Fused Braids and Centralisers of Tensor Representations of U_q(gl_N)

Abstract

We present in this paper the algebra of fused permutations and its deformation the fused Hecke algebra. The first one is defined on a set of combinatorial objects that we call fused permutations, and its deformation is defined on a set of topological objects that we call fused braids. We use these algebras to prove a Schur–Weyl duality theorem for any tensor products of any symmetrised powers of the natural representation of U_q(gl_N). Then we proceed to the study of the fused Hecke algebras and in particular, we describe explicitly the irreducible representations and the branching rules. Finally, we aim to give an algebraic description of the centralisers of the tensor products of U_q(gl_N)-representations under consideration. We exhibit a simple explicit element that we conjecture to generate the kernel from the fused Hecke algebra to the centraliser. We prove this conjecture in some cases and in particular, we obtain a description of the centraliser of any tensor products of any finite-dimensional representations of U_q(sl₂).

Appendices

Appendix A: Artin–Wedderburn Decompositions and Bratteli Diagrams

1.1 A.1 Semisimple Algebras and Algebras of the Form PAP

Artin–Wedderburn Decomposition of a Semisimple Algebra

Let A be a finite-dimensional semisimple algebra over $\mathbb {C}$. Let S be an indexing set for a complete set of pairwise non-isomorphic irreducible representations of A. Then Artin–Wedderburn theorem asserts that we have the following isomorphism of algebras:

$$ A\cong \bigoplus_{\lambda\in S} \text{End}(V_{\lambda})\ , $$

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where V_λ is a realisation of the irreducible representation corresponding to λ. The isomorphism is given naturally by sending a ∈ A to the endomorphism in End(V_λ) corresponding to the action of a on V_λ.

Algebras of the Form PAP and Their Representations

Let A be a $\mathbb {C}$-algebra with an idempotent P. Then the subset PAP = {PxP | x ∈ A} is an algebra with unit P. We recall very basic and classical facts on the algebra PAP which can be found for example in [9, §6.2].

Let ρ : A↦End(V ) be a representation of A and W := ρ(P)(V ) the image of the operator ρ(P). The subspace W is naturally a representation of the algebra PAP. Indeed W is obviously invariant under the action of any element the form ρ(PxP), and thus the action of PAP on W is given simply by restriction:

$$ \begin{array}{@{}rcl@{}} PAP & \to & \text{End}\left( W\right) \\ PxP & \mapsto & {\rho(PxP)}_{|_{W}}. \end{array} $$

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From now on, we will always remove the map ρ from the notation, and keep the same notation for an element of an algebra and its action in a given representation.

Irreducible Representations and Semisimplicity

Let {V_λ}_λ∈S be a complete set of pairwise non-isomorphic irreducible representations of A. Then the set

$$\{P(V_{\lambda})\ |\ \lambda\in S \text{ and } P(V_{\lambda})\neq 0\}$$

is a complete set of pairwise non-isomorphic irreducible representations of the algebra PAP.

Moreover, if the algebra A is a finite-dimensional semisimple algebra then the algebra PAP is also a finite-dimensional semisimple algebra and its Artin–Wedderburn decomposition is

1.2 A.2 Primitive Central Idempotents and Ideals

Let A be a finite-dimensional semisimple algebra over $\mathbb {C}$ and S an indexing set for a complete set of pairwise non-isomorphic irreducible representations of A.

Primitive Central Idempotents

Let λ ∈ S. We define E_λ as the element of A corresponding under the Artin–Wedderburn decomposition of A to $\text {Id}_{V_{\lambda }}$ in the component corresponding to λ and 0 in all other components. The set {E_λ}_λ∈S is a complete set of primitive central orthogonal idempotents of A, meaning that they are central, they sum to 1, they satisfy $E_{\lambda }E_{\lambda ^{\prime }}=\delta _{\lambda ,\lambda ^{\prime }}E_{\lambda }$ and they cannot be written as the sum of two non-zero central idempotents.

