Tensor decompositions for SL(2) and outerplanar graphs

Aleksandrs Mihailovs Department of Mathematics University of Pennsylvania Philadelphia, PA 19104-6395 mihailov@math.upenn.edu http://www.math.upenn.edu/∼mihailov/

November 24, 1997

Abstract

The main result of this article is the decomposition of tensor products of representations of SL(2) in the sum of irreducible representations parametrized by outerplanar graphs. An outerplanar graph is a graph with the vertices 0, 1, 2, …, m, edges of which can be drawn in the upper half-plane without intersections. We allow for a graph to have multiple edges, but don’t allow loops. In more detail,

ρ_{d_{1}} ⊗ ⋯ ⊗ ρ_{d_{m}} = ⊕_{G} T_{G} ,

where ρ_{d} denotes the irreducible representation of dimension d + 1, and the direct sum is taken over all the outerplanar graphs of degrees d_{0}, d_{1}, …, d_{m} with all possible values of d_{0}. T_{G} is an irreducible subrepresentation of the type ρ_{d_{0}}, and we determine explicit formulas for the basis in the space of T_{G} as well.

1 Introduction

In the classical quantum mechanics, each particle corresponds to an irreducible representations of

SL(2) of dimension 2s + 1, where s ∈ \frac{1}{2} ℤ is the spin of the particle. For example, a photon having

spin 0 corresponds to the trivial representation of dimension 1, and an electron having spin 1/2

corresponds to the standard 2-dimensional representation of SL(2):

(\begin{matrix}a & b \\ c & d \end{matrix}) (\begin{matrix}A \\ B \end{matrix}) = (\begin{matrix}aA + bB \\ cA + dB \end{matrix}) .  (1)

The projections of the spin correspond to the elements of the fixed basis of the space of the rep-

resentation, described below in section 2 just before (14). To study the systems of a few particles, one needs to know explicit formulas for the bases of the isotypic components of the tensor product of corresponding representations, and especially the basis of the isotypic component of the type of the trivial representation, i. e. the basis of the subspace of invariants of the tensor product of representations. These bases are of interest in the valency theory and a few other branches of physics as well as mathematics.

The basis of the subspace of SL(2)-invariants of V^{⊗m} = V ⊗ ⋯ ⊗ V (m times), where V is the two-dimensional linear space with the standard action (1) of SL(2), was described in the terms of the 0-outerplanar graphs in the classic work [16] of Rumer, Teller and Weyl. This theory was developed and applied to the percolation theory by Temperley and Lieb [17], to the knots theory and invariants of 3-manifolds by Jones [5], Kauffman [6], Kauffman and Lins [7], Wenzl [19], Jaeger [4], Lickorish [9], Masbaum and Vogel [11], and others, to the quantum theory by Penrose [14] and Moussouris [12], to quantum groups and the quantum link theory by Reshetikhin and Turaev [15], Ohtsuki and Yamada [13], Carter, Flath and Saito [1] and others, especially to the theory of Lusztig’s canonical bases [10] by Khovanov and Frenkel [2], Varchenko [18] and Frenkel, Varchenko and Kirillov, Jr. [3]. I am sorry that I am not able to cite everybody who made a contribution, because the literature on this topic is very extensive. I became familiar with the description of the basis of the invariants of the tensor products of any irreducible representations of SL(2) in the terms of the 0-outerplanar graphs from Kuperberg’s work [8]. Some people, and Frenkel and Khovanov [2] in particular, constructed bases not only in the invariants, but in the other components as well. The bases constructed in this work, are different. In all the previous works, the proofs used straightening algorithm. They showed that each invariant could be expressed as a linear combination of invariants corresponding to the 0-outerplanar graphs. That, after the calculation of the dimensions of an invariant space, and finding the number of 0-outerplanar graphs, confirmed that the given invariants form a basis.

In section 2 I give new proof of the result of Rumer, Teller and Weyl [16]. The method of my proof is the following. I show that the invariants corresponding to the 0-outerplanar graphs, are linearly independent. After the calculations of dimensions and numbers of 0-outerplanar graphs, it proves that the given invariants form a basis.

In section 3 I give new formulation for the extension of Rumer, Teller and Weyl’s work to the description of invariants of tensor products of any irreducible representations of SL(2), using slightly different 0-outerplanar graphs, than in [8]. Then I give the new proof of the result, following the method described in the previous paragraph. Also, I give a few new explicit formulas of those invariants.

In section 4 I give a few examples and a few new formulas of the invariants.

In section 5 I introduce bases of all the isotypic components of tensor products of any irreducible representations of SL(2), parametrized surprizingly by outerplanar graphs as well. Also, I describe the subdivisions of those bases, giving the decomposition of the isotypic component in the sum of the irreducible representations. The proofs are close to ones given in the previous sections.

Recall that all classes of equivalence of the irreducible polynomial finite dimensional representations of SL(2) are parametrized by nonnegative integers, and we can choose the natural actions ρ_{k} of SL(2) in the symmetric powers S^{k}V, k = 0, 1, 2, … as their representatives.

Definitions. An outerplanar graph is a graph with the set of vertices {0, 1, 2, …, m} ⊆ H = {z ∈ ℂ | ℑz ≥ 0}, edges of which can be drawn in the upper half-plane H without intersections. We allow for a graph to have multiple edges, but don’t allow loops. Denote O(G) the set of directed graphs, underlying undirected graph of which is G. Let G be an outerplanar graph and g ∈ O(G). For each vertix k of g denote

x_{k}(g) = x^{d_{k}^{out}(g)}y^{d_{k}^{in}(g)} ∈ S^{d_{k}}V,  (2)

where d_{k}^{out}(g) is the number of arrows in g, beginning in k, d_{k}^{in}(g) is the number of arrows, ending in k, and

d_{k} = d_{k}^{out}(g) + d_{k}^{in}(g)  (3)

is the degree of k. In other words, we put x at the beginning of each arrow of g, and y —at the end, and multiply those xs and ys in each vertix. Denote

b_{g} = x_{1}(g) ⊗ ⋯ ⊗ x_{m}(g) ∈ S^{d_{1}}V ⊗ ⋯ ⊗ S^{d_{m}}V  (4)

and for any nonnegative integer i ≤ d_{0}(G) denote

t_{G,i} = ∑_{\begin{matrix}g∈O(G) \\ d_{0}^{in}(g)=i \end{matrix}}(−1)^{inv g}b_{g},  (5)

where inv g is the number of inversions in g, i. e. , the number of arrows (k, l) in g with k > l.

