive roots and negative roots of equal cardinality. One way to do this is to simply pick an arbitrary ordered basis for the root space and call positive roots those whose first non-zero coordinate is positive. For an arbitrary finite dimensional representation of , one can, as before, simultaneously diagonalize and consider the eigenvalues of the common eigenspaces. This gives us elements of the dual space, called weights, such that , where is the eigenvalue. Unlike in the adjoint representation, the common eigenspaces of the in an arbitrary finite dimensional representation need not be one-dimensional. In other words, the weights can have multiplicity. However, if is irreducible (like the adjoint representation), the highest and the lowest weights are unique. Finally, one can define an inner product on the root space by using the Killing form i.e., . This is clear when and are roots and by linearity, one can extend this to all of .

Let us now discuss how to describe the weights of a given finite dimensional irreducible representation. Using the definition of positive roots, one can define a special class of roots called simple roots as those positive roots that cannot be written as a sum of positive roots. It turns out that simple roots span the root space and essentially encode the structure of the Lie algebra. However, they are not always a convenient basis. Often one uses a different basis called the basis of fundamental weights. First, define the so called coroots as for each simple root (not to be confused with , which is the eigenvalue ). Now, fundamental weights are defined as the duals of coroots i.e., . In the basis of fundamental weights, the components of any weight are called Dynkin labels and there is a correspondence between the Dynkin labels of the highest weight of a finite dimensional irreducible representation and Young diagrams (which are partitions of some integer). Dynkin labels of highest weights are positive integers and one can create a partition , where . As you can see, the partitions need to be of the same integer. For weights that are not the highest weight, the Dynkin labels can be negative and need not correspond to valid partitions. Corresponding to each weight, there are weight vectors and all the weight vectors span the space of the irreducible representation. For a given weight , we label the weight vector as , where label any multiplicity it may have.

Now, we are ready to discuss the Clebsch-Gordan decomposition of two irreducible representations. When we take the tensor product of two irreducible representations, we have a representation of the group by what is called the diagonal action i.e., if and are two representations, then is the diagonal action. This is in general a reducible representation and its decomposition into irreducibles is called the Clebsch-Gordan (CG) decomposition. For the unitary group, there is a rule that tells us which irreducible representations appear in the decomposition. If we use Young diagrams to label the irreps, then the Littlewood-Richardson rule gives the different irreps and the multiplicity in the decomposition. (In using Young diagrams, one can remove any column of rows.) The weights occurring in the decomposition are linear combinations of the form

where label the multiplicity of the weight and the coefficient is non-zero only if . There are systematic ways to calculate the CG coefficients for the unitary group using the Wigner-Eckart theorem and reduced Wigner coefficients.

[The story of Littlewood-Richardson (LR) rule is quite interesting. In their paper, where Littlewood and Richardson introduce this famous rule, they have a proof for a special case (when one of the partitions has only two rows). However, it is stated as a theorem in general but with a note saying something like – no simple proof has been found for the general case. It seems unlikely that they had a proof (simple or otherwise), given that the first correct proof was found about 40 years later. Shortly after the LR paper, Robinson published a claimed proof. Even though it had several gaps, which were discovered after a long time, his technique eventually led to a correct proof. His method was rediscovered by Schensted and explored further by Knuth and has become the famous RSK correspondence. It appears that the first complete proof of the LR rule was given by Schützenberger using his jeu de taquin. Other correct proofs individually by Thomas, Macdonald and Zelevinsky followed. This interesting story is recounted at the end of this paper.]

**5. Applications**

We are primarily interested in applications to quantum information theory. Here we will look at three applications.

**5.1. Schur transform**

The space has the basis , where (here has the basis ). The action of the unitary and symmetric groups can be block diagonalized and this basis can be labeled , where labels an irreducible representation of (or ) and and are states in the irreducible spaces of and respectively. The Schur transform, then, is the unitary matrix that rotates the computational basis to the block diagonal basis. In this paper, Bacon, Chuang and Harrow construct an efficient quantum circuit for the Schur transform. In order to do this, they first construct the Clebsch-Gordan (CG) transform for the unitary group and use it to decompose the action. To see how the CG transform can be used to construct the Schur transform, suppose that we have decomposed into into irreducible spaces and want to go to the action of , then given any irreducible representation label (at the level) and a state in that irreducible space, we can use the CG transform to decompose the tensor product , where stands for a state in the fundamental representation (i.e., the tensor copy). The CG transform gives us . Here the sum is over all unitary group irreducible representations that appear in the CG decomposition and is a state in the irreducible space . It turns out that keeping track of the labels for each automatically gives us a basis of the irreducible space of the symmetric group.

