Cylinder braiding for quantum linear groups

Tammo tom Dieck

1. Introduction and results

The braid group ZB_{k} associated to the Coxeter graph B_{k} has generators t, g_{1}, …, g_{k−1} and relations

tg_{1}tg_{1} = g_{1}tg_{1}t
tg_{i} = g_{i}t  i > 1
g_{i}g_{j}g_{i} = g_{j}g_{i}g_{j}  |i − j| = 1
g_{i}g_{j} = g_{j}g_{i}  |i − j| > 1.

This group is isomorphic to the group of topological braids with k strings in the cylinder (C \ 0) × [0, 1].

It is the purpose of this paper to describe a general construction of tensor representations of ZB_{k}, based on quantum linear groups.

A tensor representation of ZB_{k} is obtained from any four braid pair (X, F) on a module V over a commutative ring 𝔎. By definition, this consists of a Yang-Baxter automorphism (= R-matrix) X: V ⊗V → V ⊗V which satisfies the braid relation (X ⊗ 1)(1 ⊗ X)(X ⊗ 1) = (1 ⊗ X)(X ⊗ 1)(1 ⊗ X) on V ⊗ V ⊗ V and an automorphism F: V → V such that the four braid relation

X(F ⊗ 1)X(F ⊗ 1) = (F ⊗ 1)X(F ⊗ 1)X

holds on V ⊗ V. A representation of ZB_{k} on the k-fold tensor power V^{⊗k} is then obtained by the assignment

\begin{matrix}t & ↦ & F ⊗ 1 ⊗ ⋯ ⊗ 1 \\ g_{i} & ↦ & 1 ⊗ ⋯ ⊗ X ⊗ ⋯ ⊗ 1 = X_{(i)}, \end{matrix}

where the morphism X in X_{(i)} acts on the factors i and i + 1 .

The results of this paper are based on a fundamental four braid pair (1.3) which we now describe. Let q ∈ 𝔎^{×} (= the units of 𝔎) and write δ = q − q^{−1}. Let

P = (p_{ij} | 1 ≤ i, j ≤ n)

be a matrix with entries in 𝔎 such that p_{ii} = q and p_{ij}p_{ji} = 1 for i ≠ j. Let V be a free 𝔎-module with basis v_{1}, …, v_{n}. The assignment

(1.1) X(v_{i} ⊗ v_{j}) = \{ \begin{matrix}p_{ij}v_{j} ⊗ v_{i} & i = j \\ p_{ij}v_{j} ⊗ v_{i} & i > j \\ p_{ij}v_{j} ⊗ v_{i} + δv_{i} ⊗ v_{j} & i < j \end{matrix}

is a Yang-Baxter operator. (Compare [5, p. 171], devide by p^{1/2} and replace q by p^{−1/2}q.) The standard case p_{ij} = 1 for i ≠ j is the most interesting. We work with the multi-parameter version, since it makes some of the computations more transparent.

Let β_{j} (1 ≤ j ≤ n) and w be elements in 𝔎 such that

(1) z = β_{j}β_{n+1−j} is independent of j;

(2) β_{l}^{2} = wβ_{l} + z in case 2l = n + 1;

(3) ∏_{j} β_{j} ∈ 𝔎^{×}. Define an automorphism F: V → V by

(1.2) F (v_{j}) = \{ \begin{matrix}β_{j}v_{n+1−j} & for 2j ≤ n + 1 \\ β_{j}v_{n+1−j} + wv_{j} & for 2j > n + 1. \end{matrix}

We use the notation j′ = n + 1 − j. Here is the basic four braid pair; the proof (1.3) will be given in the next section.

(1.3) Theorem. Suppose p_{ij}p_{jk} = p_{ij′}p_{j′k} whenever i ≠ j, j′ and k ≠ j, j′. Then (X, F), defined by (1.1) and (1.2), is a four braid pair.

