BRAID GROUPS, INFINITE LIE ALGEBRAS OF

CARTAN TYPE AND RINGS OF INVARIANTS

Stephen P. Humphries

Abstract. In this paper we show that each element α of the pure braid group P_{n} or the pure symmetric automorphism group H(n) of the free group F_{n} of rank n can be represented as

α = exp(D) = id + D + \frac{D^{2}}{2!} + \frac{D^{3}}{3!} + …

where D = D(α) is an element of an infinite-dimensional Lie algebra 𝔥(n). Each such D is a derivation of the power series ring ℂ[[a_{1}, …, a_{r}]], r = n^{2} − n, which fixes the volume form a_{1} ∧⋯∧a_{r} and so 𝔥(n) is a subalgebra of S_{n^{2}−n}, the special Lie algebra of Cartan type. There is a corresponding action of these groups on ℂ[[a_{1}, …, a_{r}]] and ℂ[a_{1}, …, a_{r}]. We use the representation α = exp(D) to prove results about the ring of invariants for this action of the pure braid group. The Lie algebra 𝔥(n) is a subalgebra of a graded Lie algebra 𝔩(n); we also calculate the Poincaré series of the Lie algebra 𝔩(n) and of certain of its subalgebras, and show that these Poincaré series are rational.

§1. Introduction.

In this paper we show that each element α of the pure braid group P_{n} or the pure symmetric automorphism group H(n) of the free group F_{n} of rank n (defined below) can be represented as

α = exp(D) = id + D + \frac{D^{2}}{2!} + \frac{D^{3}}{3!} + …

where D = D(α) = log(α) is an element of an infinite-dimensional Lie algebra 𝔥(n) which is a sub-algebra of a Lie-algebra of Cartan type [Ca]. In fact each such D is a derivation of the power series ring ℂ[[a_{1}, …, a_{r}]], r = n^{2} −n, which fixes the volume form a_{1} ∧ ⋯ ∧ a_{r} and so 𝔥(n) is a subalgebra of S_{n^{2}−n}, the special Lie algebra of Cartan type. We use this representation of elements of P_{n} to prove results about the ring of invariants of the pure braid group for a certain action of P_{n} on the polynomial ring ℂ[a_{1}, …, a_{r}] that was studied in [Hu1, Hu2]. The Lie algebra 𝔥(n) is a subalgebra of a certain graded Lie algebra 𝔩(n); in sections 7-10 we calculate the Poincaré series of certain of the subalgebras of the Lie algebra 𝔩(n), and show that these Poincaré series are rational. We now explain these ideas in greater detail.

Typeset by AMS-TEX

1991 Mathematics Subject Classification. 20F36; 17B50, 17B65.

Key words and phrases. Pure braid group, Lie algebra, Poincaré series, derivation, ring of invariants.

Introduce the group H(n) =< t_{ij}|1 ≤ i ≠ j ≤ n > of pure symmetric automorphisms of the free group F_{n} =< x_{1}, …, x_{n} > of rank n. Here t_{ij} is the automorphism which conjugates x_{j} by x_{i} and fixes all x_{k} for k ≠ j. A set of relations for H(n) is given by McCool [Mc] :

t_{ij}t_{rs} = t_{rs}t_{ij} if card ({i, j, r, s}) = 4
t_{ij}t_{is} = t_{is}t_{ij} if card ({i, j, s}) = 3; and
t_{ij}t_{is}t_{js} = t_{js}t_{ij}t_{is} if card ({i, j, s}) = 3.

The symmetric group Σ_{n} also acts on F_{n} by permutations of {x_{1}, …, x_{n}}. Let Ĥ(n) =< H(n), Σ_{n} >. Then there is a split exact sequence

1 → H(n) → Ĥ(n) → Σ_{n} → 1.

Ĥ(n) is the group of symmetric automorphisms of the free group F_{n} of rank n [Mc], although this description will not concern us in this paper. We will now give a different interpretation of the groups H(n) and Ĥ(n) by making then act on a polynomial algebra.

For n, r ≥ 0 let ℂ_{(n+r)}[a_{ij}] denote the polynomial algebra

ℂ[a_{12}, a_{13}, …, a_{1,n+r}, a_{21}, a_{23}, …, a_{2,n+r}, …, a_{n+r,1}, a_{n+r,2}, …, a_{n+r,n+r−1}],

where the a_{ij} are commuting indeterminates. By [Hu1] the group H(n) acts on ℂ_{(n+r)}[a_{ij}] as follows: for λ ∈ ℤ we have

t_{kj}^{λ}(a_{ij}) = a_{ij} + λa_{ik}a_{kj}, if i ≠ j;
t_{kj}^{λ}(a_{ji}) = a_{ji} − λa_{jk}a_{ki}, if i ≠ j;
t_{kj}^{λ}(a_{hi}) = a_{hi},  if h ≠ j, i ≠ j.  (1.1)

Here t_{kj} = t_{kj}^{1}. Thus we also have t_{kj}^{λ}(a_{kj}) = a_{kj} and t_{kj}^{λ}(a_{jk}) = a_{jk}. If we let λ ∈ ℂ in (1.1), then we get elements t_{kj}^{λ} which generate a subgroup H̃(n) =< t_{kj}^{λ}|1 ≤ i ≠ j ≤ n, λ ∈ ℂ > of Aut(ℂ_{(n+r)}[a_{ij}]). Thus H̃(n) contains H(n) as a ‘discrete’ subgroup. We extend the action of H(n) and H̃(n) to the ring of formal power series ℂ_{(n+r)}[[a_{ij}]] in the obvious way. Let the symmetric group Σ_{n} act on ℂ_{(n+r)}[a_{ij}] (as a subgroup of Σ_{n+r}) by permutation of subscripts and let Ĥ(n) =< H(n), Σ_{n} >. The action of H(n) on ℂ_{(n+r)}[a_{ij}] is only faithful if r > 0, otherwise the kernel is the free normal subgroup generated by the ‘inner’ automorphisms {J_{k} = t_{k1}t_{k2} … t_{kn}|k = 1, …, n} [Hu1, Theorem 2.5]. We note as in [Hu2] that there is a natural ring involution ∗ on ℂ_{(n+r)}[a_{ij}] which commutes with the action of H̃(n); this involution is determined by its action on the generators a_{ij} which is as follows: a_{ij}^{∗} = −a_{ji}.

We now explain the ideas behind the above representation of Ĥ(n) as automorphisms of ℂ_{(n+r)}[a_{ij}]. This representation was motivated by the investigations of Magnus [Ma] who studied the action of some subgroups of the automorphism group Aut(F_{n}) on the ring F of Fricke characters. Here F is the ring generated by the traces of certain 2×2 matrices A_{1}, …, A_{n} ∈ SL(2, ℂ) which ‘generically’ generate a free group. In our approach we restrict the choice of A_{i} to being ‘generic’ transvections, while at the same time extending them to be n × n matrices. This has the advantage that the corresponding trace ring is a polynomial algebra, giving greater ease of computation, whereas in Magnus’s situation F is a quadratic extension of a polynomial algebra. The drawback is that we only obtain an action of Ĥ(n) on our ring, in contrast to the fact that the whole of Aut(F (n)) acts on F.

More specifically we have: Let T, T′ be two transvections in SL_{n+r}(ℂ_{(n+r)}[a_{ij}]). Thus relative to some choice of basis we have T = I_{n+r} + A, T′ = I_{n+r} + A′ ∈ SL_{n+r}(ℂ_{(n+r)}[a_{ij}]) where I_{n+r} is the identity element and A and A′ are rank 1 matrices over ℂ_{(n+r)}[a_{ij}] with A^{2} = A′^{2} = 0. Then trace(T T′) = n − xx^{∗} where x = x(T, T′) ∈ ℂ_{(n+r)}[a_{ij}]. Here there is a canonical choice of x over x^{∗}. Now it is possible to choose a set {T_{1}, …, T_{n+r}} of such transvections so that x(T_{i}, T_{j}) = a_{ij} for all 1 ≤ i ≠ j ≤ n+r. Then for i = 1, …, n+r and α ∈ Ĥ(n) the elements α(T_{i}) are also tranvections and we define the action of α on ℂ_{(n+r)}[a_{ij}] by requiring that

α(a_{ij}) = x(α(T_{i}), α(T_{j})).

Now the braid groups B_{n}, having standard generators σ_{1}, …, σ_{n−1} and presentation

σ_{i}σ_{i+1}σ_{i} = σ_{i+1}σ_{i}σ_{i+1}, for i = 1, …, n − 2
σ_{i}σ_{j} = σ_{j}σ_{i}, for |i − j| > 1,

are subgroups of Ĥ(n) and the pure braid groups P_{n} = H(n) ∩ B_{n} are subgroups of H(n) [Bi, p. 30]. Let Inn(n) =< J_{1}, …, J_{n} > be the subgroup of all inner automorphisms of F_{n}. Then Inn(n) is a normal subgroup of H(n) and of Ĥ(n) and we let

H(n)^{∗} = H(n)/Inn(n), Ĥ(n)^{∗} = Ĥ(n)/Inn(n).

We will first show that, for each α ∈ H̃(n), the element log(α) referred to in the first paragraph belongs to a Lie algebra 𝔩(n) defined as follows (the Lie algebra 𝔥(n) will be a subalgebra of 𝔩(n)): For 1 ≤ i, j ≤ n, an ij-word in ℂ_{(n)}[a_{rs}] is a monomial of the form

a_{ik}a_{kl}a_{lm} … a_{rs}a_{sj},  (1.2)

where 1 ≤ i, k, l, m, …, r, s, j ≤ n. Note that if i = j, then a word may be an ii word for many choices of i. For example, a_{12}a_{23}a_{31} is a 11 word as well as a 22 word. Let A(n; i, j) be the subspace of ℂ_{(n)}[a_{rs}] generated by all ij-words. Let

A(n) = {⊕ \atop 1 ≤ i ≠ j ≤ n}A(n; i, j).

We now make the A(n; i, j) (for 1 ≤ i ≠ j ≤ n) and A(n) into graded Lie algebras. Let x ∈ A(n; i, j) be an ij-word with 1 ≤ i ≠ j ≤ n. Associate to x the derivation x^{#} = D_{x} defined on the generators of ℂ_{(n)}[a_{ij}] by

D_{x}(a_{ij}) = x, and D_{x}(a_{rs}) = 0 if (r, s) ≠ (i, j).