**A dual version of the Schur transform**

A natural question is whether there is a way to approach the Schur transform using the symmetric group, rather than the unitary group. In order to do this, let us look at the representations in terms of the symmetric group. Recall that as the computational basis for , we picked a basis for each as the set and using this we construct a basis for . Given any basis vector of , define its type as the ordered set , where is the number of times appears in the tensor product. Clearly, the action of the symmetric group permutes all the vectors of a given type in a transitive fashion. This means that the subspace spanned by these vectors of type is an induced representation of — induced from the trivial representation of the Young subgroup (note that can be isomorphic for two different types). If we can block diagonalize these induced representations, we can use it to construct the Schur transform. Let us look at this in more detail. First, we want to convert the basis from to , where is the type of the vector and is the transversal of in . This transform can be done efficiently since the type and the transversal can be efficiently computed from the basis vector and vice versa. Now, we need to block diagonalize each of these induced representations. Fortunately, we can use the algorithm for the Fourier transform over by Beals (see here). There is still a problem since the Fourier transform decomposes the regular representation of the symmetric group, which is the induced representation from the trivial subgroup, not induced from . However, by using the QFTs over and , we can construct the transform for induced representations.

We describe here an algorithm to block diagonalize the induced representation from a subgroup to of an irreducible representation of , if one can perform a QFT over and (and a couple of other conditions, which we explain below). Take a state in the induced representation i.e., a state of the form where is an element of the transversal and is a state in the representation . Append it with an ancilla of dimension , where is the dimension of . Now apply the inverse QFT over to rotate this to the computational basis of . The register now contains states of the form , where is an element of . The first of the two conditions mentioned above is that we need the QFT over to rotate from this basis (which can be thought of as the computational basis) to the block diagonal basis. Next we apply the QFT over to get a basis , where is an irreducible representation of and indexes its multiplicity and is a state in the irreducible space. Now, we need the second condition. We need to be able to re-index the irrep labels so that the label register is of the form . The first register is, of course, of dimension . Now, we can return the ancilla register and we have our transform. Let us check that the two conditions are satisfied in our case i.e., when and . In Beals’ algorithm for the QFT over , the computational basis is assumed to be a word in terms of the generating set of consisting of adjacent transpositions i.e., . So, writing our pair as a sequence using this set should be easy. The second condition is also satisfied since the irreducible representations of that appear in the induced representations from to are those that are greater than the partition corresponding to (here greater is with respect to the dominance order of partitions). In fact, the multiplicities in that induced representation is given by the so called Kostka number. Therefore, if we use a labeling of partitions that respects dominance order, then we automatically satisfy the second condition. (We don’t really have to worry about the second condition since it would just lead to a slightly inefficient encoding.)

Now, the Schur transform is the following sequence of transformations. First, take the basis from to (as discussed above). Then to , where is a partition labeling an irreducible representation of , labels the multiplicity space and the representation space. We have our transform by rearranging the registers to , where is an orthonormal basis for the irreducible space of the unitary group.

**5.2. Quantum de Finetti theorem**

A version of the quantum de Finetti theorem states that given a joint density operator on systems with permutational symmetry, the partial trace over systems can be approximated as a convex sum of product states. Quantum de Finetti theorems have been applied to several areas of quantum information science such as security proof of quantum key distribution schemes against coherent attacks. Here, we discuss a proof of the quantum de Finetti theorem, which uses representation theory of Lie algebras and Schur-Weyl duality. This is from this paper of Koenig and Mitchison.

Suppose that we have a tensor product of two irreducible representations of a Lie algebra and we are interested in states in an irreducible representation inside the tensor product. Here and are the highest weights. We are specifically interested in the partial trace over the system of any state in and how we can write this as a convex sum over certain states in . Suppose that we have a state in , which is a weight vector. Since appears in the decomposition of , the weight vectors of can be written as linear combinations of the type

where the sum of the weights of and is the weight of . Now, consider the partial trace

where the integration is with respect to the Haar measure. The above equation follows from Schur’s lemma on the system. Notice that the integrand is a pure state i.e.,

This means that the partial trace is

where is the induced probability measure.