Henceforth we assume the hypothesis of (1.3).

Associated to the R-matrix X above is, by the FRT-construction, a cobraided bialgebra A = A(X) with braid form r = r_{X}: A ⊗ A → 𝔎. (See [5, VIII 5, VIII 6] or [6, 9.1] for back ground; r is called universal r-form in these references.) In our case, the algebra A is generated by T_{i}^{j}, 1 ≤ i, j ≤ n, with relations

\begin{aligned}qT_{i}^{m}T_{i}^{n} & = & p_{mn}T_{i}^{n}T_{i}^{m} \\ qT_{i}^{m}T_{j}^{m} & = & p_{ji}T_{j}^{m}T_{i}^{m} \\ p_{ji}T_{j}^{m}T_{i}^{n} & = & p_{mn}T_{i}^{n}T_{j}^{m} \\ δT_{j}^{m}T_{i}^{n} & = & p_{mn}T_{j}^{n}T_{i}^{m} − p_{ij}T_{i}^{m}T_{j}^{n} \end{aligned}

for m > n and i > j. The comultiplication μ of A is determined by μ(T_{i}^{j}) = Σ_{k}T_{i}^{k} ⊗ T_{k}^{j} and the counit ε by ε(T_{i}^{j}) = δ_{i}^{j}.

It was shown in [2] that any four braid pair (X, F) induces a so-called cylinder form f = f_{(X,F)}: A → 𝔎. We recall the notion of a cylinder form on a cobraided bialgebra (A, r). We denote the multiplication in the dual algebra C^{∗} of a coalgebra C as convolution ∗. A cylinder form on (A, r) is a 𝔎-linear map f: A → 𝔎 such that f is convolution invertible and satisfies in the convolution algebra (A ⊗ A)^{∗} the identity

(1.4) fm = (f ˆ⊗ε) ∗ rτ ∗ (ε ˆ⊗f) ∗ r,

where m: A ⊗ A → A is the multiplication, ε: A → 𝔎 the counit, f ˆ⊗g: a ⊗ b ↦ f(a)g(b), and τ(x ⊗ y) = y ⊗ x. In terms of formal notation μ(a) = ∑_{(a)} a_{1} ⊗ a_{2} and (μ ⊗ 1)μ(a) = ∑_{(a)} a_{1} ⊗ a_{2} ⊗ a_{3} for the (iterated) comultiplication, (1.4) assumes the following form

f(ab) = ∑_{(a),(b)} f(a_{1})r(b_{1} ⊗ a_{2})f(b_{2})r(a_{3} ⊗ b_{3})

T. tom Dieck 1. Introduction and results 3

for a, b ∈ A.

The quantum linear group GL_{X}(n) is a localization of A(X), and SL_{X}(n) is a quotient of A(X). In order to define them, one has to use the quantum determinant D (see [6, 9.2], [8], [9], [12] for this topic). In our case it is given as a sum over the symmetric group S_{n} (compare [3, p. 157])

(1.5) D = ∑_{σ∈S_{n}} α(σ)T_{1}^{σ(1)} ⋯ T_{n}^{σ(n)} = ∑_{σ∈S_{n}} α(σ)^{−1}T_{n}^{σ(n)} ⋯ T_{1}^{σ(1)}

with

α(σ) = ∏_{{μ<ν \atop σ(μ)>σ(ν)}}(−qp_{σ(ν)σ(μ)}).