In any representation W of A, the action of E_λ projects onto the isotopic component of W corresponding to λ. More precisely, if the decomposition of W into irreducible is then the action of E is the projection onto V m corresponding to this decomposition, that is:

Ideals and Quotients

From the decomposition (1), one sees immediately that ideals (and equivalently quotients) of A are in correspondence with subsets S^′⊂ S as follows:

$$I_{S'}:=\bigoplus_{\lambda\in S'} \text{End}(V_{\lambda})\ \ \ \ \ \ \text{and}\ \ \ \ \ \ A/I_{S'}\cong\bigoplus_{\lambda\in S\backslash S'} \text{End}(V_{\lambda})\ .$$

One set of generators of the ideal $I_{S'}$ consists of the elements $E_{\lambda }$ with λ ∈ S^′.

Since we use it in Section ??, we recall that the ideal generated by an element x ∈ A is $I_{S'}$ where S^′ is the subset of irreducible representations such that x acts as a non-zero element.

1.3 A.3 Bratteli Diagram of a Chain of Algebras

Let {A_n}_n≥ 0 be a family of algebras and assume that, for any n ≥ 0, there is a given injective map from A_n to A_n+ 1. We call these maps “inclusion maps” and we say that {A_n}_n≥ 0, sometimes denoted as follows

$$A_{0}\subset A_{1}\subset A_{2}\subset\dots\ldots\subset A_{n}\subset A_{n+1}\subset\dots\ ,$$

forms a chain of algebras. The inclusion maps allow to consider elements of A_n as elements of A_n+ 1, and more generally of A_n+k for k ≥ 1, and in turn to consider A_n as a subalgebra of A_n+ 1, and more generally of A_n+k for k ≥ 1.

From the inclusion of A_n into A_n+ 1, any representation V of A_n+ 1 can be seen as a representation of A_n by restriction; we denote this representation of A_n by $\text {Res}_{A_{n}}(V)$.

Bratteli Diagram of a Chain of Semisimple Algebras

Let {A_n}_n≥ 0 be a chain of algebras and we assume here that the algebras A_n are finite-dimensional semisimple algebras, since this will always be our setting in this paper.

For an irreducible representation V_i of A_n+ 1, the restriction to A_n decomposes by semisimplicity into a direct sum of irreducible representations, namely,

$$\text{Res}_{A_{n}}(V_{i})\cong\bigoplus m_{j} W_{j}\ ,$$

where W_j are non-isomorphic irreducible representations of A_n and the numbers m_j are called the multiplicities. The knowledge of such decomposition for any irreducible representation of A_n+ 1 for any n ≥ 0 are called the branching rules of the chain of algebras. Then the Bratteli diagram of the chain of algebras {A_n}_n≥ 0 is the following graph:

• The set of vertices is partitioned into subsets indexed by n ≥ 0. We call n the level. The vertices of level n are indexed by the (isomorphism classes of) irreducible representations of the algebras A_n.

• The edges express the branching rules of the chain of algebras and they only connect vertices of adjacent levels. Let V be an irreducible representation of A_n and V^′ an irreducible representation of A_n+ 1. Then there are m edges connecting the vertices indexed by V and V^′ if and only if V appears in the decomposition of $\text {Res}_{A_{n}}(V')$ with multiplicity m.

Graphically, we place all the vertices of a given level on an horizontal line, and we put the vertices of level n + 1 below the vertices of level n. We often think of the edges as going down from vertices of level n to vertices of level n + 1.

Let v,v^′ be two vertices of the Bratteli diagram. A path from v to v^′ is a sequence of vertices of the form $v,v_{1},\dots ,v_{k-1},v'$, for some k ≥ 1, such that at each step of the sequence, the level increases by 1. In other words, a path from v to v^′, if it exists, is obtained by starting from v and following edges only in the downward direction to reach v^′.

The partial order ≤ on the set of vertices of a Bratteli diagram is defined by setting that v ≤ v^′ if and only if v = v^′ or there is a path from v to v^′.

