2.1 The $\star $ -algebra of horizontal chord diagrams

We start with a definition.

A first class of examples of $\star $ -algebras is given by group rings.

Example 2.1 Given a group G, the group ring $\mathbb {C}[G]$ has a natural structure of $\star $ -algebra given by setting

$$\begin{align*}\left(\sum_{i=1}^{k} z_i \cdot g_i\right)^\star = \sum_{i=1}^{k}\overline{z}_i\cdot g_i^{-1}, \end{align*}$$

for all $z_1,\ldots ,z_k\in \mathbb {C}$ and $g_1,\ldots ,g_k\in G$ .

In the next example, we present a formal definition of the $\star $ -algebra of horizontal chord diagrams $\mathscr {A}^{\mathrm {h}}$ . While the reader can keep in mind the pictorial definition given in the introduction, it is useful to have an explicit definition.

Example 2.2 An horizontal chord diagram on N strands is an element of the free monoid $(\mathscr {D}^{\mathrm {h}}_N, \circ )$ generated by the pairs $(i,j)$ , with $1\leq i < j \leq N$ , called chords, with neutral element the chord-less diagram $\uparrow _N$ . We also consider the empty chord diagram $\uparrow _0 = \emptyset $ , defined as the diagram with neither chords nor strands, and set ${\mathscr {D}^{\mathrm {h}}_0 = \{ \uparrow _0 \}}$ . This monoid has a natural involution $^\star $ given by “reading the chord diagram backward,” that is,

$$\begin{align*}[(i_1,j_1)\circ \cdots \circ (i_r,j_r)]^\star = (i_r,j_r)\circ \cdots \circ (i_1,j_1). \end{align*}$$

The algebra of horizontal chord diagrams on N strands is given by the quotient algebra

where $I\subseteq \mathbb {C}[\mathscr {D}^{\mathrm {h}}_N]$ is the ideal generated by elements of the form

(2T) $$ \begin{align} (i,j)\circ (k,l) - (k,l)\circ (i,j) \end{align} $$

for $ i, j, k, l$ such that the two closed intervals $[i,j],\ [k,l]\subset \mathbb {R}$ are either disjoint or one contains the other, and also elements of the form

(4T) $$ \begin{align} (i,j)\circ (i,k) + (i,j)\circ (j,k) - (i,k)\circ (i,j) - (j,k)\circ (i,j),\quad 1\leq i < j < k \leq N \end{align} $$

(see Figure 2 for a pictorial representation). Finally, define the algebra of horizontal chord diagrams as $\mathscr {A}^{\mathrm {h}} = \bigoplus _{N\geq 0} \mathscr {A}^{\mathrm {h}}_N$ . Extending $^\star $ by anti-linearity, since $I^\star = I$ , the algebra of horizontal chord diagrams acquires the structure of complex $\star $ -algebra. This involution has actually a deeper and more abstract interpretation as the antipode of an Hopf algebra associated with the homology of some loop spaces (cf. [Reference Corfield, Sati and Schreiber3, Section 4]).

For small values of N, we can explicitly identify $\mathscr {A}^{\mathrm {h}}_N$ ; we have the following isomorphisms of $\star $ -algebras: $\mathscr {A}^{\mathrm {h}}_1 \cong \mathscr {A}^{\mathrm {h}}_0 \cong \mathbb {C}$ and $\mathscr {A}^{\mathrm {h}}_2 \cong \mathbb {C}[x]$ , where the involution on the latter algebra is defined by setting $x^\star = x$ . For $N\geq 3$ , the algebra $\mathscr {A}^{\mathrm {h}}_N$ is noncommutative. In fact, we have a natural embedding $\mathscr {A}^{\mathrm {h}}_{N-1}\subseteq \mathscr {A}^{\mathrm {h}}_{N}$ and $\mathscr {A}^{\mathrm {h}}_3$ is noncommutative; it is a direct product of the free algebra on two generators $\mathbb {C}\langle x,y\rangle $ and $\mathbb {C}[u]$ (see [Reference Chmutov, Duzhin and Mostovoy2, Proposition 5.11.1]).