It is not totally obvious that the two expressions of D define the same element of A. Since this fact is important for our applications, we indicate a proof as follows. We think of A(X) as being defined formally over the ring ZZ[p_{ij}, p_{ij}^{−1}]. The algebra has an anti-involution ι defined by ι(T_{i}^{j}) = T_{i}^{j}, ι(q) = q^{−1}, ι(p_{ij}) = p_{ij}^{−1}. It sends the first expression for D into the second one. We have the exterior algebra as a comodule over A. The exterior algebra is the algebra generated by v_{1}, …, v_{n} with relations v_{i}v_{i} = 0, v_{i}v_{j} +q^{−1}p_{ij}v_{j}v_{i} = 0 for i < j. The element v_{1}v_{2} … v_{n} = v generates a one-dimensional subcomodule with coaction μ: v ↦ D ⊗ v. We also have an anti-involution ι on the exterior algebra which is the identity on the v_{i} and defined on the coefficient ring as for A. The coaction μ is compatible with the anti-involutions μ ∘ ι = (ι ⊗ ι) ∘ μ. This compatibility and the fact that ι sends v to a multiple of v implies ι(D) = D.

In the standard case and some other cases, the quantum determinant D is a group-like central element of A (see [3, 6.3]). If this is the case, then the algebra GL(n) = GL_{X}(n) is defined as the localization of A(X) with respect to D, i. e., GL_{X}(n) = A(X)[y]/(Dy − 1). This is now a Hopf algebra, and the braid form r_{X} extends to a braid form on GL_{X}(n); see [3, 3.1] for a more general result. (In the standard case, these algebras are denoted O(GL_{q}(n)) and O(SL_{q}(n)) in [6, 9.2].)

We assume q^{1/n} ∈ 𝔎 and renormalize our R-matrix X′ = q^{−1/n}X. We assume that D is group-like and central. The quotient of A(X′) by the twosided ideal generated by D − 1 is then a Hopf algebra. We denote it by SL_{X}(n) = SL(n). The braid form r = r_{X′} passes to SL(n) and yields a braid form.

The braid form r = r_{X′} on A = A(X′) = A(X) satisfies

(1.6) r(x ⊗ D) = r(D ⊗ x) = ε(x)

for x ∈ A. By the axioms of a braid form, we have r(xy ⊗ D = r(x ⊗ D)r(y ⊗ D), since D is group-like. Therefore it suffices to prove (1.6) for x = T_{i}^{j}. But then it is a restatement of the equality β = τ in [3, 4.2].

The next proposition collects the formal results about cylinder forms.

(1.7) Proposition. Suppose that the quantum determinant is a group-like central element. Then the cylinder form f = f_{(X,F)} satisfies f(xD) = f(x)f(D) for x ∈ A. If f(D) ∈ 𝔎^{×}, then the assignment f(xD^{−l}) = f(x)f(D)^{−l} yields a cylinder form on GL(n). If f′ = f_{(X′,F)} satisfies f′(D) = 1, then f′ factors through the quotient map A(X′) → SL(n) and induces a cylinder form on SL(n).

Proof. The identity (1.4) together with (1.6) and the fact that D is group-like yields (in formal notation)

f(xD) = ∑_{(x)} f(x_{1})r(D ⊗ x_{2})f(D)r(x_{3} ⊗ D)
= ∑_{(x)} f(x_{1})ε(x_{2})f(D)ε(x_{3})
= f(x)f(D);

the last equality follows from the counit axiom. From f(xD) = f(x)f(D) we see that xD^{−s} ↦ f(x)f(D)^{−s} is a well-defined map on GL(n). By using the same property, one verifies that (1.4) still holds for the extended form on GL(n). Finally, it is clear that f vanishes on the ideal generated by D − 1 if f(D) = 1.□

In order to make use of (1.7) we have to know the value f(D). Theorem (1.8) will be obtained from a general analysis of tensor representations in section 3.

(1.8) Theorem. The cylinder form f_{(X,F)} assumes the following value on the quantum determinant

f(D) = (−q)^{−n(n−1)/2} ∏_{i>j} p_{i′j} ∏_{j} β_{j}.

In the standard case p_{ij} = 1 for i ≠ j, we have

∏_{i>j} p_{i′j} = q^{l} for n = 2l, n = 2l + 1.

If one works with X′ in place of X, then this value has to be multiplied by q^{1−n}, since the definition of f via t_{n} (see section 3) involves n(n − 1) R-matrices X.