We extend # linearly so that (x + y)^{#} = D_{x+y} = D_{x} + D_{y}, etc. Note that D_{x} = x\frac{∂}{∂a_{ij}} if x is an ij-word with i ≠ j. For example D_{a_{12}a_{23}} = a_{12}a_{23} \frac{∂}{∂a_{13}} . Let Ā(n; i, j) (for 1 ≤ i ≠ j ≤ n) and Ā(n) be the rings of formal power series for A(n; i, j) and A(n) (respectively) and extend the map # to Ā(n; i, j) and Ā(n) in the obvious way. Let 𝔩_{n}(i, j) be the Lie algebra generated by all such derivations D_{x} where x ∈ Ā(n; i, j) and let 𝔩_{n} be the Lie algebra generated by all such D_{x} where x ∈ Ā(n). We will describe convergence in these Lie algebras in section 2. Then any D ∈ 𝔩_{n} can be written as a sum

D = D_{12} + D_{13} + ⋯ + D_{n,n−1},

where D_{ij} ∈ 𝔩_{n}(i, j) for all 1 ≤ i ≠ j ≤ n. Also each D_{ij} is a sum of x_{k}^{#}’s where the x_{k}’s are ij-words. We thus have an isomorphism # : Ā(n) → 𝔩_{n}; let † : 𝔩_{n} → Ā(n) be the inverse of # so that D^{†#} = D and x^{#†} = x for all D ∈ 𝔩_{n} and x ∈ Ā(n).

We will sometimes find it convenient to think of D = ∑_{1≤i≠j≤n}^{n} D_{ij} as the n × n matrix with 0’s on the diagonal and whose ij entry is D_{ij} for i ≠ j. If α ∈ H̃(n), then D(α) will denote this matrix.

It is easily checked that if x, y ∈ 𝔩_{n}, then so does the Lie bracket [x, y]. If we further restrict to elements x ∈ 𝔩_{n} which satisfy

i) x^{∗} = −x; and ii) x fixes the volume form

a_{12} ∧ a_{13} ∧ ⋯ ∧ a_{n,n−1};

then we obtain a Lie subalgebra which we denote by 𝔥(n). We will show that log(α) ∈ 𝔥(n) if α ∈ H̃(n). Clearly 𝔩_{n} is a graded algebra (by degree). Information on the dimensions of these graded parts can be found in sections 7–10 where it is shown that 𝔩_{n} has a rational Poincaré series. In section 5 we will show that D(α) is in some sense skew-Hermitian, making α unitary. We compare this fact with Squier’s result that the Burau representation is unitary ([Sq], but see also p. 551 of [Lo1]).

We note that in [Lo1, Lo2] Long describes an action of B_{n} by diffeomorphisms on a finite-dimensional representation variety and uses Lie-algebraic methods to prove certain results about linear representations of subgroups of B_{n} and Aut(F_{n}). There seems to be no connection with the present work however.

We now explain our results about the invariant ring for the action of P_{n} on ℂ_{(n)}[a_{rs}]. For i = 1, …, n define the following matrices (transvections)

T_{i} = ( \begin{matrix}1 & 0 & … & 0 & … & 0 & 0 \\ 0 & 1 & … & 0 & … & 0 & 0 \\ ⋮ & ⋮ & ... & ⋮ & … & ⋮ & ⋮ \\ a_{i1} & a_{i2} & … & 1 & … & a_{in−1} & a_{in} \\ ⋮ & ⋮ & … & ⋮ & ... & ⋮ & ⋮ \\ 0 & 0 & … & 0 & … & 1 & 0 \\ 0 & 0 & … & 0 & … & 0 & 1 \end{matrix} )

where the non-zero off-diagonal entries occur in the ith row. These are the transvections referred to above. Let X_{n}(λ) be the characteristic polynomial of the matrix T_{1}T_{2} … T_{n} with indeterminate λ. Note that the coefficients of λ in X_{n}(λ) are polynomials over ℤ in ℂ_{(n)}[a_{ij}]. For example, if n = 4, then X_{n}(λ) = ∑_{i=0}^{4} c_{i}λ^{i} where c_{0} = c_{4} = 1 and

c_{1} = −4 − a_{12}a_{21} − a_{31}a_{13} − a_{23}a_{32} − a_{24}a_{42} − a_{41}a_{14} − a_{34}a_{43}

+ a_{32}a_{24}a_{43} + a_{21}a_{42}a_{14} + a_{21}a_{32}a_{13} + a_{31}a_{43}a_{14} − a_{21}a_{32}a_{43}a_{14};

c_{2} = 6 + 2 a_{34}a_{43} + 2 a_{23}a_{32} + 2 a_{42}a_{24} + 2 a_{12}a_{21} + 2 a_{31}a_{13} + 2 a_{4,1}a_{1,4}

+ a_{42}a_{23}a_{34} − a_{32}a_{43}a_{24} + a_{12}a_{21}a_{34}a_{43} + a_{31}a_{12}a_{23} + a_{31}a_{13}a_{42}a_{24} + a_{41}a_{34}a_{13} + a_{41}a_{12}a_{24} + a_{41}a_{14}a_{23}a_{32} − a_{23}a_{31}a_{42}a_{14} − a_{41}a_{24}a_{32}a_{13} − a_{21}a_{32}a_{13} − a_{21}a_{42}a_{14} − a_{21}a_{42}a_{34}a_{13} − a_{31}a_{43}a_{14} − a_{31}a_{43}a_{12}a_{24};

c_{3} = −4 − a_{34}a_{43} − a_{23}a_{32} − a_{42}a_{24} − a_{12}a_{21} − a_{31}a_{13} − a_{41}a_{14}

− a_{42}a_{23}a_{34} − a_{31}a_{12}a_{23} − a_{41}a_{34}a_{12}a_{23} − a_{41}a_{34}a_{13} − a_{41}a_{12}a_{24}.

In [Hu2] we proved that these coefficients are invariant under the action of P_{n}. We conjecture that they generate the invariant subring for the action of P_{n} on ℂ_{(n)}[a_{ij}]. In this paper we prove that they generate the invariant subring for the action of P_{n} on certain subrings of ℂ_{(n)}[a_{ij}] which we now define.

Let A(n; ∗) be the subring of ℂ_{(n)}[a_{ij}] generated over ℂ by all ii-words for all i = 1, ..., n. Then A(n; ∗) is generated by all ii-words (1.2) (with i = j) which have none of the indices i, k, l, m, …, r, s repeated; such ii-words we call cycles. For example, A(3; ∗) is generated by the cycles a_{12}a_{21}, a_{13}a_{31} , a_{23}a_{32}, a_{12}a_{23}a_{31}, a_{13}a_{32}a_{21}. In section 7 we will show that Ĥ(n) acts faithfully on A(n; ∗). One can check that A(n; ∗) contains each of the coefficients of X_{n}(λ) (see the proof of [Hu2, Theorem 2.8] or the proof of Lemma 6.1 b) in this paper). We will show that A(n; ∗) is a graded ring with rational Poincaré series (see sections 7 and 8). Using our representation α = exp(D) for elements of P_{n} we are able to prove

Theorem 1.1. For n = 3, 4, 5 the coefficients of λ in X_{n}(λ) generate the invariant subring for the action of P_{n} (or B_{n}) on the ring A(n; ∗).

The proof of Theorem 1.1 involves finding log(A_{ij}) explicitly, where {A_{ij}|1 ≤ i ≠ j ≤ n} are the standard generators for P_{n} [Bi, p. 20]. It will turn out that the power series log(A_{ij}) factors into a product of a finite and an infinite part (Corollary 4.4) and this is what enables us to prove Theorem 1.1.

A monomial x ∈ A(n; ∗) is of type (s_{1}, …, s_{n}) where, for 1 ≤ i ≤ n, s_{i} is the number of subscripts in x that are equal to i. For example, if n = 3, then the type of a_{12}a_{21} is (2, 2, 0). Note that if x ∈ A(n; ∗), then its type has all even entries, and that each monomial in det(M)^{k} has type (2k, 2k, …, 2k). Note that the set of monomials x of type (2k, 2k, …, 2k), for k = k(x) generates a subring of A(n; ∗). We determine the invariant subring for the action of P_{n} on this subring of A(n; ∗).

Theorem 1.2. Let n ≥ 3. If x ∈ A(n; ∗) is a finite ℂ-sum of monomials y having type (2k, 2k, …, 2k), for some k = k(y), and if x is invariant for the action of P_{n}, then x ∈ ℂ[det(M)].

We note that in [Hu2, Theorem 4.1] we showed that there is also a corresponding action of H(n) on the associative, but non-commutative, polynomial ring generated by b_{ij}, 1 ≤ i ≠ j ≤ n. Nearly everything that we have done in the commutative case has an analog in this non-commutative case, giving slightly different Lie algebras etc.; we have found no analogs of Theorems 1.1, 1.2, however.

§2. Infinite Lie algebras of Cartan type

Let R = ℂ[x_{1}, …, x_{n}] be a polynomial algebra and let R^{∗} = ℂ[[x_{1}, …, x_{n}]] be the corresponding ring of formal power series. Let W_{n} be the general Lie algebra of Cartan type, i.e. W_{n} is linearly generated by all ℂ-derivations of the form

f \frac{∂}{∂x_{i}} , i = 1, …, n

where f ∈ R^{∗} [Ca, SS, Ka]. Now W_{n} (and so any of its subalgebras) can be given a filtration

W_{n} = W_{n,−1} ⊃ W_{n,0} ⊃ W_{n,1} ⊃ ⋯ ⊃ W_{n,i} ⊃ …

Here an element of W_{n,i} is a linear sum of f \frac{∂}{∂x_{j}} ’s where f has degree at least i + 1. There is a corresponding graded algebra with graded pieces

W_{n,i}/W_{n,i+1}.

Using this grading we can define a valuation on W_{n} as follows (c.f. [Ja, p.171]): |0| = 0 and if a ≠ 0, then |a| = 2^{−i} where a ∈ W_{n,i} and a ∉ W_{n,i+1}. Then we have:

|a| ≥ 0,  |a| = 0 if and only if a = 0,
|[a, b]| ≤ |a||b|, and |a + b| ≤ max{|a|, |b|}.

This makes W_{n} into a topological algebra where a_{1} + a_{2} + … converges if and only if |a_{i}| → 0 as i → ∞.

For any derivation D ∈ W_{n} and any x ∈ R^{∗} we define

exp(D)(x) = x + D(x) + \frac{1}{2!}D^{2}(x) + \frac{1}{3!}D^{3}(x) + ….

There are four classes of irreducible graded algebras of interest: general (W_{n}), simple (S_{n}), Hamiltonian (H_{n}) and contact (K_{n}). The only one of these that concerns us is S_{n} which consists of elements which fix the volume form

x_{1} ∧ x_{2} ∧ ⋯ ∧ x_{n}.

§3. Symmetric automorphisms as exponentials

Let F be an integral domain containing the real numbers and R an F − algebra. An algebra homomorphism τ : R → R is a derived homomorphism if there are functions D_{1} = D_{1}(τ), D_{2} = D_{2}(τ), … such that

(i) D_{i}(wx) = wD_{i}(x) for all w ∈ ℝ, all i > 0 and all x ∈ R; and (ii) we have

τ^{w}(x) = x + wD_{1}(x) + w^{2}D_{2}(x) + …

for all w ∈ ℝ and all x ∈ R. The algebra homomorphism τ : R → R will be called specially derived if we also

have D_{r}(x) = \frac{1}{r!}D_{1}^{r}(x) for all r ≥ 1 and all x ∈ R.