Theorem 3. For any fixed d_{0}, d_{1}, …, d_{m}, tensors t_{G,i} parametrized by all outerplanar graphs with degrees d_{0}, d_{1}, …, d_{m} and nonnegative integers i ≤ d_{0}, form a basis in the isotypic component of the type ρ_{d_{0}} in the representation ρ_{d_{1}} ⊗ ⋯ ⊗ ρ_{d_{m}}. For any fixed outerplanar graph G, the subspace T_{G} spanned by the basis (t_{G,0}, …, t_{G,d_{0}}), are invariant; the linear homomorphism

s_{G} : S^{d_{0}}V → T_{G}, x^{i}y^{d_{0}−i} ↦ \frac{(−1)^{i}}{\binom{d_{0}}{i}} t_{G,i}  (6)

defines the isomorphism of ρ_{d_{0}} and the subrepresentation of SL(2) in T_{G}, and

S^{d_{1}}V ⊗ ⋯ ⊗ S^{d_{m}}V = ⊕_{G} T_{G} ,  (7)

where the direct sum in the right hand side is taken over all the outerplanar graphs of degrees d_{0}, d_{1}, …, d_{m} with all possible values of d_{0}.

2 Invariants of tensor powers of the standard represesentation

Let f be a field of characteristic 0, and SL(2) —the group of 2 × 2 f-matrices with determinant 1, acting on 2-dimensional linear f-space V with basis (x, y) by the standard way (1):

(\begin{matrix}a & b \\ c & d \end{matrix}) (Ax + By) = (aA + bB)x + (cA + dB)y,  (8)

i. e.

x ↦ ax + cy, y ↦ bx + dy.  (9)

Then V ⊗ V is 4-dimensional linear f-space with the basis (x ⊗ x, x ⊗ y, y ⊗ x, y ⊗ y), and SL(2) acts on V ⊗ V through the tensor product of the standard actions (9), i. e.

x ⊗ x ↦ (ax + cy) ⊗ (ax + cy),  (10)

and so on:

\begin{aligned}(\begin{matrix}a & b \\ c & d \end{matrix}) (Ax ⊗ x + Bx ⊗ y + Cy ⊗ x + Dy ⊗ y) & = (a^{2}A + ab(B + C) + b^{2}D)x ⊗ x \\ & + (acA + adB + bcC + bdD)x ⊗ y \\ & + (acA + bcB + adC + bdD)y ⊗ x \\ & + (c^{2}A + cd(B + C) + d^{2}D)y ⊗ y. \end{aligned}  (11)

Lemma 1. The subspace of SL(2)-invariants of V ⊗ V is one-dimensional, and we can choose

x ∧ y = x ⊗ y − y ⊗ x  (12)

as a basis element in that space.

Proof. From (11),

(\begin{matrix}a & b \\ c & d \end{matrix}) (x ⊗ y − y ⊗ x) = (ad − bc)(x ⊗ y − y ⊗ x) = x ⊗ y − y ⊗ x.  (13)

It means that x ∧ y is invariant.

Recall some fundamental facts about the representations of SL(2). The word representation will mean below a polynomial finite dimensional linear representation over f. Every representation of SL(2) is equivalent to a sum of irreducible representations. All classes of equivalence of the irreducible representations are parametrized by nonnegative integers, and we can choose the natural actions ρ_{k} of SL(2) in the symmetric powers S^{k}V, k = 0, 1, 2, … as their representatives. In other words, ρ_{0} is a trivial 1-dimensional representation, ρ_{1} is the standard representation (8), and for k > 0 the representation ρ_{k} acts in the (k + 1)-dimensional linear f-space of the homogeneous polynomials of degree k of two variables x and y, which we can provide with the basis of monomials (x^{k}, x^{k−1}y, …, y^{k}), such that

x^{k} ↦ (ax + cy)^{k}, x^{k−1}y ↦ (ax + cy)^{k−1}(bx + dy), …, y^{k} ↦ (bx + dy)^{k}.  (14)

By definition,

S^{2}V = (V ⊗ V)/(fx ∧ y).  (15)

It means that

ρ_{1} ⊗ ρ_{1} ≃ ρ_{2} ⊕ ρ_{0}.  (16)

As we see, the subspace of invariants is 1-dimensional.

To study decompositions of tensor products of representations, it is convenient to consider characters, i. e. traces of the representations. For a diagonal matrix ( \begin{matrix}q & 0 \\ 0 & q^{−1} \end{matrix} ) formulas (14) give us

x^{k} ↦ q^{k}x^{k}, x^{k−1}y ↦ q^{k−2}x^{k−1}y, …, y^{k} ↦ q^{−k}y^{k}.  (17)

It means that

ch ρ_{k} = q^{k} + q^{k−2} + ⋯ + q^{−k} = \frac{q^{k+1} − q^{−(k+1)}}{q − q^{−1}} .  (18)

Characters are elements of the ℤ/2-graded ring

K = ℤ[q + q^{−1}] = K^{odd} ⊕ K^{even}.  (19)

Two representations are equivalent iff they have the same characters. For any representations σ, τ we have

ch σ ⊕ τ = ch σ + ch τ  (20)

and

ch σ ⊗ τ = ch σ · ch τ.  (21)

Characters of the representations ρ_{k} with odd (even) k form a basis of ℤ-module K^{odd} (or K^{even}, correspondingly), and it is useful to know the exact formulas for the coefficients in that basis of arbitrary odd (or even) character:

∑_{odd (even) k} C_{k}q^{k} = ∑_{odd (even) k≥0}(C_{k} − C_{k+2}) ch ρ_{k}.  (22)

Lemma 2. The dimension of the subspace of SL(2)-invariants in V^{⊗m} equals to the Catalan number

c_{n} = \frac{(2n)!}{n! (n + 1)!} = \binom{2n}{n} − \binom{2n}{n − 1}  (23)

for m = 2n, and 0 otherwise.

Proof. Using formula (22), we get

(ch ρ_{1})^{m} = (q + q^{−1})^{m} = ∑_{k≡mmod2} \binom{m}{(m − k)/2}q^{k} = ∑_{k=0}^{m} c_{m}^{(k)} ch ρ_{k},  (24)

where

c_{m}^{(k)} = \begin{cases}\binom{m}{(m−k)/2} − \binom{m}{(m−k)/2−1} & if m − k is even, \\ 0 & otherwise. \end{cases}  (25)

In particular, the dimension of the subspace of invariants in V^{⊗m} is equal to c_{m}^{(0)} = c_{n} for m = 2n, and 0 otherwise.