Finally, we recall from [2] that a cylinder form f on a cobraided bialgebra (A, r) induces a cylinder braiding on the category of left A-comodules. Let μ_{M}: M → A ⊗ M, x ↦ ∑ x^{1} ⊗ x^{2} be a comodule structure on M in formal notation. Define t_{M}: M → M by x ↦ ∑ f(x^{1})x^{2}. Let z_{M,N}: M ⊗N → N ⊗M denote the braiding morphisms induced by the braid form r. Then for any two left A-comodules M and N the following identities hold

t_{M⊗N} = z_{N,M}(t_{N} ⊗ 1)z_{M,N}(t_{M} ⊗ 1) = (t_{M} ⊗ 1)z_{N,M}(t_{N} ⊗ 1)z_{M,N}.

Isomorphisms t_{M}: M → M which satisfy these identities are called a cylinder braiding on the category of left A-comudules. In particular, we obtain a four braid pair (z_{M,M}, t_{M}) for each comodule M.

We remark that the algebra of cylinder braidings can be used to construct invariant of coloured ribbon tangles in the cylinder along the lines of Turaev

T. tom Dieck 2. The four braid pair 5

2. The four braid pair

This section is devoted to the proof of (1.3). Consider the involutions σ(i, j) = (n + 1 − i, j) and τ(i, j) = (j, i) on the set of indices J = {(i, j) | 1 ≤ i, j ≤ n}. Since στστ = τστσ, they formally generate the dihedral group D_{8} of order 8. We decompose J into the orbits under the D_{8}-action. Recall the notation n+1−i = i′.

Let n = 2k. Then we have the orbit (i, i), (i′, i), (i, i′), (i, i′) of length 4. This is the orbit of (i, j) if i = j or i = j′. If i ≠ j, j′, then the orbit of (i, j) has length 8.

Let n = 2ℓ − 1. Then we have the fixed point (ℓ, ℓ). There is another orbit type of length 4, namely (ℓ, j), (j, ℓ), (j′, ℓ), (ℓ, j′) for j ≠ ℓ. In addition, we have the orbits of (i, i), i ≠ ℓ, of length 4; and for i ≠ j, j′ and i ≠ ℓ ≠ j the orbit of (i, j) of length 8.

For each D_{8}-orbit x ∈ J/D_{8} the subspace of V ⊗ V spanned by {v_{i} ⊗ v_{j} | (i, j) ∈ x} is invariant under X and F ⊗ 1 = Y.

Therefore it suffices to verify the four braid relation on these subspaces. The structure of the matrices involved depends only on the D_{8}-isomorphism type of the orbit. When we display matrices we use the antilex(icographical) order: (i, j) < (a, b) if and only if j < b or j = b and i > a. This ordering is applied to the basis {v_{i} ⊗ v_{j} | (i, j) ∈ J} of V ⊗ V.

The fixed point orbit is trivial. Therefore we have to consider three cases.

Case (I): The orbit (i, i), (j, i), (i, j), (j, j) with i < j = i′. The matrices are

Y = ( \begin{matrix}0 & β_{j} \\ β_{i} & w \end{matrix} ; \begin{matrix}0 & β_{j} \\ β_{i} & w \end{matrix} ) X = ( q ; \begin{matrix}0 & p_{ij} \\ p_{ji} & δ \end{matrix} ; q ) .

One computes Y XY X to be the matrix of 2 × 2-blocks

( \begin{matrix}0 & qp_{ij}β_{j}F_{ij} \\ qβ_{i}p_{ji}F_{ij} & (q^{2} − 1)F_{ij}^{2} + wF_{ij} \end{matrix} ) with F_{ij} = ( \begin{matrix}0 & β_{j} \\ β_{i} & w \end{matrix} ) .