2.2 Horizontal chord diagrams, permutations, and standard weight systems

To each horizontal chord diagram $C = (i_r,j_r) \circ \cdots \circ (i_1,j_1) \in \mathscr {A}^{\mathrm {h}}_N$ , there is a naturally associated permutation:

(2.1) $$ \begin{align} \sigma(C) = \tau_{i_r,j_r} \circ \cdots \circ \tau_{i_1,j_1} \in \mathfrak{S}_N, \end{align} $$

where $\tau _{i,j}$ denotes the transposition of i and j. Recall, from Example 2.1, that the group ring $\mathbb {C}[\mathfrak {S}_N]$ has a natural structure of $\star $ -algebra. Indeed, we have the following proposition.

Proof We have to prove that $\sigma $ is a well-defined ring homomorphism. That is, we have to show that the images of elements in equations (2T) and (4T) are trivial. This is immediate for the elements in equation (2T). On the other hand, a direct computation shows that

$$\begin{align*}\sigma((i,j)\circ (i,k)) = \sigma((j,k)\circ (i,j)) = ( j\ k \ i)\end{align*}$$

and that

$$\begin{align*}\sigma((i,j)\circ (j,k) ) = \sigma( (i,k)\circ (i,j)) = ( i\ k \ j).\end{align*}$$

Since $\sigma $ commutes with $\star $ , this concludes the proof.

Following [Reference Bar-Natan1, Section 2.2], the $\mathfrak {gl}_{n}$ weight system associated with the defining representation, here called standard $\mathfrak {gl}_{n}$ weight system, is the map $ W_{\mathtt {st}} : \mathscr {A}^{\mathrm {h}} \to \mathbb {C} $ which assigns to each chord diagram $C\in \mathscr {A}^{\mathrm {h}}_N$ , the integer

(2.2)

extended by $\mathbb {C}$ -linearity. We remark that here the number of cycles in a permutation also includes the trivial cycles (i.e., the fixed points).

2.3 Tensor splitting

Another important operation on $\mathscr {A}^{\mathrm {h}}$ is what we call tensor splitting. Intuitively, the $\underline {i}$ -tensor splitting, with $\underline {i} = (i_1 ,\ldots , i_N)\in \mathbb {N}^N$ , is obtained by replacing the rth strand with $i_r$ parallel strands, and replacing each chord with the sum of all possible “liftings” of said chord in the new horizontal chord diagram. More formally, we have the following definition.

Definition 2.2 ([Reference Bar-Natan1, Definition 2.2])

Let $\underline {i} = (i_1 ,\ldots , i_N)$ be an N-tuple of positive integers. The $\underline {i}$ -tensor splitting is the ring homomorphism

$$\begin{align*}\Delta^{\underline{i}}:\: \mathscr{A}^{\mathrm{h}}_{N} \to \mathscr{A}^{\mathrm{h}}_{\ell_{N+1}(\underline{i})}:\: (j,k) \mapsto \sum_{r=\ell_{j}(\underline{i}) + 1}^{\ell_{j+1}(\underline{i}) }\sum_{s=\ell_{k}(\underline{i}) + 1}^{\ell_{k+1}(\underline{i}) } (r,s),\end{align*}$$

where $\ell _{1}(\underline {i}) = 0$ and $\ell _{r}(\underline {i} ) = i_1 + \cdots +i_{r-1}$ , for $0 < r \leq N +1$ .

Let us see in an example the effect of the tensor splitting.

Example 2.4 Denote $C = (2,3) \circ (1,3)\in \mathscr {A}^{\mathrm {h}}_3$ , we explicitly compute some tensor splittings in this case; for instance, we have that

$$\begin{align*}\Delta^{(1,2,1)}(C) = (2,4) \circ(1,4) + (3,4) \circ(1,4)\in \mathscr{A}^{\mathrm{h}}_4, \end{align*}$$

and also that

$$\begin{align*}\Delta^{(3,1,1)}(C) = (4,5) \circ(1,5) + (4,5) \circ(2,5) + (4,5) \circ(3,5)\in \mathscr{A}^{\mathrm{h}}_5. \end{align*}$$

Note that $\Delta ^{\underline {1}}$ , where $\underline {1} = (1,\ldots ,1)$ , is just the identity map. We conclude this subsection with the following, useful, observation.