Catalan numbers c_{n} count the number of 0-outerplanar regular graphs of degree 1 with m = 2n vertices.

Definition 1. A 0-outerplanar graph is a graph with the set of vertices {1, 2, …, m} ⊆ H = {z ∈ ℂ | ℑz ≥ 0}, edges of which can be drawn in the upper half-plane H without intersections.

For example, here are two of c_{3} = 5 0-outerplanar regular graphs of degree 1 with 6 vertices:

[figure: 1; 2; 3; 4; 5; 6; 1; 2; 3; 4; 5; 6]

The tensor products of invariants are invariants. Edges of each 0-outerplanar regular graph of degree 1 give us n pairs of numbers from 1 to m = 2n. If we take the tensor product of the known invariants x ∧ y in each of those pairs, we get an invariant in V^{⊗m}. The count of such invariants is equal to the count of the considered 0-outerplanar graphs, i. e. with the Catalan number c_{n}. And we know that the dimension of the subspace of invariants in V^{⊗m} equals the same Catalan number c_{n}. It is natural to suppose that the constructed invariants form a basis of the space of invariants. To prove that, it is enough to show that they are linearly independent.

The standard basis B of V^{⊗m} consists of 2^{m} tensor products x_{1} ⊗ ⋯ ⊗ x_{m} with x_{1}, …, x_{m} ∈ {x, y}. We suppose that B is ordered lexicographically. Denote O(G) the set of directed graphs, underlying undirected graph of which is G. Let G be an 0-outerplanar regular graph of degree 1, and g ∈ O(G). Denote

b_{g} = x_{1}(g) ⊗ ⋯ ⊗ x_{m}(g) ∈ B  (26)

setting x_{i}(g) = x, x_{j}(g) = y for each edge (i, j) in g. In other words, we put x at the beginning of each arrow of g, and y —at its end. We can write the tensor product of x ∧ y corresponding to G that was introduced in the last paragraph as

t_{G} = ∑_{g∈O(G)}(−1)^{inv g}b_{g},  (27)

where inv g is the number of inversions in g, i. e. , the number of arrows (i, j) in g with i > j.

Theorem 1. Tensors t_{G} parametrized by all regular 0-outerplanar graphs of degree 1, form a basis in the subspace of invariants in the representation ρ_{1}^{⊗m}.

Proof. Notice that for each G exists exactly one g_{0} ∈ O(G) without inversions—with the orientation of each edge from the left to the right. Changing the orientation of the arrows of g_{0} increases b_{g} in the lexicografical order of B. It means that b_{g_{0}} is the minimal element with a non-zero coefficient in the decomposition of t_{G} in the basis B. Denote b_{G} = b_{g_{0}}. For the element of type b_{G}, we can reconstruct G, associating the left bracket with each x, the right bracket with each y, and connecting the corresponding left and right brackets. So, we have c_{n} elements b_{G} —one for each G.

To prove the linear independence of the set of t_{G}, we can show that the rank of the c_{n}×2^{m} matrix of the coefficients of t_{G} in the basis B is equal to c_{n}. To do that, we can find a non-zero c_{n} ×c_{n} minor of that matrix. Consider the c_{n} × c_{n} submatrix with rows numbered by G ordered the same way as b_{G}, and columns corresponding to b_{G}. As we noticed in the previous paragraph, b_{G} is the first element with a nonzero coefficient in the row G, and this coefficient equals 1 by definition. So, this matrix is unipotent, its determinant equals 1, that completes the proof of the linear independence of t_{G}.

3 General tensor invariants

Now we allow for a graph to have multiple edges, but don’t allow loops. Let G be an 0-outerplanar graph and g ∈ O(G). For each vertix i of g denote

x_{i}(g) = x^{d_{i}^{out}(g)}y^{d_{i}^{in}(g)} ∈ S^{d_{i}}V,  (28)

where d_{i}^{out}(g) is the number of arrows in g, beginning in i, d_{i}^{in}(g) is the number of arrows, ending in i, and

d_{i} = d_{i}^{out}(g) + d_{i}^{in}(g)  (29)

is the degree of i. In other words, we put x at the beginning of each arrow of g, and y —at the end, and multiply those xs and ys in each vertix. Denote

b_{g} = x_{1}(g) ⊗ ⋯ ⊗ x_{m}(g) ∈ S^{d_{1}}V ⊗ ⋯ ⊗ S^{d_{m}}V  (30)

and

t_{G} = ∑_{g∈O(G)}(−1)^{inv g}b_{g}.  (31)

Theorem 2. For any fixed d_{1}, …, d_{m}, tensors t_{G} parametrized by all 0-outerplanar graphs with degrees d_{1}, …, d_{m}, form a basis in the subspace of invariants in the representation ρ_{d_{1}} ⊗ ⋯ ⊗ ρ_{d_{m}}.

Proof. First, we can compare the dimension of the space of invariants and the number of 0-outer-planar graphs with the given degrees. Using characters, we get

ch ρ_{d_{1}} ⊗ ⋯ ⊗ ρ_{d_{m}} = ∏_{i=1}^{m}(q^{d_{i}} + q^{d_{i}−2} + ⋯ + q^{−d_{i}}) = ∑_{k≡|d|mod2} C_{d}^{(k)}q^{k} = ∑_{k=0}^{|d|} c_{d}^{(k)} ch ρ_{k},  (32)

where d = (d_{1}, …, d_{m}), |d| = d_{1} + ⋯ + d_{m} and

c_{d}^{(k)} = \begin{cases}C_{d}^{(k)} − C_{d}^{(k+2)} & if |d| − k is even, \\ 0 & otherwise. \end{cases}  (33)

Applying the Clebsh-Gordon formula to the last two items of the product, we get

ch ρ_{d_{m−1}} · ch ρ_{d_{m}} = ch ρ_{d_{m−1}+d_{m}} + ch ρ_{d_{m−1}+d_{m}−2} + ⋯ + ch ρ_{|d_{m}−d_{m−1}|}.  (34)

It gives us the recursion relation

c_{d}^{(k)} = c_{(d_{1},...,d_{m−2},d_{m−1}+d_{m})}^{(k)} + c_{(d_{1},...,d_{m−2},d_{m−1}+d_{m−2}−2)}^{(k)} + ⋯ + c_{(d_{1},...,d_{m−2},|d_{m}−d_{m−1}|)}^{(k)}  (35)

for any k, and in particular, for k = 0 defining the dimension of the space of invariants. That relation makes it possible to decrease m, and for m = 1 we have the initial conditions

c_{(d)}^{(k)} = δ_{dk},  (36)

because the representations ρ_{d} and ρ_{k} are irreducible.