One obtains XY XY by transposition of Y XY X and the changes p_{ab} ↦ p_{ba}, β_{a} ↦ β_{a′}. This shows the four braid relation this case.

Case (II). The orbit (ℓ, i), (i, ℓ), (i′, ℓ), (ℓ, i′) for i < ℓ if n = 2ℓ − 1. The matrices are (j = i′)

Y = ( β_{ℓ} ; \begin{matrix}0 & β_{j} \\ β_{i} & w \end{matrix} ; β_{ℓ} ) X = ( \begin{matrix}0 & p_{iℓ} \\ p_{ℓi} & δ \end{matrix} ; \begin{matrix}0 & p_{ℓj} \\ p_{jℓ} & δ \end{matrix} ) .

Recall that β_{ℓ}^{2} = β_{ℓ}w + β_{i}β_{j}. (Strangely enough, Y is now an R-matrix and X has almost the form of the Y in case (I).) The necessity of the condition p_{iℓ}p_{ℓi} = p_{jℓ}p_{ℓj} results from the comparison of the 2 × 2-blocks on the codiagonal of XY XY = Y XY X. If we invoke this condition and use β_{ℓ}^{2} = β_{ℓ}w + z, we compute XY XY to be the matrix of 2 × 2-blocks

( \begin{matrix}0 & β_{j}β_{ℓ}p_{ℓj}P_{jℓ}^{iℓ} \\ β_{i}β_{ℓ}p_{jℓ}P_{ℓi}^{ℓj} & β_{ℓ}w(P_{jℓ}^{ℓj})^{2} + δβ_{i}β_{j}P_{jℓ}^{ℓj} \end{matrix} ) with P_{ab}^{cd} = ( \begin{matrix}0 & p_{cd} \\ p_{ab} & δ \end{matrix} ) .

Again, this matrix does not change under transposition and the changes as in case (I).

Case (III). The orbit of length eight for a < b ≤ [\frac{n}{2}] is (b, a), (b′, a), (a, b), (a′, b), (a, b′), (a′, b′), (b, a′), (b′, a′). For simplicity of notation we use n = 4, (a, b) = (1, 2). The matrices are in 2 × 2-block form

Y = ( \begin{matrix}Y_{23} \\ & Y_{14} \\ & & Y_{14} \\ & & & Y_{23} \end{matrix} ) Y_{ab} = ( \begin{matrix}0 & β_{b} \\ β_{a} & w \end{matrix} )

X = ( \begin{matrix}& & p_{12} \\ & & & & p_{13} \\ p_{21} & & δ \\ & & & & & & p_{24} \\ & p_{31} & & & δ \\ & & & & & & & p_{34} \\ & & & p_{42} & & & δ \\ & & & & & p_{43} & & δ \end{matrix} ) .

The necessity of the condition p_{ai}p_{ib} = p_{ai′}p_{i′b} results from the comparison of the 2 × 2-blocks on the codiagonal of Y XY X and XY XY. If we invoke this condition, then Y XY X turns out to be the following matrix of 2 × 2-blocks.

p_{13}p_{34}β_{4}Y_{23}

p_{24}p_{43}Y_{14}  δp_{24}β_{3}Y_{24}

p_{34}p_{42}β_{2}Y_{14}  wY_{14}  δp_{34}(zI + wY_{24})

p_{43}p_{31}β_{1}Y_{23} δp_{42}β_{2}Y_{13} δp_{43}(zI + wY_{13}) δ^{2}zI + (w + δ^{2}w)Y_{23}

Here I is a unit matrix. Again, transposition and the changes of case (I) do not change the matrix.

T. tom Dieck 3. The cylinder twist in the tensor representation 7

3. The cylinder twist in the tensor representation

Let E_{ij}: V → V be the linear map defined by E_{ij}(v_{r}) = δ_{ir}v_{j}. We use the same symbol E_{ij} for its matrix, it has 1 in position (j, i) and 0 in the other positions. If M: V → V is any linear map with matrix M = (m_{ab}), then

(3.1) E_{ji}ME_{ℓk} = m_{jk}E_{ℓi}.