Dimensions

We often add the following numerical information to the Bratteli diagram of the chain {A_n}_n≥ 0: next to each vertex, we indicate the dimension of the corresponding irreducible representation.

Obviously, for n ≥ 1 and any vertex V of level n, this dimension can be obtained from the preceding level. Indeed it is the sum of the dimensions of the irreducible representations of level n − 1 connected to V (counted with multiplicity indicated by the number of edges).

Moreover, from the Artin–Wedderburn decomposition of the algebras A_n, we have that the dimension of the algebra A_n is the sum of the squares of the dimensions appearing at level n.

Example A.1

A standard example of a Bratteli diagram is the Young diagram, corresponding to the poset of partitions partially ordered by inclusion. It is the Bratteli diagram associated to the chain $\{\mathbb {C}[\mathfrak {S}_{n}]\}_{n\geq 0}$ of the complex group algebras of the symmetric groups. The first levels are given in Appendix A.

1.4 A.4 Quotients of Bratteli Diagrams and Chains of Ideals

We keep our setting of a chain {A_n}_n≥ 0 of finite-dimensional semisimple algebras. Let $\mathcal {D}$ be the Bratteli diagram of the chain {A_n}_n≥ 0 and let S be a subset of vertices of $\mathcal {D}$.

Quotients of Bratteli Diagrams

We denote by 〈S〉 the set of vertices v^′ such that there exists v ∈ S with v ≤ v^′ (i.e. all vertices in S and all vertices connected by a path to them).

Definition A.2

We define $\overline {\mathcal {D}}_{S}$ to be the diagram obtained from $\mathcal {D}$ by removing all vertices v^′∈〈S〉 and keeping only the edges of $\mathcal {D}$ which connect the remaining vertices.

We call the resulting diagram $\overline {\mathcal {D}}_{S}$ the quotient of $\mathcal {D}$ generated by S.

Obviously, the quotient $\overline {\mathcal {D}}_{S}$ depends only on the set of vertices 〈S〉 generated by S. Hence, several choices of S can lead to the same quotient. There is a unique minimal choice S_min, which is the set of minimal elements (for the partial order ≤) in 〈S〉. We call S_min the minimal generating set for the quotient $\overline {\mathcal {D}}_{S}$.

Example A.3

A standard example of a quotient of a Bratteli diagram is the following. Take the Bratteli diagram of the chain $\{\mathbb {C}[\mathfrak {S}_{n}]\}_{n\geq 0}$ (see Appendix A below) and make the quotient generated by the vertex labelled by the partition . It is easy to see that the remaining vertices are the partitions with no more than two lines. The quotient is equal to the Bratteli diagram of the chain of Temperley–Lieb algebras.

Representation-Theoretic Meaning and Chains of Ideals

We will explain the name Bratteli diagram for $\overline {\mathcal {D}}_{S}$, and its representation-theoretic meaning.

For every n ≥ 0, let S_n be the set of vertices of level n inside 〈S〉 (that is, the vertices of level n which have been removed from $\mathcal {D}$). To S_n corresponds an ideal I_n of A_n. We have that {I_n}_n≥ 0 forms a chain of ideals in {A_n}_n≥ 0:

$$ I_{0}\subset I_{1}\subset I_{2}\subset\dots\ldots\subset I_{n}\subset I_{n+1}\subset\dots\ , $$

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which means that I_n, seen as a subset of A_n+ 1 using the inclusion map, is contained in I_n+ 1 for any n ≥ 0. This chain property follows from the fact that, by definition of 〈S〉, every edge starting from S_n ends in S_n+ 1 (see Remark A.4 below).

Now the quotient $\overline {\mathcal {D}}_{S}$ has the following meaning:

• The vertices of level n are in bijection with the irreducible representations of A_n/I_n.