Denote c_{d} the number of 0-outerplanar graphs with degrees of vertices d = (d_{1}, ..., d_{m}). For any 0-outerplanar graph with the given degrees, contracting m to m − 1, and deleting after that all the loops at the last vertix, we get an 0-outerplanar graph with the degrees (d_{1}, …, d_{m−2}, d_{m−1} + d_{m} − 2a_{m−1,m}), where a_{m−1,m} denotes the number of edges between m − 1 and m —each edge has two ends—it explains the coefficient 2. The number a_{m−1,m} can be any integer between 0 and min{d_{m−1}, d_{m}}. That means the last degree in the new 0-outerplanar graph can be d_{m−1} + d_{m}, d_{m−1} + d_{m} − 2, …, |d_{m} − d_{m−1}|. This procedure is reversible: for any 0-outerplanar graph with admissible new degrees we can construct an 0-outerplanar graph with degrees d, moving the last d_{m} ends of edges from m − 1 to m and adding a_{m−1,m} edges between m − 1 and m. I wrote ’the last’ bearing in mind the natural order on the ends of edges of a vertix i of an 0-outerplanar graph: we can suppose that a small half-circle with the center in i, is lying in the upper half-plane H, which intersects with each edge ending in i, exactly in one point, and we can order these points on the half-circle clockwise, and these ends of edges—correspondingly. So, we have a bijection between the set of 0-outerplanar graphs with degrees d, and the union of the sets of 0-outerplanar graphs of the mentioned above degrees. It gives us the same recursion relation as (35) for c_{d}^{(k)}:

c_{d} = c_{(d_{1},...,d_{m−2},d_{m−1}+d_{m})} + c_{(d_{1},...,d_{m−2},d_{m−1}+d_{m−2}−2)} + ⋯ + c_{(d_{1},...,d_{m−2},|d_{m}−d_{m−1}|)}.  (37)

For m = 1 we have only one 0-outerplanar graph—with one vertix 1 and without edges. So, we have the same initial conditions as (36) for c_{(d)}^{(0)}:

c_{(d)} = δ_{d0}.  (38)

Thus, we proved that the dimension c_{d}^{(0)} of the space of invariants is equal to the number c_{d} of the 0-outerplanar graphs with the given degrees d.

The proof of the linear independence of the set of t_{G} is exactly the same as for the case of regular 0-outerplanar graphs of degree 1. Denote B the standard basis of S^{d_{1}}V ⊗ ⋯ ⊗ S^{d_{m}}V, consisting of (d_{1} + 1) ⋯ (d_{m} + 1) tensor products x_{1} ⊗ ⋯ ⊗ x_{m} with x_{i} ∈ {x^{d_{i}}, x^{d_{i}−1}y, …, y^{d_{i}}} for i = 1, …, m. We suppose that B is ordered lexicographically. We have b_{g} ∈ B for all g ∈ O(G). For each G exists exactly one g_{0} ∈ O(G) without inversions - with orientation each edge from the left to the right. Changing the orientation of the arrows of g_{0} increases b_{g} in the lexicografical order of B. It means that b_{g_{0}} is the minimal element with a non-zero coefficient in the decomposition of t_{G} in the basis B. Denote b_{G} = b_{g_{0}}. In any vertix i of g_{0} we have, first of all the incoming arrows, and then - all the arrows coming out, in the order of the ends of the edges described above. Thus, for an element of type

b_{G} = x^{d_{1}^{out}}y^{d_{1}^{in}} ⊗ ⋯ ⊗ x^{d_{m}^{out}}y^{d_{m}^{in}},  (39)

we can reconstruct G, associating the sequence of d_{i}^{in} in the right brackets followed by d_{i}^{out} in the left brackets, with each vertix i , and the connecting corresponding left and right brackets. So, we have c_{d} elements b_{G} —one for each G. Consider the c_{d} × c_{d} submatrix with rows numbered by G ordered the same way as b_{G}, and columns corresponding to b_{G}, of the c_{d} × |B| matrix of the coefficients of t_{G} in the basis B. As we noticed, b_{G} is the first element with a nonzero coefficient in the row G, and this coefficient equals 1 by definition. So, this matrix is unipotent, its determinant equals 1, that completes the proof of the linear independence of t_{G}.

To complete the proof of the theorem, we have to show that the tensors t_{G} are SL(2)-invariant. The representation ρ_{d_{1}} ⊗ ⋯ ⊗ ρ_{d_{m}} is a subrepresentation of ρ^{⊗m} in (SV)^{⊗m}, where SV = f[x, y] and ρ denotes the standard changing of variables. (SV)^{⊗m} is an algebra, and ρ^{⊗m} commutates with its multiplication

(a_{1} ⊗ ⋯ ⊗ a_{m})(b_{1} ⊗ ⋯ ⊗ b_{m}) = a_{1}b_{1} ⊗ ⋯ ⊗ a_{m}b_{m},  (40)

because ρ commutates with the multiplication in SV. In particular, a product of invariants is invariant. It follows from the definition that

t_{G} = ∏_{(i,j)∈E(g_{0})} ι_{ij}(x ∧ y),  (41)

where E(g_{0}) denotes the set of the arrows of g_{0}, and

ι_{ij} : V ⊗ V → (SV)^{⊗m}, v ⊗ w ↦ 1^{⊗(i−1)} ⊗ v ⊗ 1^{⊗(j−i−1)} ⊗ w ⊗ 1^{⊗(m−j)},  (42)

i. e. it puts v on the i-th place and w on the j-th place of the tensor product. Now, ι_{ij} commutate with the actions of SL(2), x ∧ y is invariant, and t_{G}, the product of invariants, is invariant.

Proposition 1. Let E(G) be the set of the edges of the 0-outerplanar graph G with m vertices. For any disjoint union

E(G) = E_{1} ⨿ ⋯ ⨿ E_{k}  (43)

we have

t_{G} = t_{G_{1}} … t_{G_{k}},  (44)

where G_{1}, …, G_{k} are the 0-outerplanar graphs with m vertices, with the sets of the edges E_{1}, …, E_{k} correspondingly.

Proof. It follows from (41) immediately.

4 Examples

If m = 1, we have exactly one 0-outerplanar graph: with the vertix 1 and without edges. It means that the subspace of invariants of ρ_{k} is nontrivial iff k = 0. In that case, the corresponding invariant is 1 ∈ f.