We also use

E(r) = ∑_{j=1}^{r−1} E_{jj}, 1 ≤ r ≤ n.

This is the zero matrix for r = 1.

In order to organize the following computations, we need a suitable notation for matrices in tensor products. Suppose U and W are 𝔎-modules with basis u_{1}, …, u_{r} and w_{1}, …, w_{s}. We call

u_{1} ⊗ w_{1}, …, u_{r} ⊗ w_{1}; u_{1} ⊗ w_{2}, …, u_{r} ⊗ w_{2}; … u_{r} ⊗ w_{s}

the antilex(icographical) basis of U ⊗W. This ordering of a basis is an associative process. In particular, V (k) := V^{⊗k} has for any k ∈ IN a well-defined antilex basis. In the sequel, we always use the antilex basis, unless the contrary is explicitly stated.

Suppose f: U → U has matrix A. Then f ⊗ 1_{W}: U ⊗ W → U ⊗ W has the block diagonal matrix Dia(A, …, A) with rank (W) diagonal terms A.

Suppose g: W → W has matrix B = (b_{ij}). Then 1_{U} ⊗ f: U ⊗ W → U ⊗ W has the block matrix B, where b_{ij} now stands for the diagonal block b_{ij}I(u), with I(u) the unit matrix of size u = rank (U).

The matrix X in (1.1) is in this notation the n × n-block matrix with blocks of size n

(3.2) X = ( \begin{matrix}qE_{11} + δE(1) & & & p_{12}E_{12} & & & p_{13}E_{13} & & … & & p_{1n}E_{1n} \\ p_{21}E_{21} & & & qE_{22} + δE(2) & & & p_{23}E_{23} & & … & & p_{2n}E_{2n} \\ . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\ p_{n1}E_{n1} & & & p_{n2}E_{n2} & & & p_{n3}E_{n3} & & … & & qE_{nn} + δE(n) \end{matrix} ) .

By the remarks above, the matrix

X_{k} = 1_{V (k−2)} ⊗ X

on V^{⊗k} has a similar structure, but each entry is now a diagonal block of size n^{k−2}.

We still use the notation as in (3.2). Then E_{ij} is a block-matrix of size n, and its entries are blocks of size n^{k−2}. In this block notation (3.1) still holds.

Basis objects for our inductive computations are bottom-right triangular block-matrices with blocks of size m, or BRTB(m)-matrices for short. These are, by definition, n × n-matrices T = (T_{ij}) with:

(1) T_{ij} = 0 for i + j ≤ n.

(2) T_{ij} is a block of size m.

(3) The blocks T_{i,n+1−i} on the codiagonal are diagonal matrices.

The next proposition provides us with the basic inductive structure of BRTBmatrices with regard to the matrix X.

(3.3) Proposition. Let T = (T_{ij}) be a BRTB(n^{k−2})-matrix. Then

T̃ = (T̃_{ab}) = X_{k}(T ⊗ 1)X_{k}

is a BRTB(n^{k−1})-matrix.

Proof. We use the remarks above about the structure of matrices in tensor products, the structure of X_{k}, and the relation (3.1). Denote the n × n-matrix X in (3.2) by (e_{ab}). Then, by matrix multiplication,

(3.4)  T̃_{ab} = ∑_{j=1}^{n} e_{aj}T e_{jb}.

We use (3.1) with e_{aj} = p_{aj}E_{aj}, for a ≠ j, and e_{aa} = qE_{aa} + δ ∑_{j=1}^{a−1} E_{jj}. Suppose a + b ≤ n. Then, by assumption, T_{ab} = 0, and (3.1) gives T̃_{ab} = 0.