• The edges give the branching rules for the following restriction procedure: a representation V^′ of A_n+ 1/I_n+ 1 can be seen as a representation of A_n+ 1 where I_n+ 1 acts as 0. Thus, from the inclusion A_n ⊂ A_n+ 1, we can form the representation $\text {Res}_{A_{n}}(V')$ of A_n. Now by the inclusion I_n ⊂ I_n+ 1, we have that I_n obviously acts as 0, and therefore $\text {Res}_{A_{n}}(V')$ can be seen as a representation of A_n/I_n.

Remark A.4

• The family of quotients {A_n/I_n}_n≥ 0 does not necessarily form a chain of algebras, even if the ideals I_n form a chain of ideals. In fact, one can check that the quotients A_n/I_n form a chain of algebras for the natural inclusion maps x + I_n↦x + I_n+ 1 if and only if we have, for any n ≥ 0, I_n = I_n+ 1 ∩ A_n as subsets of A_n+ 1. This is stronger than the chain property for the ideals I_n.

• The property I_n = I_n+ 1 ∩ A_n ensuring that the quotients form a chain of algebras can be seen easily in the Bratteli diagram. We have that I_n = I_n+ 1 ∩ A_n if and only if:

  $$v\in S_{n}\ \ \ \ \Longleftrightarrow\ \ \ \ \forall v' \text{of level} n+1, v\leq v' \text{implies} v'\in S_{n+1}.$$

  We note that the weaker property I_n ⊂ I_n+ 1 is equivalent to the single implication ⇒.

Algebraic Description of the Quotients A _n/I _n

To make explicit the situation we need in this paper, assume that the set S_min contains vertices of a single level, say N + 1 (the general case can be obtained by partitioning the set S_min according to the level, and applying this procedure to each part).

Denote by X a generator of the ideal I_N+ 1 of A_N+ 1 (for example the sum of the central idempotents corresponding to S_min). Then we have that for any n ≥ N + 1, the ideal I_n of A_n is generated by the element X^↑n which is the element X seen as an element of A_n by the inclusion (indeed, this element X^↑n is non-zero precisely in the correct set of irreducible representations of A_n, by definition of S_min and of a Bratteli diagram).

As a conclusion, we note that for all n ≥ N + 1, the algebra A_n/I_n is the quotient of A_n over the relation X^↑n = 0. We refer to Example 4.8 for the well-known example of Temperley–Lieb algebras.

Appendix B: Examples

2.1 B.1 The Chain of Hecke Algebras H _n(q)

The Hecke algebra H_n(q) is the particular case of the algebra H_k,n(q) where $\mathbf {k}=(1,1,\dots )$ is the infinite sequence of 1’s. The first levels of the Bratteli diagram for the chain of Hecke algebras {H_n(q)}_n≥ 0 is shown below.

The shaded areas indicate the connections between the Hecke algebras H_n(q) and the centralisers of the representations (corresponding to $\mathbf {k}=(1,1,1,\dots )$) of U_q(gl_N). Namely, by deleting the vertices included in the shaded area labelled gl_N together with the edges touching them, we obtain the Bratteli diagram of the centralisers of U_q(gl_N). For example, if N = 2, the quotiented Bratteli diagram is the Bratteli diagram of the Temperley–Lieb algebras.

2.2 B.2 The Chain of Algebras H _k,n(q) When $\mathbf {k}=(2,2,2,\dots )$

When $\mathbf {k}=(2,2,2,\dots )$ is the infinite sequence of 2’s, the Bratteli diagram for the chain of algebras {H_k,n(q)}_n≥ 0 begins as:

Note that there is no arrow from to even if μ ⊂ λ since λ/μ contains two boxes in the same column.

The shaded area indicates the connections between the fused Hecke algebras H_k,n(q) and the centraliser of the representations (corresponding to $\mathbf {k}=(2,2,2,\dots )$) of U_q(gl_N), as in the preceding example.

2.3 B.3 The Chain of Algebras H _k,n(q) When $\mathbf {k}=(3,1,1,1,\dots )$

When $\mathbf {k}=(3,1,1,1\dots )$, the Bratteli diagram for the chain of algebras {H_k,n(q)}_n≥ 0 begins as (the shaded areas have a similar meaning as in the preceding examples):