Corollary 1. The subspace of invariants of S^{d_{1}}V ⊗S^{d_{2}}V is nontrivial iff d_{1} = d_{2} = a for an integer a ≥ 0. In that case, the space of invariants is one-dimensional with the basis element

(x ∧ y)^{a} = ∑_{i=0}^{a}(−1)^{i}\binom{a}{i}x^{a−i}y^{i} ⊗ x^{i}y^{a−i} ∈ S^{a}V ⊗ S^{a}V.  (45)

Proof. If m = 2, the degrees d_{1}, d_{2} of any 0-outerplanar graph G are equal. For any integer a there is a unique 0-outerplanar graph with two vertices of degrees d_{1} = d_{2} = a. Formula (45) follows from (41).

[figure: 1; a; 1; 2; a; c; b; 1; 2; 3]

Corollary 2. The subspace of invariants of S^{d_{1}}V ⊗ S^{d_{2}}V ⊗ S^{d_{3}}V is nontrivial iff the integers d_{1}, d_{2} and d_{3} could be the lengths of the sides of a triangle (perhaps, degenerate, i. e. sides can have zero lenghth, or a vertix can be situated in the opposite side) with an even perimeter. In that case, the space of invariants is one-dimensional with the basis element

t_{G} = ((x ∧ y)^{a} ⊗ 1)(x ⊗ 1 ⊗ y − y ⊗ 1 ⊗ x)^{b}(1 ⊗ (x ∧ y)^{c}),  (46)

where

a = \frac{d_{1} + d_{2} − d_{3}}{2} , b = \frac{d_{1} + d_{3} − d_{2}}{2} , c = \frac{d_{2} + d_{3} − d_{1}}{2} .  (47)

Proof. If m = 3, we have a 0-outerplanar graph with a edges between 1 and 2, with b edges between 1 and 3, and c edges between 2 and 3 for any nonnegative integers a, b, and c. In this case

d_{1} = a + b, d_{2} = a + c, d_{3} = b + c,  (48)

and conversely (47). Formula (46) follows from (41).

In the degenerate cases some of a, b, c equal 0. If all of them equal 0, we have a trivial representation. If two of them equal 0, say b = c = 0, the situation is the same as for m = 2:

t_{G} = (x ∧ y)^{a} ⊗ 1 = ∑_{i=0}^{a}(−1)^{i}\binom{a}{i}x^{a−i}y^{i} ⊗ x^{i}y^{a−i} ⊗ 1 ∈ S^{a}V ⊗ S^{a}V ⊗ f.  (49)

If one of them equals 0, say b = 0, it follows from Corollary 1 and Proposition 1, that

\begin{aligned}t_{G} & = ((x ∧ y)^{a} ⊗ 1)(1 ⊗ (x ∧ y)^{c}) \\ & = ∑_{i=0}^{a} ∑_{j=0}^{c}(−1)^{i+j}\binom{a}{i}\binom{c}{j}x^{a−i}y^{i} ⊗ x^{c+i−j}y^{a−i+j} ⊗ x^{j}y^{c−j} ∈ S^{a}V ⊗ S^{a+c}V ⊗ S^{c}V. \end{aligned}  (50)

It is a special case of the following

Proposition 2. Let Γ_{G} denote the graph without multiple edges, with the same vertices as G, in which an edge between two vertices exists iff there are edges between these vertices in G. Then

t_{G} = ∑_{i_{1}=0}^{a_{1}} ⋯ ∑_{i_{N} =0}^{a_{N}}(−1)^{i_{1}+···+i_{N}} \binom{a_{1}}{i_{1}} … \binom{a_{N}}{i_{N}}b_{i_{1},...,i_{N}} ,  (51)

where N is the number of edges of Γ_{G}, and for each edge i_{k} of Γ_{G} we denote a_{k} the number of the edges of G connecting the same vertices, and b_{i_{1},...,i_{N}} = b_{g} for any directed graph g ∈ O(G) with the exactly i_{1} inverted arrows connecting the vertices of the 1-st edge of Γ_{G}, …, the exactly i_{N} inverted arrows connecting the vertices of the N-th edge of Γ_{G}. If Γ_{G} is a tree, i. e. a graph (not necessary connected) without cycles, then (51) is the decomposition of t_{G} in the elements of the basis B.

Proof. It follows from Corollary 1 and Proposition 1, that formula (51) is true for any 0-outerplanar graph G. We have to prove that for the considered case in which Γ(G) is a tree, tensors b_{i_{1},...,i_{N}} for different i_{1}, …, i_{N} are not equal. We’ll use the induction on N. For N = 0 we have only one item in the right hand side of (51), so the statement is true. Let N > 0 and we know that the statement is true for N − 1. Choose the vertix k of Γ(G) with degree 1 (it exists because Γ(G) is a tree). Then k-th component of b_{i_{1},...,i_{N}} equals by formula (28) to

x_{k} = \begin{cases}x^{a_{K}−i_{K}} y^{i_{K}} & if k < l, \\ x^{i_{K}} y^{a_{K}−i_{K}} & if k > l, \end{cases}  (52)

where K = {k, l} is the unique edge of Γ(G) having an end k, and another its end is l. It means that b_{i_{1},...,i_{N}} are different for different i_{K}. And if i_{K} are the same, then by induction, considering G without edges connecting k and l, we see that b_{i_{1},...,i_{N}} without k-th component, and without some factor on l-th place, are different for different i_{1}, …, i_{N}. They will differ on the same place after adding equal k-th components and multiplying l-th component on equal factors.

If Γ(G) is not a tree, some b_{...} in formula (51) can be the same. For instance, let a = b = c = 1. Then

(−1)^{inv g}b_{g} = \begin{cases}+x^{2} ⊗ xy ⊗ y^{2} & if g = \xymatrix{1 \ar[r] \ar[rr] & 2 \ar[r] & 3}, \\ −xy ⊗ xy ⊗ xy & if g = \xymatrix{1 \ar[r] & 2 \ar[r] & 3 \ar[ll]}, \\ −x^{2} ⊗ y^{2} ⊗ xy & if g = \xymatrix{1 \ar[r] \ar[rr] & 2 & 3 \ar[l]}, \\ +xy ⊗ y^{2} ⊗ x^{2} & if g = \xymatrix{1 \ar[r] & 2 & 3 \ar[l] \ar[ll]} , \\ −xy ⊗ x^{2} ⊗ y^{2} & if g = \xymatrix{1 \ar[rr] & 2 \ar[l] \ar[r] & 3}, \\ +y^{2} ⊗ x^{2} ⊗ xy & if g = \xymatrix{1 & 2 \ar[l] \ar[r] & 3 \ar[ll]}, \\ +xy ⊗ xy ⊗ xy & if g = \xymatrix{1 \ar[rr] & 2 \ar[l] & 3 \ar[l]}, \\ −y^{2} ⊗ xy ⊗ x^{2} & if g = \xymatrix{1 & 2 \ar[l] & 3 \ar[l] \ar[ll]} . \end{cases}  (53)

The items corresponding to cycles, cancel, and t_{G} is the sum of the other items. It is a special case of the following

Corollary 3. In the situation of the Corollary 2 for d_{1} = d_{2} = d_{3} = 2a,

t_{G} = (x^{2} ∧ xy ∧ y^{2})^{a}.  (54)

Proof. For a = 1 see (53), and for the other a it follows from this special case and Proposition 1.