It remains to show that T̃_{ab} for a+b = n+1 is a diagonal matrix. From (3.1) and (3.4) we obtain T̃_{ab} = ∑_{j} p_{aj}p_{jb}T_{ab}E_{jj}, since δE(a)T e_{ab} = 0 and e_{ab}T δE(b) = 0 by the BRTB(n^{k−2})-property of T. This is the diagonal block matrix

(3.5) Dia(p_{a1}p_{1b}T_{ab}, p_{a2}p_{2b}T_{ab}, …, p_{an}p_{nb}T_{ab}).

We apply this result to the tensor representation obtained from the four braid pair (1.3). We set t(1) = F and define inductively

t(k) = X_{k}(t(k − 1) ⊗ 1)X_{k}.

We can apply (3.3) to the matrices t(k). We determine the diagonal blocks on the codiagonal of t(k). Since we are working in V^{⊗k}, we use a multi-index notation for the basis and other terms. A multi-index is a function i: {1, …, k} → {1, …, n}. Let A(k, n) denote the set of these functions. We write

v_{i} = v_{i(1)} ⊗ ⋯ ⊗ v_{i(k)} for i ∈ A(k, n).

The diagonal positions of the diagonal block t(k)_{a,n+1−a} of t(k) are indexed by multi-indices i with i(k) = a; here i denotes the columns of t(k). Let d(i) denote this diagonal term. From (3.5) we obtain by induction on k:

(3.6) d(i) = β_{i(k)} ∏_{j=1}^{k−1} p_{i(k)i(j)}p_{i(j)i(k)′}.

The cylinder twist t_{k}: V^{⊗k} → V^{⊗k} is, by definition, the automorphism defined inductively by

t_{1} = t(1) and t_{k} = (t_{k−1} ⊗ 1)t(k).

(3.7) Theorem. The matrix of t_{k} is a BRTB(1)-matrix. The codiagonal element ℓ(i) in column i ∈ A(k, n) is

ℓ(i) = ∏_{a<b} p_{i(b)i(a)}p_{i(a)i(b)′} · ∏_{a=1}^{k} β_{i(a)}.

Proof. The blocks of t_{k} are t_{k−1}t(k)_{ab}. This immediately gives that t_{k} is a BRTB(1)-matrix. The value of the codiagonal element is obtained inductively from (3.6). □

We mention the standard special case p_{ij} = 1 for i ≠ j. Let Or(n) be the orbit set of {1, …, n} under the involution i ↦ n + 1 − i. Let s(x) be the order of the isotropy group of x ∈ Or(n). In that case we have

(3.8) ℓ(i) = q^{λ(i)} ∏_{a=1}^{k} β_{i(a)} with λ(i) = ∑_{x∈Or(n)} s(x)\binom{|i^{−1}(x)|}{2}.

It may be helpful to see a specific example for these computations. We consider the case n = 3 and p_{ij} = 1 for i ≠ j. Then we obtain the matrix

t(2) = \begin{aligned}& & & & & & qβ_{3} \\ & & & & & & & β_{3} \\ & & & & & & & & qβ_{3} \\ & & & β_{2} & & & & δβ_{3} \\ & & & & q^{2}β_{2} \\ & & & & & β_{2} & & δβ_{2} \\ qβ_{1} & & & & & & w & & qδβ_{3} \\ & β_{1} & & δβ_{1} & & δβ_{2} & & w + δ^{2}β_{2} \\ & & qβ_{1} & & & & qδβ_{1} & & q^{2}w \end{aligned}

and finally the matrix for t_{2} = (t_{1} ⊗ 1)t(2) in the antilex basis.

qβ_{3}^{2}
β_{2}β_{3}
qβ_{1}β_{3}  qβ_{3}w
β_{2}β_{3}  δβ_{2}β_{3}
q^{2}β_{2}^{2}
β_{1}β_{2}  wβ_{2}  δβ_{2}^{2}
qβ_{1}β_{3}  qδβ_{1}β_{3}  q^{2}wβ_{3}
β_{1}β_{2}  δβ_{1}β_{2}  δβ_{2}^{2}  wβ_{2} + δ^{2}β_{2}^{2}
qβ_{1}^{2}  qβ_{1}w  q^{2}wβ_{1}  qδβ_{1}β_{3} + q^{2}w^{2}