Using the definition of t_{ij}^{w}, w ∈ ℂ given in §1 and induction on the degree of a monomial x ∈ ℂ_{(n)}[a_{ij}] one can show that each t_{ij}^{w} is a derived homomorphism of ℂ_{(n)}[a_{ij}].

Lemma 3.1. Let τ : R → R be a derived homomorphism. Then (i) The function D_{1} : R → R is a derivation. (ii) Suppose that a group G of F -algebra homomorphisms of R is generated by

a set of specially derived homomorphisms τ_{1}, …, τ_{q}. Then a subring R′ of R is a G-invariant subring of R if R′ is invariant under all D_{i} = D_{1}(τ_{i}), for all i = 1, …, q.

(iii) Suppose that a group G of ring homomorphisms of R is generated by a set of specially derived homomorphisms τ_{1}, …, τ_{q}. Then the ring of invariants for the action of G on R is the intersection ∩_{i=1}^{q} ker(D_{1}(τ_{i})).

Proof. (i) First we show that D = D_{1}(τ) is linear: for any w ∈ ℝ we have

x + y + wD(x) + wD(y) + w^{2}() + … = α^{w}(x) + α^{w}(y)
= α^{w}(x + y)
= x + y + wD(x + y) + w^{2}() + …,

from which we deduce that D(x + y) = D(x) + D(y) (by comparing coefficients of w in the above expression). Next for any w ∈ ℝ we have

α^{w}(xy) = xy + wD(xy) + w^{2}() + …
= α^{w}(x)α^{w}(y)
= (x + wD(x) + w^{2}() + …)(y + wD(y) + w^{2}() + …)
= xy + wyD(x) + wxD(y) + w^{2}() + …

from which we deduce that D(xy) = yD(x) + xD(y). This gives (i).

For (ii): Suppose that R′ is invariant under all D_{1}(τ_{i}). Let x ∈ R′. It will suffice to show that τ_{i}(x) ∈ R′ for all i = 1, …, q. In fact since

τ_{i}(x) = x + D_{1}(τ_{i})(x) + D_{2}(τ_{i})(x) + …

it will suffice to show that D_{r}(τ_{i})(x) ∈ R′ for all i and all r. But D_{r}(τ_{i})(x) = \frac{1}{r!}D_{1}^{r}(τ_{i})(x) since each τ_{i}, i = 1, …, q is specially derived and so (ii) follows.

(iii) Clearly, any x ∈ ∩_{i=1}^{q} ker(D_{1}(τ_{i})) is fixed by the action of G since τ_{i}(x) = x + D_{1}(τ_{i})(x) + D_{1}^{2}(τ_{i})(x)/2! + ⋯ = x. Now suppose that x is an invariant. Then for any w ∈ ℝ we have τ_{i}^{w}(x) = x = x + wD_{1}(τ_{i})(x) + w^{2}D_{2}(τ_{i})(x) + … for all i = 1, …, q if and only if D_{1}(τ_{i})(x) = −wD_{2}(τ_{i})(x) − … for all i = 1, …, q. Thus D_{1}(τ_{i})(x) = 0 for all i = 1, …, q. □

Lemma 3.2. Any derived homomorphism of R is a specially derived homomorphism of R. In particular for any w ∈ ℝ the generators t_{ij}^{w} of H̃(n) are specially derived ring homomorphisms of the ring ℂ_{(n)}[[a_{ij}]] or any subring of ℂ_{(n)}[[a_{ij}]] which is a H̃(n)-module.

Proof. Let t be a derived homomorphism of R and let T = D_{1}(t). For each w ∈ ℝ, and all x ∈ R we will want to show that

t^{w}(x) = x + wT (x) + (1/2!)w^{2}T^{2}(x) + (1/3!)w^{3}T^{3}(x) + ….

Now for x ∈ R we think of t^{w}(x) as a function of w ∈ ℝ and let

D_{r}(x) = \frac{1}{r!} \frac{d^{r}}{dw^{r}} (t^{w}(x)) |_{w=0}

for r ≥ 0. Then D_{0} = id, D_{r}(ux) = uD_{r}(x) for all u ∈ ℝ and x ∈ R and we have

t^{w}(x) = x + wD_{1}(x) + w^{2}D_{2}(x) + w^{3}D_{3}(x) + w^{4}D_{4}(x) + …

and so

t^{2w}(x) = t^{w}(t^{w}(x))
= t^{w}(x + wD_{1}(x) + w^{2}D_{2}(x) + w^{3}D_{3}(x) + w^{4}D_{4}(x) + …)
= x + wD_{1}(x) + w^{2}D_{2}(x) + w^{3}D_{3}(x) + w^{4}D_{4}(x) + …

+ w(D_{1}(x) + wD_{1}D_{1}(x) + w^{2}D_{1}D_{2}(x) + w^{3}D_{1}D_{3}(x) + w^{4}D_{1}D_{4}(x) + …)

+ w^{2}(D_{2}(x) + wD_{2}D_{1}(x) + w^{2}D_{2}D_{2}(x) + w^{3}D_{2}D_{3}(x) + w^{4}D_{2}D_{4}(x) + …)

+ …
= x + (2w)D_{1}(x) + (2w)^{2}D_{2}(x) + (2w)^{3}D_{3}(x) + (2w)^{4}D_{4}(x) + ….

Since this is true for all w ∈ ℝ, equating coefficients of powers of w in this equation gives:

2D_{2}(x) + D_{1}D_{1}(x) = 4D_{2}(x),
2D_{3}(x) + D_{2}D_{1}(x) + D_{1}D_{2}(x) = 8D_{3}(x), …

and we solve to obtain D_{2}(x) = \frac{1}{2}D_{1}^{2}(x) and D_{3}(x) = \frac{1}{6}D_{1}^{3}(x) as required. This starts a proof by induction. Now assume, inductively, that D_{r}(x) = \frac{1}{r!}D_{1}^{r}(x) for r < k > 2. In general, equating the coefficients of w^{k} in the above equation, one obtains

∑_{i=0}^{k} D_{i}D_{k−i}(x) = 2^{k}D_{k}(x),

from which the inductive hypothesis gives

\frac{1}{k!} ∑_{i=1}^{k−1} \binom{k}{i}D_{1}^{k}(x) = ∑_{i=1}^{k−1} \frac{1}{i!(k − i)!}D_{1}^{k}(x) = (2^{k} − 2)D_{k}(x).

Using induction and the fact that ∑_{i=0}^{k} \binom{k}{i} = 2^{k} gives the result. □

Lemma 3.3. The algebras 𝔥(n), 𝔩_{n}(i, j) and 𝔩_{n} (and so Ā(n; i, j) and Ā(n)) are Lie algebras. If α ∈ H̃(n), then D(α) ∈ 𝔥(n).

Proof. We first show that 𝔩_{n}(i, j) and 𝔩_{n} are Lie algebras i.e. that 𝔩_{n}(i, j) and 𝔩_{n} are closed under the Lie bracket. Suppose that s ∈ Ā(n; i, j) and that y ∈ Ā(n; r, s). Now it is easily seen that

[D_{x}, D_{y}] = D_{D_{x}(y)} − D_{D_{y}(x)}.

Since y ∈ Ā(n; r, s) we see that D_{x}(y) ∈ Ā(n; r, s) and similarly that D_{y}(x) ∈ Ā(n; i, j) since y ∈ Ā(n; r, s). Thus we have [D_{x}, D_{y}] ∈ 𝔩_{n}, for any (i, j), (r, s), and [D_{x}, D_{y}] ∈ 𝔩_{n}(i, j), if (i, j) = (r, s) and so 𝔩_{n}(i, j) and 𝔩_{n} are Lie algebras.

For 𝔥(n) we will need Lemma 3.4. For all x, y ∈ Ā(n) we have D_{x}(y)^{∗} = −D_{x^{∗}}(y^{∗}).

Proof. Since D_{x}(y) is linear in both x and y we need only consider the case where x and y are monomials in A(n). In this situation there are four cases to check depending on the parity of the degrees of x and y. One easily checks each such case. □

Thus if x^{∗} = −x and y^{∗} = −y, then by Lemma 3.4 we have

D_{x}(y)^{∗} = −D_{x^{∗}}(y^{∗}) = −D_{−x}(−y) = −D_{x}(y),

as required for condition (i) in the definition of 𝔥(n). Further, (for (ii)) if x and y both fix the volume form, then so does the Lie bracket [x, y]. Thus 𝔥(n) is also a Lie algebra.

Now if α ∈ H̃(n), then α is a product of the generators t_{ij}^{w}. By the Campbell-Hausdorff formula [Ja, p. 172] and the fact that 𝔥(n) is a Lie algebra we thus need only show that D(t_{ij}^{w}) ∈ 𝔥(n) for all 1 ≤ i ≠ j ≤ n and all w ∈ ℂ. Now from the action of t_{ij}^{w} on ℂ_{(n)}[a_{ij}] given by (1.1) we see that D(t_{ij}^{w}) = wD(t_{ij}) and

D(t_{ij}) = ∑_{\begin{matrix}k=1 \\ k≠i,j \end{matrix}}^{n} D_{a_{ki}a_{ij}} − D_{a_{ji}a_{ik}}.  (3.1)

This shows that D(t_{ij}^{w})^{∗} = −D(t_{ij}^{w}).

If ω is a form on ℂ_{(n)}[a_{ij}], then saying that ω is fixed by some element exp(D) where D ∈ 𝔥(n), is equivalent to the condition that D(ω) = 0. Let ω be the (n^{2} − n)-form

da_{12} ∧ da_{13} ∧ ⋯ ∧ da_{n,n−1}.

Thus to say that some D ∈ 𝔥(n) fixes ω is the same as saying that if D = D_{12} + D_{13} + ⋯ + D_{n,n−1}, and b_{ij} = (D_{ij})^{†} for all 1 ≤ i, j ≤ n, then

∑_{1≤i≠j≤n} \frac{∂}{∂a_{ij}} b_{ij} = 0.

It thus follows from (3.1) that D(t_{ij}^{w})(da_{12} ∧ da_{13} ∧ ⋯ ∧ da_{n,n−1}) = 0 for all 1 ≤ i ≠ j ≤ n and so D(t_{ij}^{w}) ∈ 𝔥(n) as required. □

For α ∈ H̃(n) the derivation D(α) is determined by the logarithm formula:

D(α)^{†} = ∑_{k=1}^{∞} ∑_{1≤i≠j≤n} \frac{(−1)^{k+1}}{k} (α − 1)^{k}a_{ij}.

This converges in our chosen topology since (α − 1)^{k}a_{ij} is either 0 or has degree at least k + 1 .