5 Tensor decompositions

In this section we use slightly different outerplanar graphs than before: with vertices starting from 0 instead of 1:

Definition 2. An outerplanar graph is a graph with the set of vertices {0, 1, 2, …, m} ⊆ H = {z ∈ ℂ | ℑz ≥ 0}, edges of which can be drawn in the upper half-plane H without intersections.

We still allow for a graph to have multiple edges, but don’t allow loops. Also we’ll use the same formulas (28–30) to define x_{i}(g) and b_{g} for an directed graph g ∈ O(G). Note that formula (30) for b_{g} doesn’t include x_{0}(g). For any nonnegative integer i ≤ d_{0}(G) denote

t_{G,i} = ∑_{\begin{matrix}g∈O(G) \\ d_{0}^{in}(g)=i \end{matrix}}(−1)^{inv g}b_{g}.  (55)

Theorem 3. For any fixed d_{0}, d_{1}, …, d_{m}, tensors t_{G,i} parametrized by all outerplanar graphs with degrees d_{0}, d_{1}, …, d_{m} and nonnegative integers i ≤ d_{0}, form a basis in the isotypic component of the type ρ_{d_{0}} in the representation ρ_{d_{1}} ⊗ ⋯ ⊗ ρ_{d_{m}}. For any fixed outerplanar graph G, the subspace T_{G} spanned by the basis (t_{G,0}, …, t_{G,d_{0}}), are invariant; the linear homomorphism

s_{G} : S^{d_{0}}V → T_{G}, x^{i}y^{d_{0}−i} ↦ \frac{(−1)^{i}}{\binom{d_{0}}{i}} t_{G,i}  (56)

defines the isomorphism of ρ_{d_{0}} and the subrepresentation of SL(2) in T_{G}, and

S^{d_{1}}V ⊗ ⋯ ⊗ S^{d_{m}}V = ⊕_{G} T_{G} ,  (57)

where the direct sum in the right hand side is taken over all the outerplanar graphs of degrees d_{0}, d_{1}, …, d_{m} with all possible values of d_{0}.

Proof. First we’ll compare the multiplicity of ρ_{d_{0}} with the count of the outerplanar graphs. By formulas (32–33), this multiplicity equals

c_{d_{1},...,d_{m}}^{(d_{0})} = \begin{cases}C_{d_{1},...,d_{m}}^{(d_{0})} − C_{d_{1},...,d_{m}}^{(d_{0}+2)} & if d_{0} + d_{1} + ⋯ + d_{m} is even, \\ 0 & otherwise, \end{cases}  (58)

where the coefficients C_{·}^{·} are defined by the generating function

∑_{k} C_{d_{1},...,d_{m}}^{(k)}q^{k} = ∏_{i=1}^{m}(q^{d_{i}} + q^{d_{i}−2} + ⋯ + q^{−d_{i}}).  (59)

We can easily transform an outerplanar graph to a 0-outerplanar graph and backwards, shifting it to the right, or to the left, correspondingly. Thus the number of the outerplanar graphs with degrees d_{0}, …, d_{m} is equal to the number of the 0-outerplanar graphs with the same degrees, but shifted: the degree of 1 must be d_{0}, …, the degree of m + 1 must be d_{m}, i. e. equal to

c_{d_{0},d_{1},...,d_{m}} = c_{d_{0},d_{1},...,d_{m}}^{(0)}.  (60)

This equality was proved on the first step of the proof of Theorem 2. Now

\begin{aligned}c_{d_{1},...,d_{m}}^{(d_{0})} & = C_{d_{1},...,d_{m}}^{(d_{0})} − C_{d_{1},...,d_{m}}^{(d_{0}+2)} = C_{d_{1},...,d_{m}}^{(d_{0})} − C_{d_{1},...,d_{m}}^{(−d_{0}−2)} \\ & = res_{0}(q^{−d_{0}−1} − q^{d_{0}+1}) ∏_{i=1}^{m}(q^{d_{i}} + q^{d_{i}−2} + ⋯ + q^{−d_{i}}) \\ & = res_{0}(q^{−1} − q) ∏_{i=0}^{m}(q^{d_{i}} + q^{d_{i}−2} + ⋯ + q^{−d_{i}}) \\ & = C_{d_{0},d_{1},...,d_{m}}^{(0)} − C_{d_{0},d_{1},...,d_{m}}^{(2)} = c_{d_{0},d_{1},...,d_{m}}^{0} = c_{d_{0},d_{1},...,d_{m}}, \end{aligned}  (61)

where res_{0} denotes a residue in 0. It means that the number of outerplanar graphs of degrees d_{0}, d_{1}, …, d_{m} coincides with the multiplicity of ρ_{d_{0}} in ρ_{d_{1}} ⊗ ⋯ ⊗ ρ_{d_{m}}. Notice, that using formula (60) we are able to obtain that result very easily from the following equality as well:

mult_{ρ} τ = dim Inv(ρ^{∗} ⊗ τ)  (62)

for any irreducible representation ρ and representation τ, where mult_{ρ} τ is the multiplicity of ρ in τ, ρ^{∗} denotes the representation conjugated to ρ, and dim Inv is the dimension of the subspace of invariants. In our case

c_{d_{1},...,d_{m}}^{(d_{0})} = mult_{ρ_{d_{0}}} ρ_{d_{1}} ⊗ ⋯ ⊗ ρ_{d_{m}} = dim Inv(ρ_{d_{0}} ⊗ ρ_{d_{1}} ⊗ ⋯ ⊗ ρ_{d_{m}}) = c_{d_{0},d_{1},...,d_{m}}^{(0)}. (63)