2.4 Young symmetrizers

A partition of $n\in \mathbb {N}$ is a (nonempty) ordered collection of positive integers ${\lambda = (n_1,\ldots , n_k)}$ , with $n_1 \geq n_2 \geq \dots \geq n_k> 0$ , which add up to n. If $\lambda $ is a partition of n, we shall write $\lambda \vdash n$ or, if it is understood that $\lambda $ is a partition, $|\lambda | = n$ . The (positive) integers $n_1,\ldots , n_k$ are called parts, and the number of parts, which is k in the case at hand, is called the length of the partition.

The Young diagram associated with a partition $\lambda = (n_1,\ldots , n_k)$ is a finite collection of boxes arranged in k (left-justified) rows of length (from top to bottom) $n_1,\ldots , n_k$ , respectively. Conversely, to each Young diagram, we can associate a partition whose parts are the length of its rows. We denote both a partition and its associated Young diagram by the same symbol.

Example 2.6 The partition $\lambda = (5,3,1,1)$ is a partition of $10$ whose length is $4$ . The corresponding Young diagram $\lambda $ is the following.

An essential tool for building irreducible representations of $\mathfrak {sl}_n$ , and thus of $\mathfrak {gl}_{n}$ , are Young symmetrizers (cf. [Reference Fulton and Harris4, pp. 45–46]).

Given a Young diagram of shape $\lambda = (n_1, n_2,\ldots , n_k)$ , there is a canonical (standard) Young tableau associated with it, which is the following tableau.

To get hold of how the canonical tableau is defined, we give a concrete example.

Example 2.7 The canonical Young tableau of shape $\lambda = (5,3,1,1)$ is the following.

Let $T^\lambda $ be the set of all tableaux of shape $\lambda =(n_1,\ldots ,n_k)$ . A permutation $\sigma \in \mathfrak {S}_{|\lambda |}$ acts on each tableau $t^\lambda \in T^\lambda $ by permuting the labels, denote by $\sigma .t^\lambda $ the resulting tableau. Fixed a tableau $t^\lambda $ , we can define two subgroups of $\mathfrak {S}_{|\lambda |}$ ; that is, the row stabilizer

$$\begin{align*}\mathfrak{R}_{t^{\lambda}} = \left\{\ \sigma \in \mathfrak{S}_{|\lambda|}\ \left\vert\ \begin{array}{c}\text{the set of labels on corresponding} \\ \text{rows of } \sigma.t^\lambda \text{ and } t^\lambda \text{ are the same}\end{array} \right.\right\}, \end{align*}$$

and the column stabilizer

$$\begin{align*}\mathfrak{C}_{t^{\lambda}} = \left\{\ \sigma \in \mathfrak{S}_{|\lambda|}\ \left\vert\ \begin{array}{c}\text{the set of labels on corresponding} \\ \text{columns of } \sigma.t^\lambda \text{ and } t^\lambda \text{ are the same}\end{array} \right.\right\}. \end{align*}$$

Using these subgroups, we can associate with each tableau its Young symmetrizer.

Definition 2.4 The unnormalized Young symmetrizer $\tilde {c}_{t^{\lambda }}\in \mathbb {C}[\mathfrak {S}_{|\lambda |}]$ associated with the tableau $t^\lambda $ is defined as $\tilde {c}_{t^{\lambda }} := a_{t^{\lambda }} b_{t^{\lambda }}$ , where

$$\begin{align*}a_{t^{\lambda}} = \sum_{\sigma\in \mathfrak{R}_{t^{\lambda}}} \sigma\qquad \qquad \qquad b_{t^{\lambda}} = \sum_{\sigma \in \mathfrak{C}_{t^{\lambda}}} \mathrm{sign}(\sigma)\sigma.\end{align*}$$

The Young symmetrizer $ c_{t^{\lambda }}\in \mathbb {C}[\mathfrak {S}_{|\lambda |}]$ is a rescaling of $\tilde {c}_{t^{\lambda }}$ by a positive rational number, in such a way that ${c}_{t^{\lambda }} {c}_{t^{\lambda }} = {c}_{t^{\lambda }}$ (cf. [Reference Fulton and Harris4, Lemma 4.6]).