Thus 𝔥(n) is contained in the special Lie algebra of Cartan type. However there are more restrictions on the D(α) where α ∈ H̃(n): let M = (a_{ij}) be the n × n matrix of indeterminates, where we put a_{ii} = 0 for i = 1, …, n.

Lemma 3.5. If α ∈ H̃(n), then D(α) fixes the 0-form det(M).

Proof. By [Hu1, Corollary 2.6] we see that det(M) is invariant under the action of Ĥ(n) and so the result follows. □

We further remark that it is easily shown that D(α), α ∈ H̃(n), fixes each of the (n^{2} − n − 1)-forms da_{12} ∧ da_{13} ∧ ⋯ ∧ d̂a_{ij} ∧ ⋯ ∧ da_{nn−1} (again we just consider the action of the standard generators t_{rs}).

§4 Calculation of some D(α)

We first note that since

log(α) = log(α)_{12} + log(α)_{13} + ⋯ + log(α)_{nn−1},

and

log(α)_{ij} = (α − 1)a_{ij} − (α − 1)^{2}a_{ij}/2 + (α − 1)^{3}a_{ij}/3 − (α − 1)^{4}a_{ij}/4 + …,

for 1 ≤ i ≠ j ≤ n, then we can write

D(α) = (α − 1)M − (α − 1)^{2}M/2 + (α − 1)^{3}M/3 − (α − 1)^{4}M/4 + ….

If we are thinking of the pure braids as diffeomorphisms of D_{n}, the disc with n holes, then the generators A_{ij}, 1 ≤ i ≠ j ≤ n, are Dehn twists about certain simple closed curves containing 2 holes in their interior [Bi]. We will be interested in calculating these D(A_{ij}); we start this calculation in the following:

Proposition 4.1. Let α be the Dehn twist about the curve surrounding the first

n − 1 holes of the disc D_{n}. Then D(α) = ( \begin{matrix}0 & N \\ −N^{∗t} & 0 \end{matrix} ) where N is an (n − 1) × 1

matrix; more specifically, N = log(N^{′∗t}) ( a_{1n} ; a_{2n} ; ⋮ ; a_{n−1,n} ) where t denotes transpose and

N′ is the (n − 1) × (n − 1) principal submatrix of (T_{1}T_{2} … T_{n−1})^{−1}.

If α is the Dehn twist about the curve surrounding the first k < n holes of the disc D_{n}, then

D(α) = ( \begin{matrix}\begin{matrix}0 & … & 0 \\ ⋮ & ... & ⋮ \\ 0 & … & 0 \end{matrix} & P ( a_{1,k+1} ; ⋮ ; a_{k,k+1} ) … P ( a_{1n} ; ⋮ ; a_{kn} ) \\ − (P ( a_{1,k+1} ; ⋮ ; a_{k,k+1} ) … P ( a_{1n} ; ⋮ ; a_{kn} ))^{∗t} & \begin{matrix}0 & … & 0 \\ ⋮ & ... & ⋮ \\ 0 & … & 0 \end{matrix} \end{matrix} ) ,

where P = log(N^{′∗t}) and N′ is the k × k principal submatrix of (T_{1}T_{2} … T_{k})^{−1}.

Proof. Let α be the Dehn twist about the curve surrounding the first n − 1 holes. Since D(α)^{∗t} = −D(α) and α fixes all a_{ij} for 1 ≤ i, j < n (so that D(α)_{ij} = 0

for such i, j) we see that D(α) = ( \begin{matrix}0 & N \\ −N^{∗t} & 0 \end{matrix} ) where N is an (n − 1) × 1 matrix.

Now, as we have noted above,

D(α) = (α − 1)M − (α − 1)^{2}M/2 + (α − 1)^{3}M/3 − (α − 1)^{4}M/4 + ….

But α, as an automorphism of the free group F_{n}, has the effect of conjugating each of x_{1}, …, x_{n−1} by the element (x_{1} … x_{n−1}) and so a calculation using [Hu2, Lemma 2.3] easily shows that

α ( a_{1n} ; a_{2n} ; ⋮ ; a_{n−1,n} ) = N^{′∗t} ( a_{1n} ; a_{2n} ; ⋮ ; a_{n−1n} )

BRAID GROUPS, INFINITE LIE ALGEBRAS AND RINGS OF INVARIANTS 11

for N′ as described in the Theorem. Since N′ is a matrix over the subring generated by all a_{ij} for 1 ≤ i ≠ j < n we see that α(N′) = N′ and so we have

D(α) = (N^{′∗t} − I)M − (N^{′∗t} − I)^{2}M/2 − (N^{′∗t} − I)^{3}M/3 − …
= log(N^{′∗t})M

and the first result is proved. The second follows immediately since for 1 ≤ i ≤ k < m ≤ n the element D(α)_{km}^{†} is equal to D(α)_{i,k+1}^{†} with m replacing k + 1 in the indices of every monomial of D(α)_{i,k+1}^{†}. □

In order to prove Theorems 1.1 and 1.2 we will need an even more specific calculation for the D(A_{ij}). First we note that the braid group generators σ_{1}, …, σ_{n−1} have the form σ_{i} = t_{ii+1}(i, i + 1). This is the action (on the left) as automorphisms of ℂ_{(n)}[a_{ij}]. Also we note that for a permutation π ∈ Σ_{n} we have

πt_{ij}π^{−1} = t_{i′j′} where i′ = π(i) and j′ = π(j).

Example 4.2. We first investigate log(A_{12}). Then by Proposition 4.1 we need only consider n = 3. Using 4.1 we have P = log(N′^{∗t}) where

N′ = ( \begin{matrix}1 & −a_{12} \\ −a_{21} & 1 + a_{12} a_{21} \end{matrix} ) and D(σ_{1}^{2}) = ( \begin{matrix}{0 \atop 0} {0 \atop 0} & P ( \begin{matrix}a_{13} \\ a_{23} \end{matrix} ) \\ ( a_{31} a_{32} ) P^{∗t} & 0 \end{matrix} ) .

Now one calculates using the logarithm formula that

log(σ_{1}^{2}) = log(t_{12}t_{21}) = log(exp(T_{12}) exp(T_{21}))

= (1/1)(D_{321} + D_{312} − D_{123} − D_{213})
+(1/2)(D_{3121} − D_{3212} + D_{1213} − D_{2123})
+(1/6)(−D_{32121} − D_{31212} + D_{12123} + D_{21213})
+(1/12)(−D_{312121} + D_{321212} − D_{121213} + D_{212123})

+(1/30)(D_{3212121} + D_{3121212} − D_{2121213} − D_{1212123})

+(1/60)(D_{31212121} − D_{32121212} + …)
+(1/140)(−D_{321212121} − D_{312121212} + …)
+(1/280)(−D_{3121212121} + D_{3212121212} + …)
+(1/630)(D_{32121212121} + D_{31212121212} + …) + …

Here we are using the following notation: D_{irst...uj} = a_{ir}a_{rs}a_{st} … a_{uj} \frac{∂}{∂a_{ij}}, and so for example, we have D_{32121212121} = (a_{12}a_{21})^{4}D_{321} etc. We find, somewhat surprisingly, that there is a factorization into a finite and an infinite part (to be proved below):

\begin{matrix}D(σ_{1}^{2})^{†} = (a_{32}a_{21} + a_{31}a_{12}a_{21}/2 + a_{31}a_{12} − a_{32}a_{21}a_{12}/2 \\ −a_{21}a_{13} − a_{12}a_{23} + a_{12}a_{21}a_{13}/2 − a_{21}a_{12}a_{23}/2) × (1 − a_{12}a_{21}/6 \\ +(a_{12}a_{21})^{2}/30 − (a_{12}a_{21})^{3}/140 + (a_{12}a_{21})^{4}/630 − …). \end{matrix}  (4.1)

Here the coefficients 1, 6, 30, 140, 630, 2772, 12012, 51480, … in the denominators give the sequence β_{mv} where m = v + 2 and

β_{mv} = (−1)^{m+v}\binom{m + v}{m}\binom{m}{v}\frac{1}{v + 1} = \frac{(2n + 1)!}{(n!)^{2}}.

See [Jo, p. 449].

If we let K = (1−a_{12}a_{21}/6+(a_{12}a_{21})^{2}/30−(a_{12}a_{21})^{3}/140+(a_{12}a_{21})^{4}/630+…), then we can write

D(σ_{1}^{2})^{†} = (D_{31} + D_{32} + D_{13} + D_{23})K

where

D_{31} = a_{32}a_{21} + a_{31}a_{12}a_{21}/2,
D_{13} = −a_{12}a_{23} + a_{12}a_{21}a_{13}/2,
D_{32} = a_{31}a_{12} − a_{32}a_{21}a_{12}/2,
D_{23} = −a_{21}a_{13} − a_{21}a_{12}a_{23}/2.

Proof. To see why we have the factorization (4.1) we let T_{ij} = log(t_{ij}). Then σ_{1}^{2} = t_{12}t_{21} = exp(T_{12}) exp(T_{21}) and so by the Campbel-Hausdorff formula [Ja p. 172] we see that log(σ_{1}^{2}) is an infinite sum of rational multiples of Lie brackets involving only T_{12} and T_{21}. But we have the simple calculation:

[T_{12}, T_{21}] = a_{12}a_{21} (a_{31} \frac{∂}{∂a_{31}} + a_{13} \frac{∂}{∂a_{13}} − a_{32} \frac{∂}{∂a_{32}} − a_{23} \frac{∂}{∂a_{23}} )

which in turn can be used to show that

[[T_{12}, T_{21}], T_{21}] = −2a_{12}a_{21}T_{21},
[[T_{12}, T_{21}], T_{12}] = 2a_{12}a_{21}T_{12},
[[T_{12}, T_{21}], T_{12}, T_{12}] = 0,
[[T_{12}, T_{21}], T_{21}, T_{21}] = 0 etc.

This shows that log(σ_{1}^{2}) is a sum

log(σ_{1}^{2}) = KT_{12} + LT_{21} + M[T_{21}, T_{21}]

where K, L and M are power series in a_{12}a_{21} with rational coefficients. Since σ_{1}^{2} = t_{12}t_{21} and σ_{1}^{−2} = t_{21}^{−1}t_{12}^{−1} and σ_{1}^{2} fixes the coefficients of the characteristic polynomial of T_{1}T_{2}T_{3} one easily finds that K = L = 2M. Thus

log(σ_{1}^{2}) = KT_{12} + KT_{21} + (K/2)[T_{12}, T_{21}].  (4.2)

We find K explicitly by considering the action of σ_{1}^{2}. Now if we let D = log(σ_{1}^{2}), then we have

σ_{1}^{2}(a_{13}) = a_{13} − a_{12}a_{23} + a_{12}a_{21}a_{13}
= a_{13} + D(a_{13}) + (1/2!)D^{2}(a_{13}) + (1/3!)D^{3}(a_{13}) + ….