On the second step of the proof we’ll show that the set of t_{G,0} is linearly independent. For each G exists exactly one g_{0} ∈ O(G) without inversions - with orientation each edge from the left to the right. Changing the orientation of the arrows of g_{0} with non-zero ends increases b_{g} in the lexicografical order of B. It means that b_{g_{0}} is the minimal element with a non-zero coefficient in the decomposition of t_{G,0} in the basis B. Denote b_{G} = b_{g_{0}}. Note that for the 0-outerplanar graph G′, obtained from G by shifting it to the right on 1, we have

b_{G′} = x^{d_{0}} ⊗ b_{G}.  (64)

In the proof of Theorem 2, we checked that all c_{d_{0},d_{1},...,d_{m}} elements b_{G′} are different for all the 0outerplanar graphs of degrees d_{0}, d_{1}, …, d_{m}. Thus, by formula (64), all c_{d_{0},d_{1},...,d_{m}} elements b_{G} are different for all the outerplanar graphs of degrees d_{0}, d_{1}, …, d_{m}. Consider the c_{d_{0},d_{1},...,d_{m}}×c_{d_{0},d_{1},...,d_{m}} submatrix with rows numbered by G ordered the same way as b_{G}, and columns corresponding to b_{G}, of the c_{d_{0},d_{1},...,d_{m}} × |B| matrix of the coefficients of t_{G,0} in the basis B. As we noticed, b_{G} is the first element with a nonzero coefficient in the row G, and this coefficient equals 1 by definition. So, this matrix is unipotent, its determinant equals 1, that completes the proof of the linear independence of t_{G,0}.

Third step: T_{G}. First consider the case of a star with the center in 0, i. e. G having degrees d_{0} = m, d_{1} = ⋯ = d_{m} = 1:

[figure: …; 0; 1; 2; m − 1; m]

In this case

t_{G,i} = (−1)^{i}\binom{m}{i} Sym(x^{⊗i} ⊗ y^{⊗(m−i)}),  (65)

where x^{⊗i} = x ⊗ ⋯ ⊗ x (m times) and y^{⊗(m−i)} = y ⊗ ⋯ ⊗ y (m − i times) analogously, and

Sym(x_{1} ⊗ ⋯ ⊗ x_{m}) = \frac{1}{m!} ∑_{σ∈S_{m}} x_{σ(1)} ⊗ ⋯ ⊗ x_{σ(m)},  (66)

where S_{m} is the symmetric group of permutations of 1, …, m. Thus

T_{G} = S_{m}V = Sym(V^{⊗m}).  (67)

By the definition of the actions of SL(2) in S^{m}V and V^{⊗m}, the homomorphism

s_{G} : S^{m}V → S_{m}V, x^{i}y^{m−i} ↦ Sym(x^{⊗i} ⊗ y^{⊗(m−i)})  (68)

is intertwining, therefore s_{G} is an isomorphism and T_{G} = S_{m}V is an invariant subspace.

Now let G be a star with the center in 0 and multiple edges, i. e.

d_{0} = d_{1} + ⋯ + d_{m}.  (69)

Suppose for simplicity that all d_{1}, …, d_{m} are not zero. Denote G̃ the covering G star with the the same degree d_{0} of 0 but without multiple edges, i. e. with d_{1} = ⋯ = d_{d_{0}} = 1. Consider a projection

Pr = pr_{d_{1}} ⊗ ⋯ ⊗ pr_{d_{m}} : V^{d_{0}} → S^{d_{1}}V ⊗ ⋯ ⊗ S^{d_{m}},  (70)

where

pr_{d} : V^{⊗d} → S^{d}V  (71)

is the projection defining S^{d}. One can check that for any nonnegative integer i ≤ d_{0},

t_{G,i} = Pr(t_{G̃,i}).  (72)

Hense

T_{G} = Pr(TG̃).  (73)

By definitions (70–71), the projection Pr is intertwining with the SL(2)-actions. Therefore, T_{G} is an invariant subspace since TG̃ is an invariant subspace. The representation of SL(2) in T_{G} is a factor-representation of the irreducible representation of SL(2) in TG̃, and T_{G} is non-zero space, because it contains t_{G,0} that is not zero since the set of the tensors t_{G,0} is linearly independent—it was stated on the first step of the proof of the theorem. It means that the representation of SL(2) in T_{G} is equivalent ρ_{d_{0}}, and

s_{G} = Pr ∘ sG̃  (74)

is the corresponding isomorphism.

Adding a few vertices of degrees 0 to G doesn’t change the situation, because of the canonical isomorphism:

α : S^{d_{1}}V ⊗ ⋯ ⊗ S^{d_{m}}V → 1^{⊗n_{0}} ⊗ S^{d_{1}}V ⊗ 1^{⊗n_{1}} ⊗ ⋯ ⊗ 1^{⊗n_{m−1}} ⊗ S^{d_{m}} ⊗ 1^{⊗n_{m}},  (75)

adding 1’s to every place corresponding to a vertix of degree 0. In particular, we have

t_{G_{+},i} = α(t_{G,i}), T_{G_{+}} = α(T_{G}),  (76)

and

s_{G_{+}} = α ∘ s_{G},  (77)

where G_{+} is an outerplanar graph obtained from G by adding n_{0} vertices of degree 0 between 0 and 1, …, n_{k} vertices of degree 0 between k and k + 1, …, d_{m} vertices of degree 0 after m.

Now, one can check that for an arbitrary outerplanar graph G, and for any nonnegative integer i ≤ d_{0},

t_{G,i} = t_{G_{∗},i} t_{G_{0}},  (78)

where G_{0} is the 0-outerplanar graph obtained from G by deleting the vertex 0 together with all the edges ending in 0, and G_{∗} is the star obtained from G by deleting all the edges of G_{0}. Since we stated in Theorem 2 that t_{G_{0}} is invariant, and the multiplication is intertwining, it follows from (78) that the linear homomorphism

μ : T_{G_{∗}} → T_{G}, t ↦ t t_{G_{0}}  (79)

is intertwining and surjective. Further, T_{G} is nonzero, because it contains t_{G,0} ≠ 0, see above. Since the representation of SL(2) in T_{G_{∗}} is irreducible, and equivalent ρ_{d_{0}}, the representation of SL(2) in T_{G}, equivalent its factor-representation, is equivalent ρ_{d_{0}} as well, and

s_{G} = μ ∘ s_{G_{∗}}  (80)

is the corresponding isomorphism.

Final step. We already know that all the subspaces T_{G} are invariant, and the count of them is exactly the same as necessary for (57). Now we can use the following

Lemma 3. For any isotypic representation T of SL(2) and for any q ∈ f that is not a root of unity, all the eigenspaces of T ( \begin{matrix}q & 0 \\ 0 & q^{−1} \end{matrix} ) have the same dimension equal to the number of the irreducible components of T.