Example 2.10 Consider the partition $\lambda \vdash 3$ given by $(2,1)$ , the corresponding Young diagram $\lambda $ is the following.

Two examples of Young tableau of shape $\lambda $ are the following.

Observe that $s^\lambda $ is standard (actually, it is the canonical tableau), whereas $t^\lambda $ is not. In these cases, we have

$$\begin{align*}\mathfrak{R}_{t^{\lambda}} = \{ \mathrm{id}, (1,2) \},\qquad \mathfrak{C}_{t^{\lambda}} = \{ \mathrm{id}, (1,3) \} \end{align*}$$

and

$$\begin{align*}\mathfrak{R}_{s^\lambda} = \{ \mathrm{id}, (2,3) \},\qquad \mathfrak{C}_{s^\lambda} = \{ \mathrm{id}, (1,3) \}, \end{align*}$$

respectively. It follows that

$$\begin{align*}\tilde{c}_{t^{\lambda}} = (\mathrm{id} + (1,2))(\mathrm{id} - (1,3)) =\mathrm{id} + (1,2) - (1,3) - (1,3,2)\end{align*}$$

and

$$\begin{align*}\tilde{c}_{s^\lambda} = (\mathrm{id} + (2,3))(\mathrm{id} - (1,3)) = \mathrm{id} + (1,2) - (1,3) - (1,2,3).\end{align*}$$

The next subsection is dedicated to the description of irreducible $\mathfrak {gl}_{n}$ -representations in terms of partitions. Our main scope there is to set some notation, for the general theory, the reader may refer to [Reference Fulton and Harris4, Chapter 15].

2.5 Representations of $\mathfrak {gl}_{n}$

Denote by $\rho _{\mathtt {st}}$ the defining representation of $\mathfrak {gl}_{n}$ , that is the natural action of ${\mathfrak {gl}_{n} = \mathrm { End}(\mathbb {C}^n)}$ on $\mathbb {C}^n$ , and denote by $\rho ^{\mathfrak {sl}}_{\mathtt {st}}$ the defining representation of $\mathfrak {sl}_n$ , which is the restriction of $\rho _{\mathtt {st}}$ to $\mathfrak {sl}_n$ .

It is well known that (finite-dimensional) irreducible representation of $\mathfrak {sl}_n$ are associated with a partition of length at most n [Reference Fulton and Harris4, Section 15.3]. The representation corresponding to a partition $\lambda $ can be described explicitly. Consider the natural action (by permutation of the tensor factors) of the symmetric group $\mathfrak {S}_k$ on the tensor product of k copies of $\mathbb {C}^n$ . This action commutes with the action of $\mathfrak {sl}_n$ , given by $(\rho _{\mathtt {st}}^{\mathfrak {sl}})^{\otimes k}$ . In particular, we can consider the images of the action of a Young symmetrizer $c_{t^{\lambda }}\in \mathbb {C}[\mathfrak {S}_k]$ (with $|\lambda | = k$ ) in $\mathbb {C}^{n} \otimes \dots \otimes \mathbb {C}^{n}$ .

The restriction of $(\rho _{\mathtt {st}}^{\mathfrak {sl}})^{\otimes k}$ to the Weyl module $\mathbb {S}_\lambda (\mathbb {C}^n)$ associated with $\lambda \vdash k$ is isomorphic to the irreducible representation of $\mathfrak {sl}_n$ with highest weight $w_{\lambda }$ , denoted by $\rho ^{\mathfrak {sl}}_\lambda $ . We can extend $\rho ^{\mathfrak {sl}}_\lambda $ to a representation of $\mathfrak {gl}_{n} =\mathfrak {sl}_n\oplus \mathbb {C}\langle \mathrm { Id}_{\mathbb {C}} \rangle $ by making $ \mathrm {I}_n$ act trivially on $V_\lambda $ . For $z\in \mathbb {C}$ , consider the representation $\rho _{z}:\mathfrak {gl}_{n} \to \mathrm {End}(\mathbb {C})$ defined by

$$\begin{align*}\rho_{z}(g + k \mathrm{Id}_{\mathbb{C}})[x] = zk x,\quad \forall g\in \mathfrak{sl}_n,\ k, x\in\mathbb{C}\,. \end{align*}$$