Now consider the set , where is the set of weights occurring in and let be the projector onto the space spanned by these weight vectors. We have that

But since is the highest weight, we can replace by i.e., . This means that we can “approximate” the partial trace of exactly with states from . In general, suppose we have another set of states (instead of the ones in ) and we start from another state (instead of ). Let the projector onto this new set of states be , then we can define the states (normalized appropriately), where . The integral can be used to approximate the partial trace and the error (i.e., the trace norm of the difference) in approximation is upper bounded by . The error is first related (via the gentle measurement lemma and triangle inequality) to

By plugging back the expression for and applying Schur’s lemma, this can be calculated to be

Defining as the supremum of over all states , we get the upper bound. This expression for the error gives us the exponential de Finetti theorem. For this, we take the representation with highest weight to be , to be and to be .

**5.3. Spectrum of a density operator**

The spectrum of many copies of a density operator turns out to be related to Schur-Weyl duality. There are many consequences of this and several related problems. But here, we discuss only one aspect of it. It was originally proved by Keyl and Werner. Hayashi and Matsumoto gave a proof using Schur-Weyl duality. The following is from Christandl and Mitchison.

If is a density operator (i.e., a positive self-adjoint operator with unit trace) in a Hilbert space of dimension , then the spectrum of is closely related to Schur-Weyl duality. Since the operator commutes with permutations of tensor copies, it “lives” in the irreducible spaces of the unitary group. It turns out that as increases it mostly lives in only one particular space. That irreducible representation (or rather its normalized version) tends to the spectrum of . If is a partition of , then the normalized version of is . To state this result more precisely,

Theorem 3Suppose the spectrum of is and is the projection onto the isotypic space corresponding to the Young frame , then

where is the usual Kullback-Leibler distance between probability distributions.

*Proof:* First pick the eigenbasis of as the basis of each tensor copy of . As basis vectors of , we have . Each such basis vector corresponds to a partition by counting the number of times a certain occurs in it. Now, consider the projector onto the isotypic space of some in Schur-Weyl duality. This can be given by picking a standard Young tableau of shape .

where and are subsets of permutations of which permute the integers in each row and column respectively. It can be seen that this projector has non-zero overlap with only those basis vectors whose corresponding partition is dominated by (in the dominance order mentioned earlier). If not, then there would be two boxes in the same column which have the same integer and the projector would be zero. Therefore, we have that if is dominated by , then Tr is non-zero. This is easy to bound by using standard bounds on the dimensions of these irreducible representations.

**5.4. Permanents, determinants and immanants**

The last thing I want to mention is a way to describe immanants using Schur-Weyl duality. The permanent of an matrix is defined as

In order to relate this to Schur-Weyl duality, let us rewrite in terms of an fold tensor power.

Now recall that the projector onto the isotypic space of the trivial representation of the symmetric group is given by

where is the representation of the symmetric group on (i.e., the action that permutes the tensor factors). Using this, we have

We can generalize this to other isotypic spaces of the symmetric group to get the determinant and the immanents.

where is the projector into the isotypic space of the alternating irrep of . For the immanants, we have

where is the projector onto the irrep space of and is the dimension of the symmetric group irrep .

Let us now work only with immanants (since they are more general). The last equation can be written as

First, we can rewrite this by as

where is the permutation module corresponding to the partition . It is the induced representation from the trivial representation of the Young subgroup corresponding to the partition . To obtain the above equation, we have used the fact that

Now we can also block diagonalize to obtain

where is the identity on the symmetric group space and is the part in the unitary group irreducible space. This can finally be written as

where is the projection on to the multiplicity space of the irrep in the permutation module . This space has dimension equal to the Kostka number .

If we replace the permutation module by another one, then we get immanants of replicated matrices. First, define, for any type (where ) the matrix obtained by repeating the first row and column times, repeating the second row and column times etc. Then the immanant of this matrix is

where is the projection onto the isotypic space of in the permutation module corresponding to the type . To see this, note that the immanant of this replicated matrix is

The rest of it follows in the same way as before.

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