The space V ⊗ V decomposes into the second symmetric and exterior power V ⊗ V = S^{2}V ⊕ Λ^{2}V. A basis of S^{2}V is v_{22}; u_{12}, u_{23}; v_{11}, u_{13}, v_{33} with v_{ij} = v_{i} ⊗ v_{j} and u_{ij} = v_{ij} + q^{−1}v_{ji}. The subspaces S^{2}V and Λ^{2}V are t_{2}-invariant. The matrix of t_{2} on S^{2}V in the basis above is a block diagonal matrix with blocks

(q^{2}β_{2}^{2}), \binom{qβ_{2}β_{3}}{qβ_{1}β_{2} qδβ_{2}^{2} + wβ_{2}} , ( \begin{matrix}& & qβ_{3}^{2} \\ & q^{2}β_{1}β_{3} & q^{2}wβ_{3} \\ qβ_{1}^{2} & (1 + q^{2})wβ_{1} & q^{2}w^{2} + qδβ_{1}β_{3} \end{matrix} ) .

The roots of the characteristic polynomial are q^{2}β_{2}^{2}, −β_{1}β_{3} and q^{2}(β_{2} − w)^{2}, with multiplicities 3, 2, and 1, respectively. Thus the matrix satisfies a relation of degree three.

The space Λ^{2}V has the basis w_{ij} = v_{ij} − qv_{ji}, i < j. The restriction of t_{2} to Λ^{2}V has with respect to the basis w_{12}, w_{13}, w_{23} the matrix

( \begin{matrix}& & −q^{−1}β_{2}β_{3} \\ & −β_{1}β_{3} \\ −q^{−1}β_{1}β_{2} & & −β_{1}β_{3} + q^{−2}β_{2}^{2} \end{matrix} ) .

The characteristic polynomial has the roots −β_{1}β_{3}, q^{−2}β_{2}^{2} with multiplicities 2, 1.

Proof of (1.8). We apply the preceding results in order to compute the value of the cylinder form on the quantum determinant D. For this purpose we have to recall from [2] the definition of the cylinder form f: A → 𝔎. Suppose t_{k}(v_{i}) = ∑_{j} F_{i}^{j}v_{j}, where i and j are multi-indices of length k. The cylinder form is then defined by f(T_{i}^{j}) = F_{i}^{j}.

We see from Theorem (3.7) that f(T_{n}^{σ(n)} ⋯ T_{1}^{σ(1)}) = 0 if (σ(n), …, σ(1)) ≠ (1, …, n). This claim is proved as follows: We have to consider the column (n, …, 1) of the matrix (3.7) for t_{n}. Among the (σ(n), …, σ(1)) the element (1, …, n) is the largest one in the antilex order. The claim thus follows from the BRTB-structure of the matrix. Thus only the term f(T_{n}^{1}T_{n−1}^{2} ⋯ T_{1}^{n}) contributes to f(D). The value for f(D), as stated in (1.8), is now a direct consequence of (3.7). Note that there is some cancellation of p_{ij}-factors due to the hypothesis of (1.3). □

By dualization and linear algebra (see e. g. [10], [12]) one can obtain from the results of this paper cylinder braidings on suitable categories of modules over quantum enveloping algebras (say integrable modules over U_{q}(sl_{n}) in the sense of [7]).

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T. tom Dieck 3. The cylinder twist in the tensor representation 11

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Version of February 9, 2009

Tammo tom Dieck Mathematisches Institut Bunsenstraße 3/5 D – 37073 G”ottingen tammo@uni-math.gwdg.de