Each irreducible representation of $\mathfrak {sl}_n\oplus \mathbb {C}\langle \mathrm {Id}_{\mathbb {C}} \rangle = \mathfrak {gl}_n$ is isomorphic to a tensor product of the form $\rho _\lambda \otimes \rho _{z}$ (see [Reference Fulton and Harris4, Section 15.5]), where the tensor product of two representations, say $\rho $ and $\rho '$ , is given by

$$ \begin{align*}(\rho \otimes \rho')(g)[v_1\otimes v_2] = \rho(g)[v_1] \otimes v_2 + v_1 \otimes \rho'(g)[v_2] .\end{align*} $$

We refer to the Young diagram representation $\rho _\lambda \otimes \rho _1$ simply as $\lambda $ . The symmetric and exterior power of the defining representation are associated with the partitions of the form $(k)$ and $(1,\ldots ,1)$ , respectively.

2.6 General Lie algebra weight systems

In this subsection, we recall the general construction of Lie algebra weight system, and then we specialize this construction to the case at hand (see [Reference Bar-Natan1, Section 2], [Reference Chmutov, Duzhin and Mostovoy2, Chapter 6], [Reference Jackson and Moffatt5, Chapter 14], and the references therein for a comprehensive overview).

The main ingredients in the definitions of Lie algebra weight system on $\mathscr {A}^{\mathrm {h}}_N$ are:

1. (1) a (finite-dimensional complex) Lie algebra $\mathfrak {g}$ ;

2. (2) an $\mathrm {ad}$ -invariant nondegenerate bi-linear form $\langle \cdot , \cdot \rangle $ on $\mathfrak {g}$ ; and

3. (3) an ordered collection of finite-dimensional $\mathfrak {g}$ -representations $\underline {\rho } = (\rho _1, \ldots , \rho _N)$ , called labeling where $\rho _i:\mathfrak {g} \to \mathrm {End}(V_i)$ for each $i=1,\ldots ,N$ .

The basic idea is to associate with each horizontal chord diagram $C\in \mathscr {A}^{\mathrm {h}}_N$ an element in $\mathrm {End}(V_1 \otimes \cdots \otimes V_N)$ , and then take the trace to obtain a complex number.

First, fix an orthonormal basis (with respect to $\langle \cdot , \cdot \rangle $ ) for $\mathfrak {g}$ , say $e_1,\ldots ,e_d$ . This choice, and the fact that $\langle \cdot , \cdot \rangle $ is nondegenerate, allow us to identify the aforementioned bilinear form with the element

$$ \begin{align*}\sum_{i=1}^{\mathrm{dim}(\mathfrak{g})} e_i \otimes e_i \in \mathfrak{g} \otimes \mathfrak{g} .\end{align*} $$

We can decompose, by creating a unique minima on each chord, each horizontal chord diagram in local pieces like the ones shown in Figure 3, which correspond to the illustrated maps. Interpreting horizontal juxtaposition as tensor product, and vertical composition as the usual composition, we obtain the desired map. (Crossings between chords and Wilson lines have no meaning.) More explicitly, to each chord $C_{i,j}^{N} = [(i,j)]\in \mathscr {A}^{\mathrm {h}}_N$ , we are associating the element

$$\begin{align*}\widetilde{W}_{\underline{\rho}}(C_{i,j}^{N}) = \sum_{r=1}^{\mathrm{dim}(\mathfrak{g})} \mathrm{Id}_{V_1}\otimes \cdots \otimes \overset{i\text{th pos.}}{\rho_i(e_r)} \otimes \cdots \otimes \overset{j\text{th pos.}}{\rho_j(e_r)}\otimes \cdots \otimes \mathrm{Id}_{V_N}\, .\end{align*}$$

Figure 3

Dictionary between graphical and abstract construction of Lie algebra weight systems.