On the other hand a calculation using (4.2) shows that

a_{13} − a_{12}a_{23} + a_{12}a_{21}a_{13} = a_{13} + Ka_{12}(−a_{23} + a_{21}a_{13}/2)/1!+

K^{2}a_{13}(a_{12}a_{21})^{1}(1 + a_{12}a_{21}/4)/2!+
K^{3}a_{12}(a_{12}a_{21})^{1}(1 + a_{12}a_{21}/4)(−a_{23} + a_{21}a_{13}/2)/3!+
K^{4}a_{13}(a_{12}a_{21})^{2}(1 + a_{12}a_{21}/4)^{2}/4!+
K^{5}a_{12}(a_{12}a_{21})^{2}(1 + a_{12}a_{21}/4)^{2}(−a_{23} + a_{21}a_{13}/2)/5!+
K^{6}a_{13}(a_{12}a_{21})^{3}(1 + a_{12}a_{21}/4)^{3}/6!+
K^{7}a_{12}(a_{12}a_{21})^{3}(1 + a_{12}a_{21}/4)^{3}(−a_{23} + a_{212}a_{13}/2)/7!+

K^{8}a_{13}(a_{12}a_{21})^{4}(1 + a_{12}a_{21}/4)^{4}/8! + ….

BRAID GROUPS, INFINITE LIE ALGEBRAS AND RINGS OF INVARIANTS 13

Differentiating this equation with respect to a_{23} gives

1 = ∑_{i=0}^{∞} \frac{K^{2i+1} (a_{12}a_{21} + \frac{(a_{12}a_{21})^{2}}{4} )^{i}}{(2i + 1)!} .

Since the right-hand side looks something like the power series for sin(x) this equation can be solved explicitly to get

K = \begin{matrix}arcsinh & (\sqrt{a_{12}a_{21} + \frac{(a_{12}a_{21})^{2}}{4}} ) \\ √ & \overline{a_{12}a_{21} + \frac{(a_{12}a_{21})^{2}}{4}} \end{matrix}.

Using the formulae [Be, pp. 267-268]

arcsinh (a\sqrt{1 + b^{2}} + b\sqrt{1 + a^{2}}) = arcsinh(a) + arcsinh(b)

and

arcsinh(x) = ∑_{n=0}^{∞}(−1)^{n} \frac{1 · 3 · 5 · . . . · (2n − 1)}{2 · 4 · 6 · . . . · 2n · (2n + 1)}x^{2n+1}

with a = b = \frac{\sqrt{a_{12}a_{21}}}{2} we see that

K = \frac{∑_{n=0}^{∞}(−1)^{n} \frac{1·3·5·...·(2n−1)}{2·4·6·...·2n·(2n+1)·2^{2n}} x^{n}}{\sqrt{1 + \frac{x}{4}}} .

where x = a_{12}a_{21}. Now _{\frac{1}{\sqrt{1+ \frac{x}{4}}}} = ∑_{n=0}^{∞} ( \begin{matrix}−\frac{1}{2} \\ n \end{matrix} ) (\frac{x}{4})^{n} and so the coefficient of x^{k} in K is

∑_{\begin{matrix}i,j=0 \\ i+j=k \end{matrix}}^{k}(−1)^{i} \frac{1 · 3 · 5 · . . . · (2i − 1)}{2 · 4 · 6 · . . . · 2i · (2i + 1) · 2^{2i}} ( \begin{matrix}−\frac{1}{2} \\ j \end{matrix} ) (\frac{1}{4})^{j}

= \frac{1}{4^{k}} ∑_{\begin{matrix}i,j=0 \\ i+j=k \end{matrix}}^{k}(−1)^{i} \frac{1 · 3 · 5 · . . . · (2i − 1)}{2 · 4 · 6 · . . . · 2i · (2i + 1)} ( \begin{matrix}−\frac{1}{2} \\ j \end{matrix} )

= \frac{(−1)^{k}}{8^{k}} ∑_{i=0}^{k} \frac{1 · 3 · 5 · . . . · (2i − 1) · 1 · 3 · 5 · . . . · (2(k − i) − 1)}{i! · (k − i)! · (2i + 1)}

= \frac{1}{4^{k}} ∑_{i=0}^{k} ( \begin{matrix}−\frac{1}{2} \\ i \end{matrix} ) ( \begin{matrix}−\frac{1}{2} \\ k − i \end{matrix} ) \frac{1}{(2i + 1)}

= \frac{(−1)^{k}}{4^{2k}} ∑_{i=0}^{k} ( \begin{matrix}2i \\ i \end{matrix} ) ( \begin{matrix}2(k − i) \\ k − i \end{matrix} ) \frac{1}{(2i + 1)}.

Now by [Ri, Ch. 3, problem 10 (a)] we see that this last expression is equal to \frac{(n!)^{2}}{(2n+1)!}, as required. □

The above calculation can also be made using the matrices

N′ = ( \begin{matrix}1 & −a_{12} \\ −a_{21} & 1 + a_{12} a_{21} \end{matrix} )

and P = log(N′^{∗t}) in Example 4.2.

Lemma 4.3. If α ∈ Ĥ(n), β ∈ H̃(n) with β = exp(E), then αβα^{−1} = exp(F) where

F = ∑_{1≤i≠j≤n} F_{ij} with F_{ij} = αEα^{−1}(a_{ij}).

Proof. We have

F = log(αβα^{−1}) = αlog(β)α^{−1} = αEα^{−1}

and so F_{ij} = αEα^{−1}(a_{ij}), as required. □

Corollary 4.4. For each of the generators A_{ij} of P_{n} the derivation log(A_{ij}) can be expressed as log(A_{ij}) = E_{ij1}E_{ij2} with E_{ij1} a polynomial and E_{ij2} a power series satisfying A_{ij}(E_{ij2}) = E_{ij2}

Proof. By [Bi, p.20] each A_{ij} is a conjugate of σ_{i}^{2}; but σ_{i}σ_{i+1}σ_{i}σ_{i+1}^{−1}σ_{i}^{−1} = σ_{i+1} for i = 1, …, n − 2. Thus each A_{ij} is a conjugate of A_{12} by some element of B_{n} and so the result follows from Lemma 4.3 and Example 4.2. □

We call E_{ij1} the finite part of log(A_{ij}) and write it as log(A_{ij})_{f}.

Example 4.5. For n = 4 we now find that (with A_{13} = σ_{1}σ_{2}^{2}σ_{1}^{−1})

log(A_{13})_{f} = −a_{13}a_{32} − a_{13}a_{31}a_{12}/2 − a_{13}a_{34} + a_{13}a_{31}a_{14}/2+ a_{23}a_{31} − a_{13}a_{31}a_{21}/2 + a_{21}a_{13} + a_{13}a_{31}a_{23}/2 − a_{31}a_{12} + a_{13}a_{31}a_{32}/2 − a_{31}a_{14} − a_{13}a_{31}a_{34}/2 + a_{43}a_{31} + a_{13}a_{31}a_{41}/2 + a_{41}a_{13} − a_{13}a_{31}a_{43}/2

+ a_{41}a_{13}a_{32} + a_{23}a_{31}a_{14} − a_{43}a_{31}a_{12} − a_{21}a_{13}a_{34},

and that

log(A_{24})_{f} = −a_{24}a_{43} − a_{24}a_{42}a_{23}/2 − a_{24}a_{41} + a_{24}a_{42}a_{21}/2+ a_{34}a_{42} − a_{24}a_{42}a_{32}/2 + a_{32}a_{24} + a_{24}a_{42}a_{34}/2 − a_{42}a_{23} + a_{24}a_{42}a_{43}/2 − a_{42}a_{21} − a_{24}a_{42}a_{41}/2 + a_{14}a_{42} + a_{24}a_{42}a_{12}/2 + a_{12}a_{24} − a_{24}a_{42}a_{14}/2

+ a_{12}a_{24}a_{43} + a_{34}a_{42}a_{21} − a_{14}a_{42}a_{23} − a_{32}a_{24}a_{41}.

For 1 ≤ i ≤ j ≤ n and 1 ≤ k ≤ n, k ≠ i, j, let ϵ_{k} = 1 if k < i or k > j; otherwise let ϵ_{k} = −1. The general log(A_{ij}) can now be calculated, giving:

Lemma 4.6. For 1 ≤ i < j ≤ n, log(A_{ij})_{f} is given by

\begin{aligned}∑_{\begin{matrix}k=1 \\ k≠i,j \end{matrix}}^{n} & (a_{ki}a_{ij} + a_{kj}a_{ji} − a_{ji}a_{ik} − a_{ij}a_{jk} + \frac{a_{ij}a_{ji}}{2} ϵ_{k} (a_{ki} + a_{ik} − a_{kj} − a_{jk})) \\ & + ∑_{\begin{matrix}i<s<j \\ r<i or r>j \end{matrix}} (a_{ri}a_{ij}a_{js} + a_{sj}a_{ji}a_{ir} − a_{rj}a_{ji}a_{is} − a_{si}a_{ij}a_{jr}) . □ \end{aligned}

§5. Unitary action.

In this section we investigate the nature of a general log(α) or D(α) for α ∈ H̃(n).

Proposition 5.1. Let α ∈ H̃(n). If D = D(α) is the n × n matrix representing log(α), then there is a n × n matrix U over the algebraic closure of ℂ_{(n)}[a_{ij}] such that UDU^{∗t} is a diagonal matrix

Proof. It is easy to see that one can take the involution ∗ to be complex conjugation (e.g. for each r, s we can take a_{rs} = a_{rs1} + ia_{rs2}, a_{sr} = −a_{rs1} + ia_{rs2} where a_{rs1} and a_{rs2} are algebraically independent variables which are invariant under ∗ and i^{∗} = −i). Thus if E is any normal matrix over ℂ_{(n)}[[a_{ij}]], then there is a unitary matrix U such that UEU^{∗t} is diagonal with diagonal entries e such that e = ie′ for some ‘real’ e′. Now D^{∗t} = −D (so that D is skew-Hermitian and so normal) and so there is a unitary matrix U such that UDU^{∗t} = Δ is a diagonal matrix. with Δ = iΔ′, where Δ′ is invariant under ∗. □

We have already noted that log(α)^{∗} = − log(α) for α ∈ H̃(n). Thus if we define α^{∗} = exp(log(α)^{∗}), we have

Proposition 5.1. For α ∈ H̃(n) we have αα^{∗} = id. □

§6. The invariant ring for the action of P_{n}

In this section we prove Theorems 1.1 and 1.2.

Let (λ_{ij}) = (λ_{12}, λ_{13}, …, λ_{1n}, λ_{21}, λ_{23}, …, λ_{2n}, …, λ_{n1}, λ_{n2}, …, λ_{n−1,n}) and for n ≥ 3 let

x = ∑_{(λ_{ij})} (∏_{i≠j} a_{ij}^{λ_{ij}} ) Q((λ_{ij}))

where Q((λ_{ij})) ∈ ℂ, be invariant under the action of P_{n}. Here the sum is over all n(n − 1)-tuples (λ_{ij}) = (λ_{12}, …, λ_{n,n−1}) but only a finite number of the Q((λ_{ij})) are non-zero.