Proof. For ρ_{d}, the basis elements x^{d}, …, x^{i}y^{d−i}, …, y^{d} are the eigenvectors of ρ_{d}( \begin{matrix}q & 0 \\ 0 & q^{−1} \end{matrix} ) with the eigenvalues q^{d}, …, q^{d−2i}, …, q^{−d} that are all different if q is not a root of unity. Therefore, for T being a direct sum of n representations equivalent to ρ_{d}, decomposing each component in the sum of the indicated above eigenlines (i. e. eigenspaces of dimension 1), we get the decomposition of the space of T in the sum of n eigenlines with the eigenvalue q^{d}, …, n eigenlines with the eigenvalue q^{d−2i}, …, n eigenlines with the eigenvalue q^{−d}. It means that the dimension of each eigenspace is n.

Consider the sum of T_{G} for all G with fixed degrees d_{0}, d_{1}, …, d_{m}. It is a space of isotypic representation of the type ρ_{d_{0}}. It follows from formula (56) proved above, that each t_{G,i} is an eigenvector with the eigenvalue q^{d−2i} for the matrix in Lemma 3. In particular, the eigenspace with the eigenvalue q^{−d} contains c_{d_{0},d_{1},...,d_{m}}-dimensional space spanned by t_{G,0}. By Lemma 3, the number of irreducible components in the considered space is not less than c_{d_{0},d_{1},...,d_{m}}. We know that the number of the irreducible components of the type ρ_{d_{0}} in ρ_{d_{1}} ⊗ ⋯ ⊗ ρ_{d_{m}} is c_{d_{0},d_{1},...,d_{m}}. It means that the considered sum of c_{d_{0},d_{1},...,d_{m}} subspaces T_{G} is direct, and it is the space of the isotypic component of the type ρ_{d_{0}} in ρ_{d_{1}} ⊗ ⋯ ⊗ ρ_{d_{m}}. Taking the direct sum of the isotypic components, we get (57).

Proposition 3. Let E(G) be the set of the edges of the outerplanar graph G with m vertices. For any disjoint union

E(G) = E_{1} ⨿ ⋯ ⨿ E_{k}  (81)

and for any nonnegative integer i ≤ d_{0}(G) we have

t_{G,i} = ∑_{i_{1}+···+i_{k}=i} t_{G_{1},i_{1}} … t_{G_{k},i_{k}},  (82)

where G_{1}, …, G_{k} are the 0-outerplanar graphs with m vertices, with the sets of the edges E_{1}, …, E_{k} correspondingly, and we suppose in the sum (82) that i_{1} ≤ d_{0}(G_{1}), …, i_{k} ≤ d_{0}(G_{k}).

Proof. For a star without multiple edges, see (65), it follows directly from the the definition (55) of t_{G,i}. Using formula (78) and Proposition 1 we obtain the general case.

Acnowledgements. I would like to thank Fan Chung Graham for her encouragement and series of very useful suggestions; Alexandre Kirillov for familiarizing me with Greg Kuperberg’s work and giving me the opportunity to present my work at his seminar October 2, 1996; Greg Kuperberg himself for writing a nice article and giving me a lot of references; Murray Gerstenhaber and Michael Khovanov for useful discussions, and last but not least, my Gorgeous and Brilliant Wife, Bette, for her total support and love.

References

[1] J. S. Carter, D. E. Flath, and M. Saito, The Classical and Quantum 6j-symbols, Math. Notes 43, Princeton Univ. Press, Princeton (1995).

[2] I. B. Frenkel and M. G. Khovanov, Canonical bases in tensor products and graphical calculus for U_{q}(sl_{2}), Duke Math. J. 87 (1997), 409–480.

[3] I. Frenkel, A. Kirillov, Jr. and A. Varchenko, Canonical basis and homology of local systems. Preprint (1997).

[4] F. Jaeger, Confluent reduction of cubic plane maps, Graph Theory and Combinatorics International Conference. Marseille (1990).

[5] V. F. R. Jones, Index of subfactors, Invent. Math. 72 (1987), 1–25.

[6] L. H. Kauffman, State models and the Jones polynomial, Topology 26 (1987), 395–407.

[7] L. H. Kauffman and S. L. Lins, Temperley-Lieb recoupling theory and invariants of 3manifolds. Princeton University Press, Princeton, N. J. (1994).

[8] G. Kuperberg, Spiders for rank 2 algebras, Comm. Math Phys. 180 (1996), 109–151.

[9] W. B. R. Lickorish, Calculations with the Temperley-Lieb algebra, Comment. Math. Helv. 67 (1992), 571–591.

[10] G. Lusztig, Canonical bases arising from quantized enveloping algebras, J. Amer. Math. Soc. 3 (1990), 447–498.

[11] G. Masbaum and P. Vogel, 3-valent graphs and the Kauffman bracket, Pacific J. Math. 164 (1994), 361–381.

[12] J. P. Moussouris, Vector coupling coefficients and spin networks, Advances in Twistor Theory, Res. Notes Math. 37, Pitman, Boston (1979), 313–317.

[13] T. Ohtsuki ans S. Yamada, Quantum su(3) invariants via linear skein theory. Preprint (1995).

[14] R.Penrose, Angular momentum: An approach to combinatorial space-time, Quantum Theory and Beyond, ed. by T. A. Bastin, Cambridge Univ. Press, Cambridge (1969).

[15] N. Yu. Reshetikhin and V. G. Turaev, Ribbon graphs and their invariants derived from quantum groups, Comm. Math. Phys. 127 (1990), 1–26.

[16] G. Rumer, E. Teller, and H. Weyl, Eine für die Valenztheorie geeignete Basis der binaren Vektorinvarianten, Nachr. Ges. Wiss. Göttingen Math. -Phys. Kl. (1932), 499–504.

[17] H. N. V. Temperley and E. H. Lieb, Relations between the ‘percolation’ and ‘coloring’ problem and other graph-theoretical problems associated with regular planar lattices: some exact results for the ‘percolation’ problem, Proc. Roy. Soc. London, Ser. A 322 (1971), 251– 280.

[18] A. Varchenko, Multidimensional hypergeometric functions and representation theory of quantum groups. Adv. Ser. Math. Phys. 21, World Scientific, River Edge, N. J. (1995).

[19] H. Wenzl, On sequences of projections, C. R. Math. Rep. Acad. Sci. Canada IX (1987), 5–9.