It can be shown that $\widetilde {W}_{{\underline {\rho }}}$ induces a well-defined morphism of $\star $ -algebras between $\mathscr {A}^{\mathrm {h}}_N$ and $\mathrm {End}(V^{\otimes N})$ (cf. [Reference Bar-Natan1, Section 2.1], [Reference Jackson and Moffatt5, Sections 14.2 and 14.3]). We can now give the definition of Lie algebra weight system.

Definition 2.6 Given a metrized Lie algebra $(\mathfrak {g},\langle \cdot , \cdot \rangle )$ and an ordered collection ${\underline {\rho } = (\rho _1, \ldots , \rho _N)}$ of $\mathfrak {g}$ -representations, the corresponding Lie algebra weight system is given by

$$\begin{align*}W_{\underline{\rho}} (C) = \mathrm{Tr}(\widetilde{W}_{\underline{\rho}} (C)), \end{align*}$$

for each $C\in \mathscr {A}^{\mathrm {h}}_N$ .

The $\mathfrak {gl}_{n}$ -weight system $W_{\mathtt {st}}$ , associated with the bilinear form $\langle A , B \rangle = \mathrm {Tr}( AB)$ and to the defining representation, admits a simple combinatorial description in terms of permutations, as recalled in equation (2.2) (cf. [Reference Bar-Natan1]). There exists a similar description for more general $\mathfrak {gl}_{n}$ -weight system—compare with [Reference Nagasato6].

Definition 2.7 Let $i_{0}\in \{1,\ldots ,N\}$ , $0\leq k \leq N-i_0$ , and $\lambda $ be a partition of k. The small Young symmetrizer $c_{(N,i_0,\lambda )}\in \mathbb {C}[\mathfrak {S}_N]$ is the image of the Young symmetrizer $c_\lambda \in \mathbb {C} [\mathfrak {S}_k]$ via the inclusion

$$\begin{align*}\mathbb{C}[\mathfrak{S}_k] \overset{\cong}{\longrightarrow} \mathbb{C}[\mathfrak{S}_{\{i_0 + 1 ,\ldots, i_0 + k\}}] \subset \mathbb{C}[\mathfrak{S}_N]\ .\end{align*}$$

Given a vector of $\mathfrak {gl}_{n}$ -representations $\underline {\rho } = (\lambda ^1,\ldots ,\lambda ^N)$ , we can define the Young symmetrizer associated with $\underline {\rho }$ as the product

The weight system $W_{\underline {\rho }}$ can be also defined as the following composition:

$$\begin{align*}\mathscr{A}^{\mathrm{h}}_N \overset{\Delta^{(\vert \lambda^1\vert ,\ldots, \vert \lambda^N\vert )}}{\longrightarrow} \mathscr{A}^{\mathrm{h}}_{\vert {\underline{\rho}}\vert} \overset{\sigma}{\longrightarrow} \mathbb{C}[\mathfrak{S}_{\vert {\underline{\rho}}\vert}] \overset{c_{\underline{\rho}}}{\longrightarrow} \mathbb{C}[\mathfrak{S}_{\vert {\underline{\rho}}\vert}] \overset{w_{\mathtt{st}}}{\longrightarrow} \mathbb{C}\ ,\end{align*}$$

where —cf. [Reference Nagasato6, Reference Nagasato and Takamuki7]. Roughly speaking, we are writing each $\lambda ^i$ as the composition of the representation $\rho _{\mathtt {st}}^{\otimes \vert \lambda ^i\vert }$ —encoded by $\Delta ^{(\vert \lambda ^1\vert ,\ldots , \vert \lambda ^N\vert )}$ , cf. [Reference Bar-Natan1, Definition 2.2]—and the projection onto $\mathbb {S}_{\lambda }(\mathbb {C})$ —encoded by the multiplication by $c_{\underline {\rho }}$ —and taking the trace by applying $w_{\mathtt { st}}$ .