The proof will be similar to the proof of Theorem 6 of [Hu2] in that we will show that each Q((λ_{ij})) is a rational sum of ’simpler’ Q((μ_{ij}))’s. Specifically we will show that each such Q((λ_{ij})) is a rational sum of Q((μ_{ij}))’s where the only non-zero μ_{ij}’s are for (i, j) in the set

E(n) = {(2, 1), (3, 1), …, (n, 1)} ∪ {(3, 2), (4, 3), (5, 4), …, (n, n − 1)} ∪ {(1, n)}.

Let the characteristic polynomial of T_{1}T_{2} … T_{n} be given as χ_{n}(λ) = λ^{n} + c_{n−1}λ^{n−1} + ⋯ + c_{1}λ + 1. Then ℂ[c_{1}, …, c_{n−1}] is contained in the invariant ring for the action of P_{n} [Hu2, Theorem 2.8]. We need to show that x ∈ ℂ[c_{1}, …, c_{n−1}] under the hypotheses of Theorems 1.1 and 1.2.

Lemma 6.1. a) The only non-zero cycles a_{ik}a_{kl}a_{lm} … a_{rs}a_{si}, where the pairs of indices (i, k), (k, l), (l, m), …, (r, s), (s, i) all belong to E(n), are of the form

a_{1n}a_{n,n−1}a_{n−1,n−2} … a_{r+1,r}a_{r1},

for some 2 ≤ r ≤ n.

b) For each such cycle a_{1n}a_{n,n−1}a_{n−1,n−2} … a_{r+1,r}a_{r1} there is an element z of ℚ[c_{1}, …, c_{n−1}] such that a_{1n}a_{n,n−1}a_{n−1,n−2} … a_{r+1,r}a_{r1} is a monomial of z but no other a_{1n}a_{n,n−1}a_{n−1,n−2} … a_{s+1,s}a_{s1}(s ≠ r) is a monomial of z.

Proof. a) One first notes that 1 has to be an index in any such cycle z. But then a_{1n} has to be a factor. If the cycle has length 2, then it must be a_{1n}a_{n1} else we must have z = a_{1n}a_{n,n−1} . . .. If the cycle has length 3, then it must be a_{1n}a_{n,n−1}a_{n−1,1} else we must have z = a_{1n}a_{n,n−1}a_{n−1,n−2} . . .. Continuing in this way gives the result. We will call these cycles allowable.

b) As we noted in the proof of [Hu2, Theorem 2.8] the matrix λI_{n} − T_{1}T_{2} … T_{n} can be row-reduced to the following matrix, which thus has the same determinant as λI_{n} − T_{1}T_{2} … T_{n}:

( \begin{matrix}λ − 1 & −λa_{12} & … & & … & −λa_{1,n−1} & −λa_{1n} \\ −a_{21} & λ − 1 & … & & … & −λa_{2,n−1} & −λa_{2n} \\ ⋮ & ⋮ & ... & ⋮ & & ⋮ & ⋮ \\ −a_{i1} & −a_{i2} & … & λ − 1 & … & −λa_{i,n−1} & −λa_{in} \\ ⋮ & ⋮ & & ⋮ & ... & ⋮ & ⋮ \\ −a_{n−1,1} & −a_{n−1,2} & … & & … & λ − 1 & −λa_{n−1,n} \\ −a_{n1} & −a_{n2} & … & & … & −a_{n,n−1} & λ − 1 \end{matrix} ) .

From this we see that all the allowable cycles a_{1n}a_{n,n−1}a_{n−1,n−2} … a_{r+1,r}a_{r1}, r = 2, …, n, occur as monomials of c_{1} with non-zero rational coefficients; that all the allowable cycles a_{1n}a_{n,n−1}a_{n−1,n−2} … a_{r+1,r}a_{r1}, r = 3, …, n, occur as monomials of c_{2} with non-zero rational coefficients (but the cycle a_{1n}a_{n,n−1}a_{n−1,n−2} … a_{32}a_{21} does not occur in c_{2}), that all the allowable cycles a_{1n}a_{n,n−1}a_{n−1,n−2} … a_{r+1,r}a_{r1} for r = 4, …, n occur as monomials of c_{3} with non-zero rational coefficients (but none of a_{1n}a_{n,n−1}a_{n−1,n−2} … a_{r+1,r}a_{r1}, where r = 2, 3, occur in c_{3}), etc. The result b) follows. □

Corollary 6.2. If x ∈ A(n; ∗), then there is a unique y ∈ ℂ[c_{1}, …, c_{n−1}] such that no monomial of x − y (with non-zero coefficient) is a product of allowable cycles.

Proof. By Lemma 6.1 a) and b) any product of allowable cycles is a monomial in some unique element y of ℤ[c_{1}, …, c_{n−1}], such that none of the other monomials of y is a product of allowable cycles. The result follows. □

Thus, given x ∈ A(n; ∗) as above, there is a unique y ∈ ℂ[c_{1}, …, c_{n−1}] such that x − y satisfies the condition of Corollary 6.2. Suppose that

x − y = ∑_{(λ_{ij})} (∏_{i≠j} a_{ij}^{λ_{ij}} ) Q((λ_{ij})),

where Q((λ_{ij})) ∈ ℂ. But if it has been proved (as we do below) that Q((λ_{ij})) is a rational sum of Q((μ_{ij}))’s where the only non-zero μ_{ij}’s are for (i, j) ∈ E(n), then each such Q((μ_{ij})) must be 0 and so all Q((λ_{ij}))’s are 0, showing that x − y = 0; thus that x = y ∈ ℂ[c_{1}, …, c_{n−1}], as required. This thus indicates the strategy of the proof.

We now proceed with the details of the proof that each Q((λ_{ij})) is a rational sum of Q((μ_{ij}))’s where the only non-zero μ_{ij}’s are for (i, j) ∈ E(n). Recall that by Lemma 4.6

log(A_{ii+1}) = K_{i,i+1}( ∑_{{k=1 \atop k≠i,i+1}}^{n} D_{k,i+1,i} + D_{k,i,i+1} − D_{i,i+1,k} − D_{i+1,i,k}

+ (D_{k,i,i+1,i} − D_{k,i+1,i,i+1} + D_{i,1,i,k} − D_{i+1,i,i+1,k}) /2)

where D_{i,j,k} = a_{ij}a_{jk} \frac{∂}{∂a_{ik}} etc., K_{i,i+1} is a power series in a_{i,i+1}a_{i+1,i} and that if x is invariant for P_{n}, then log(A_{ij})_{f}(x) = 0 for all 0 < i ≠ j ≤ n by Lemma 3.1 (iii). For i = 1 we use the above to write down the condition log(A_{12})_{f}(x) = 0. By looking at the coefficient of the monomial ∏_{1≤i≠j≤n} a_{ij}^{μ_{ij}} in the equation log(A_{12})_{f}(x) = 0 we obtain the following equation:

∑_{i=3}^{n}(μ_{i1} + 1)Q(μ_{i1} + 1, μ_{i2} − 1, μ_{21} − 1) + (μ_{i2} + 1)Q(μ_{i2} + 1, μ_{i1} − 1, μ_{12} − 1)

− (μ_{1i} + 1)Q(μ_{1i} + 1, μ_{12} − 1, μ_{2i} − 1) − (μ_{2i} + 1)Q(μ_{2i} + 1, μ_{1i} − 1, μ_{21} − 1)

+ (μ_{i1} + μ_{1i} − μ_{i2} − μ_{2i})Q(μ_{12} − 1, μ_{21} − 1)/2 = 0.

(6.1)

Here we are using a shorthand where for example Q(μ_{i1} + 1, μ_{i2} − 1, μ_{21} − 1) means Q((μ_{ij})) with μ_{i1} + 1 replacing μ_{i1}, μ_{i2} − 1 replacing μ_{i2}, μ_{21} − 1 replacing μ_{21} and all other μ_{rs} the same. We of course have the convention that Q((μ_{ij})) = 0 if μ_{rs} < 0 for any r, s. We refer to (6.1) as the log(A_{12})_{f} equation and in general we refer to the equation obtained in this way using log(A_{ij}) as the log(A_{ij})_{f} equation.

The equation (6.1) is obtained in the following way: to each monomial a_{ij}a_{jk} in log(A_{rs})_{f}^{#} corresponds the term (μ_{ik} + 1)Q(μ_{ik} + 1, μ_{ij} − 1, μ_{jk} − 1) in (6.1); such terms we call type 1 terms. To each monomial a_{ri}a_{ij}a_{js} in log(A_{rs})_{f}^{#} (with r, i, j, s distinct) corresponds the term (μ_{rs} + 1)Q(μ_{rs} + 1, μ_{ri} − 1, μ_{ij} − 1, μ_{js} − 1) in (6.1); such terms we call type 2 terms. To each monomial a_{rs}a_{sr}a_{uv} in log(A_{rs})_{f}^{#} (with {r, s} ∩ {u, v} ∈ {{u}, {v}}) corresponds the term μ_{uv}Q(μ_{rs} − 1, μ_{sr} − 1) in (6.1); such terms we call type 3 terms. See Lemma 4.6 for the fact that this covers all such monomials of each log(A_{rs})_{f}^{#}.

Now note that the n-cycle (1, 2, …, n) appears in the braid group in the following way: the action on ℂ_{(n)}[a_{ij}] of the word σ_{1}σ_{2} … σ_{n−1} is given by

σ_{1}σ_{2} … σ_{n−1} = t_{12}(1, 2)t_{23}(2, 3)t_{34}(3, 4) … t_{n−1,n}(n − 1, n)
= t_{12}t_{13}(1, 2)(2, 3)t_{34}(3, 4) … t_{n−1,n}(n − 1, n)
= t_{12}t_{13}t_{14} … t_{1n}(1, 2)(2, 3)(3, 4) … (n − 1, n)
= (1, 2, …, n).

Here t_{12}t_{13}t_{14} … t_{1n} = id since it is inner.

Since A_{i,i+1} = (1, 2, …, n)^{i−1}A_{12}(1, 2, …, n)^{−i+1} for i = 1, …, n−1 we see that the corresponding equation for log(A_{i,i+1})_{f}(x) = 0(i < n) is obtained from (6.1) by permuting subscripts using the n-cycle (1, 2, …, n) in such a way that 1 is sent to i. We denote the equation so obtained by (6.1.i).

Thus, for example (6.1.2) is

∑_{\begin{matrix}i=1 \\ i≠2,3 \end{matrix}}^{n}(μ_{i2} + 1)Q(μ_{i2} + 1, μ_{i3} − 1, μ_{32} − 1) + (μ_{i3} + 1)Q(μ_{i3} + 1, μ_{i2} − 1, μ_{23} − 1)

−(μ_{2i} + 1)Q(μ_{2i} + 1, μ_{23} − 1, μ_{3i} − 1) − (μ_{3i} + 1)Q(μ_{3i} + 1, μ_{2i} − 1, μ_{32} − 1) +(μ_{i2} + μ_{2i} − μ_{i3} − μ_{3i})Q(μ_{23} − 1, μ_{32} − 1)/2 = 0. (6.1.2)

We first prove Theorem 1.1 in the case n = 3, and while doing so we will also indicate how to do the simplest kinds of ‘reductions’ in general. Note that (μ_{12} + 1)Q(μ_{12} + 1, μ_{13} − 1, μ_{32} − 1) is a term in (6.1.2); in fact all other terms of (6.1.2) have the form Q((μ_{12}, …)). Thus if λ_{12} > 0 in some Q((λ_{ij})), then using (6.1.2) we can express Q((λ_{i,j})) as a ℚ-linear sum of Q((μ_{i,j}))’s where μ_{12} < λ_{12}. Continuing in this way, using (6.1.2), we see that we can express any Q((λ_{i,j})) as a ℚ-linear sum of Q((μ_{i,j}))’s where μ_{12} = 0. We refer to this process of decreasing λ_{12} as ‘reducing λ_{12} to 0.’ We now thus need only consider such Q((μ_{i,j}))’s.

Since we can now assume that λ_{12} = 0 in all Q((λ_{ij})) we see that (6.1) is

∑_{i=3}^{n}(μ_{i1}+1)Q(μ_{i1}+1, μ_{i2}−1, μ_{21}−1)−(μ_{2i}+1)Q(μ_{2i}+1, μ_{1i}−1, μ_{21}−1) = 0 (6.2)

In this equation all terms have the same value (zero) of μ_{12}. The type 1 term

(μ_{23} + 1)Q(μ_{23} + 1, μ_{13} − 1, μ_{21} − 1)

occurs in this equation and so if μ_{23} > 0, then using (6.2) we can express Q((μ_{ij})) as a ℚ-linear sum of Q((λ_{ij}))’s where λ_{23} < μ_{23} (while keeping λ_{12} = 0). Continuing in this way, using (6.1) with λ_{12} = 0, we see that we can express Q((μ_{i,j})) as a ℚ-linear sum of Q((λ_{i,j}))’s where λ_{12} = λ_{23} = 0 i.e. we can reduce λ_{23} to 0 while keeping λ_{12} = 0. We note that the n = 3 case of Theorem 1.1 now follows from the argument explained after Corollary 6.2.

For the cases n > 3 we will need to use sums of the equations corresponding to the log(A_{ij})_{f}. We will denote such an equation by

log(A_{rs})_{f} + log(A_{tu})_{f} + ….

Lemma 6.3. a) The type 1 term (μ_{rt} + 1)Q(μ_{rt} + 1, μ_{rs} − 1, μ_{st} − 1) occurs in both the equations log(A_{rs})_{f} and log(A_{st})_{f}, but with opposite signs.

b) The only type 1 terms in the equation

log(A_{12})_{f} + log(A_{13})_{f}+ ⋯ + log(A_{1,n−1})_{f}+
log(A_{23})_{f} + log(A_{24})_{f}+ ⋯ + log(A_{2,n−1})_{f}+
⋯ + log(A_{n−2,n−1})_{f}

are exactly those of the form (μ_{rt} + 1)Q(μ_{rt} + 1, μ_{rs} − 1, μ_{st} − 1) where only one of r, s, t is n.

Proof. a) follows from Lemma 4.6 while b) follows from a). □

We now continue with the case n = 4 of Theorem 1.1. As in the n = 3 case we can first use the A_{14} equation (6.1.4) to reduce μ_{12} to 0 and then we can use the A_{12} equation (with μ_{12} = 0) to reduce μ_{23} to 0. Now using Lemma 6.3 with μ_{12} = μ_{23} = 0 the equation for log(A_{12})_{f} + log(A_{13})_{f} + log(A_{23})_{f} is

(μ_{41} + 1)Q(μ_{41} + 1, μ_{42} − 1, μ_{21} − 1) − (μ_{24} + 1)Q(μ_{24} + 1, μ_{21} − 1, μ_{14} − 1)+ (μ_{42} + 1)Q(μ_{42} + 1, μ_{43} − 1, μ_{32} − 1) − (μ_{34} + 1)Q(μ_{34} + 1, μ_{31} − 1, μ_{14} − 1)+ (μ_{43} + 1)Q(μ_{43} + 1, μ_{41} − 1, μ_{13} − 1) − (μ_{34} + 1)Q(μ_{34} + 1, μ_{32} − 1, μ_{24} − 1)+ (μ_{14} + 1)Q(μ_{14} + 1, μ_{13} − 1, μ_{34} − 1) + (μ_{41} + 1)Q(μ_{41} + 1, μ_{43} − 1, μ_{31} − 1)+

(μ_{42}+1)Q(μ_{42}+1, μ_{41}−1, μ_{13}−1, μ_{32}−1)−(μ_{24}+1)Q(μ_{24}+1, μ_{21}−1, μ_{13}−1, μ_{34}−1) + type 3 terms = 0.

This sum thus has (μ_{42} + 1)Q(μ_{42} + 1, μ_{43} − 1, μ_{32} − 1) in it but no terms like (μ_{12} + 1)Q(μ_{12} + 1, …) or (μ_{23} + 1)Q(μ_{23} + 1, …) and so we can use this equation to reduce μ_{42} to 0 (while keeping μ_{12} = μ_{23} = 0).

Next with μ_{12} = μ_{23} = μ_{42} = 0 we use the equation for log(A_{14})_{f} to reduce μ_{13} to 0. We can do this (while keeping μ_{12} = μ_{23} = μ_{42} = 0) since this equation is

(μ_{34} + 1)Q(μ_{34} + 1, μ_{31} − 1, μ_{14} − 1) + (μ_{31} + 1)Q(μ_{31} + 1, μ_{34} − 1, μ_{41} − 1)− (μ_{43} + 1)Q(μ_{43} + 1, μ_{41} − 1, μ_{13} − 1) − (μ_{13} + 1)Q(μ_{13} + 1, μ_{14} − 1, μ_{43} − 1)+ (μ_{24} + 1)Q(μ_{24} + 1, μ_{21} − 1, μ_{14} − 1) − (μ_{21} + 1)Q(μ_{21} + 1, μ_{24} − 1, μ_{41} − 1)

+ type 2 terms + type 3 terms = 0.

Similarly (with μ_{12} = μ_{23} = μ_{42} = μ_{13} = 0) we now use the equation for log(A_{13})_{f}, namely

(μ_{34} + 1)Q(μ_{34} + 1, μ_{31} − 1, μ_{14} − 1) − (μ_{41} + 1)Q(μ_{41} + 1, μ_{43} − 1, μ_{31} − 1) = 0,

to reduce μ_{34} to 0 (while keeping μ_{12} = μ_{23} = μ_{42} = μ_{13} = 0).

Lastly we use the equation for log(A_{12})_{f} (with μ_{12} = μ_{23} = μ_{42} = μ_{13} = μ_{34} = 0), namely,

(μ_{24} + 1)Q(μ_{24} + 1, μ_{21} − 1, μ_{14} − 1) − (μ_{31} + 1)Q(μ_{31} + 1, μ_{32} − 1, μ_{21} − 1) = 0,

to reduce μ_{24} to 0. The n = 4 case of Theorem 1.1 now follows from the argument following Corollary 6.2.

We now briefly indicate how to do the n = 5 case; we again show that we can reduce to the argument following Corollary 6.2. From the n = 4 case we can assume that each Q((μ_{i,j})) is a sum of Q((λ_{ij}))’s where λ_{ij} = 0 for all (ij) ∈ E(4). We first use the log(A_{12})_{f} + log(A_{13})_{f} + log(A_{14})_{f} + log(A_{23})_{f} + log(A_{24})_{f} + log(A_{34})_{f} equation to reduce μ_{42} to 0 (see Lemma 6.3). In this situation the equations for log(A_{12})_{f} and log(A_{13})_{f} are

(μ_{31} + 1)Q(μ_{31} + 1, μ_{32} − 1, μ_{21} − 1) − (μ_{24} + 1)Q(μ_{24} + 1, μ_{21} − 1, μ_{14} − 1)

− (μ_{25} + 1)Q(μ_{25} + 1, μ_{21} − 1, μ_{15} − 1) = 0;

(μ_{41} + 1)Q(μ_{41} + 1, μ_{43} − 1, μ_{31} − 1) − (μ_{34} + 1)Q(μ_{34} + 1, μ_{31} − 1, μ_{14} − 1) + (μ_{51} + 1)Q(μ_{51} + 1, μ_{53} − 1, μ_{31} − 1) − (μ_{35} + 1)Q(μ_{35} + 1, μ_{31} − 1, μ_{15} − 1) = 0.

The log(A_{45})_{f} equation is

(μ_{15} + 1)Q(μ_{15} + 1, μ_{14} − 1, μ_{45} − 1) + (μ_{14} + 1)Q(μ_{14} + 1, μ_{15} − 1, μ_{54} − 1)+

(μ_{24} + 1)Q(μ_{24} + 1, μ_{25} − 1, μ_{54} − 1) + (μ_{34} + 1)Q(μ_{34} + 1, μ_{35} − 1, μ_{54} − 1)−

(μ_{51} + 1)Q(μ_{51} + 1, μ_{54} − 1, μ_{51} − 1) − (μ_{41} + 1)Q(μ_{41} + 1, μ_{45} − 1, μ_{51} − 1)−

(μ_{53} + 1)Q(μ_{53} + 1, μ_{54} − 1, μ_{43} − 1) − (μ_{43} + 1)Q(μ_{43} + 1, μ_{45} − 1, μ_{53} − 1)

+ type 3 − terms = 0.

We use the log(A_{12})_{f} and log(A_{13})_{f} equations to get rid of the (μ_{24} + 1)Q(μ_{24} + 1, μ_{25} − 1, μ_{54} − 1) and (μ_{34} + 1)Q(μ_{34} + 1, μ_{35} − 1, μ_{54} − 1) terms in this equation. This gives us an equation which has the term (μ_{53} + 1)Q(μ_{53} + 1, μ_{54} − 1, μ_{43} − 1) in it, and which we can use to reduce μ_{53} to 0 while not changing any of the μ_{ij} previously reduced to 0.

Now with μ_{52} = 0 and μ_{53} = 0 the log(A_{15})_{f} equation is

(μ_{25} + 1)Q(μ_{25} + 1, μ_{21} − 1, μ_{15} − 1) + (μ_{21} + 1)Q(μ_{21} + 1, μ_{25} − 1, μ_{51} − 1)+

(μ_{35} + 1)Q(μ_{35} + 1, μ_{31} − 1, μ_{15} − 1) + (μ_{31} + 1)Q(μ_{31} + 1, μ_{35} − 1, μ_{51} − 1)+

(μ_{45} + 1)Q(μ_{45} + 1, μ_{41} − 1, μ_{15} − 1) + (μ_{41} + 1)Q(μ_{41} + 1, μ_{45} − 1, μ_{51} − 1)−

(μ_{54} + 1)Q(μ_{54} + 1, μ_{51} − 1, μ_{14} − 1) − (μ_{14} + 1)Q(μ_{14} + 1, μ_{15} − 1, μ_{54} − 1)

+ type 3 − terms = 0.

We use this equation to reduce μ_{14} to 0.

Lastly with μ_{14} = 0 we now use the log(A_{12})_{f} and log(A_{13})_{f} equations to reduce μ_{25} to and μ_{35} to 0 (respectively). This does the n = 5 case and so completes the proof of Theorem 1.1. □

For the proof of Theorem 1.2 we define

F (n) = {(i, j)|1 ≤ i < j ≤ n, (i, j) ≠ (1, n)}.

We now prove Theorem 1.2. We first note that since x ∈ A(n; ∗) for any non-zero Q((λ_{ij})) we must have ∑_{i=1}^{n} λ_{ij} = ∑_{i=1}^{n} λ_{ji} for j = 1, …, n. It follows that if each monomial y of x has type (2k, 2k, …, 2k), where k = k(y) and y corresponds to Q((λ_{ij})), then

∑_{i=1}^{n} λ_{ij} = ∑_{i=1}^{n} λ_{ji} = ∑_{i=1}^{n} λ_{mi} = ∑_{i=1}^{n} λ_{im}

for all j, m = 1, …, n. We show that each Q((λ_{ij})) is a sum of Q((μ_{ij}))’s where μ_{ij} = 0 for all (i, j) ∈ F (n). We first show that Theorem 1.2 will follow if this is the case. Since each monomial y in x is of type (2k, 2k, …, 2k) for some k = k(y) it easily follows from the above equations that the only non-zero λ_{ij} in such a Q((λ_{ij})) are those where (i, j) = (1, n) or |i − j| = 1. Thus the corresponding monomial in x is also a monomial in det(M)^{k}. Theorem 1.2 will now follow since we can make the coefficient of that monomial 0 in x by adding a suitable multiple of det(M)^{k} to x.

We now show that each Q((λ_{ij})) is a sum of Q((μ_{ij}))’s where μ_{ij} = 0 for all (i, j) ∈ F (n). We proceed by induction on n, the case n ≤ 5 following from the proof of Theorem 1.1 for n ≤ 5 since F (n) ⊂ E(n).

First we need to examine the equation log(A_{ij})_{f} more closely. The types of its individual monomial terms correspond to monomials having types (s_{1}, …, s_{n}) all of whose coordinates s_{r}, 1 ≤ r ≤ n, are the same except for s_{i} and s_{j}, which are either (s_{i}, s_{j}), (s_{i} +2, s_{j}) or (s_{i}, s_{j} +2). Thus from the equation log(A_{ij})_{f} we obtain three equations, two of which are

\begin{matrix}∑_{\begin{matrix}r=1 \\ r≠i,j \end{matrix}}^{n}(λ_{r,j} + 1)Q(λ_{r,j} + 1, & λ_{4,i} − 1, λ_{i,j} − 1) \\ − (λ_{jr} + 1)Q(λ_{j,r} + 1, λ_{j,i} − 1, λ_{i,r} − 1) = 0; (i, j.1) \end{matrix}

\begin{aligned}∑_{\begin{matrix}r=1 \\ r≠i,j \end{matrix}}^{n}(λ_{ri} + 1)Q(λ_{ri} + 1, λ_{rj} & − 1, λ_{ji} − 1) \\ & − (λ_{ir} + 1)Q(λ_{ir} + 1, λ_{ij} − 1, λ_{jr} − 1) = 0. (i, j.2) \end{aligned}

Now with F (n − 1) and 1 < r < n − 1 equation (1, r.1) is

(μ_{r,n−1}+1)Q(μ_{r,n−1}+1, μ_{r1}−1, μ_{1,n−1}−1)+(μ_{r,n}+1)Q(μ_{r,n}+1, μ_{r,1}−1, μ_{1,n}−1) = 0

while (n − 1, n.2) is

\begin{matrix}∑_{r=1}^{n−2}(λ_{r,n−1} + 1)Q(λ_{r,n−1} + 1, & λ_{4n} − 1, λ_{n,n−1} − 1)− \\ (λ_{n−1,r} + 1)Q(λ_{n−1,r} + 1, λ_{n−1,n} − 1, λ_{nr} − 1) = 0. \end{matrix}

Using the equations (1, r.1), 1 < r < n − 1, we can remove the unwanted terms Q(λ_{r,n−1} +1, λ_{rn} −1, λ_{n,n−1} −1) (for r=2, …,n-2) from this latter equation and so obtain an equation which enables us to reduce λ_{1,n−1} to 0. Now with λ_{1,n−1} = 0 the only non-zero λ_{1r} is λ_{1n} and since the row and column sums for the λ_{ij} are equal we see that λ_{rn} = 0 for r = 2, …, n − 1. Thus each Q((λ_{ij})) is a sum of Q((μ_{ij}))’s where μ_{ij} = 0 for all (i, j) ∈ F (n) and so Theorem 1.2 follows by induction. □

§7. Poincaré series: statement of results.

Recall that for fixed 1 ≤ i ≠ j ≤ n, A(n; i, j) is the vector subspace of ℂ_{(n)}[a_{ij}] generated by all ij-words. Note that A(n; i, j) has a natural grading by degree:

A(n; i, j) = ⨁_{k=0}^{∞} A_{k}(n; i, j),

and also that A(n; i, i) is a graded commutative ring (we will forget the Lie algebra structure for the moment). Here each A_{k}(n; i, j) is a vector space of dimension p_{k}(n; i, j). In this section we calculate the Poincaré series:

P (n; i, j) = ∑_{k=0}^{∞} p_{k}t^{k},

where p_{k} = p_{k}(n) = p_{k}(n; i, j). Clearly P (n; i, j) = P (n; r, s) if and only if card({i, j}) = card({r, s}).

Theorem 7.1. The Poincaré series P (n; i, j) is a rational function of t where the denominator is a product of cyclotomic polynomials.

Examples 7.2.

P (3; 1, 1) = \frac{t^{6} − t^{5} − t^{4} + 2t^{2}}{(1 + t + t^{2})(1 + t)^{2}(1 − t)^{4}}
= 2t^{2} + 2t^{3} + 5t^{4} + 6t^{5} + 11t^{6} + 12t^{7} + 20t^{8} + 22t^{9} + 32t^{10} + 36t^{11}

+ 49t^{12} + 54t^{13} + 71t^{14} + 78t^{15} + 98t^{16} + 108t^{17} + 132t^{18} + …

P (3; 1, 2) = \frac{t}{(1 + t + t^{2})(1 + t)^{2}(1 − t)^{4}}
= t + t^{2} + 3t^{3} + 4t^{4} + 7t^{5} + 9t^{6} + 14t^{7} + 17t^{8} + 24t^{9} + 29t^{10} + 38t^{11}

+ 45t^{12} + 57t^{13} + 66t^{14} + 81t^{15} + 93t^{16} + 111t^{17} + 126t^{18} + …

P (4; 1, 1) = \frac{3t^{2} − 3t^{4} + 6t^{5} + 7t^{6} − 8t^{7} − 6t^{8} − 4t^{9} + 13t^{10}}{(1 + t + t^{2})^{2}(1 + t)^{5}(1 + t^{2})(1 − t)^{9}}

\frac{8t^{11} − 8t^{13} − 14t^{14} + 6t^{15} + 6t^{16} + 6t^{17} − 3t^{18} − 6t^{19} + 3t^{20}}{(1 + t + t^{2})^{2}(1 + t)^{5}(1 + t^{2})(1 − t)^{9}}

= 3t^{2} + 6t^{3} + 18t^{4} + 42t^{5} + 94t^{6} + 180t^{7} + 348t^{8} + 602t^{9} + 1047t^{10} + …

P (4; 1, 2) = t + 2t^{2} + 7t^{3} + 18t^{4} + 42t^{5} + 88t^{6} + 176t^{7} + 324t^{8} + 581t^{9}+

984t^{10} + 1631t^{11} + 2594t^{12} + 4055t^{13} + 6140t^{14} + 9164t^{15} + ….

We are also interested in the following subrings of A(n; i, i): Let

B(n; i) = {x ∈ A(n; i, i)|x^{∗} = x}

be the fixed “symmetric” subspace of A(n; i, i) for the involution ∗. The ring B(n; i) is a graded subring of A(n; i, i):

B(n; i) = ⨁_{k=0}^{∞} B_{k}(n; i).

Let B(n; i) = ∑_{k=0}^{∞} q_{k}t^{k} be the corresponding Poincaré series where q_{k} = q_{k}(n) = dim(B_{k}(n; i)). Another graded subspace is the subspace

C(n; i) = ⨁_{k=0}^{∞} C_{k}(n; i)

of B(n; i) generated by all monomials which are invariant under ∗. Let R(n; i) = ∑_{k=0}^{∞} r_{k}t^{k} be its Poincaré series where r_{k} = r_{k}(n) = dim(C_{k}(n; i)). Lastly let

D(n; i) = {x ∈ A(n; i, i)|x^{∗} = −x}

be the “anti-symmetric” subspace of A(n; i, i) for ∗. Again D(n; i) has a grading:

D(n; i) = ⨁_{k=0}^{∞} D_{k}(n; i).

Let S(n; i) = ∑_{k=0}^{∞} s_{k}t^{k} be the Poincaré series with s_{k} = s_{k}(n) = dim(D_{k}(n; i)).

Theorem 7.3. The Poincaré series R(n; i) is a rational function of t^{2} where the denominator is equal to a product of cyclotomic polynomials of degree no greater than n(n − 1) in t.

As will be seen in the proof of Theorem 7.3 the monomial generators of C(n; i) correspond naturally to weighted, rooted (i.e. with base point), connected subgraphs of the (un-directed) complete graph on n vertices, there being exactly dim(C_{2k}(n; i)) such subgraphs of weight k.

Examples 7.4.

R(3; 1) = \frac{t^{4} − 2t^{2}}{(t^{2} − 1)^{3}}

= 2t^{2} + 5t^{4} + 9t^{6} + 14t^{8} + 20t^{10} + 27t^{12} + 35t^{14} + 44t^{16} + 54t^{18} + 65t^{20}+

77t^{22} + 90t^{24} + 104t^{26} + 119t^{28} + 135t^{30} + 152t^{32} + 170t^{34} + …

R(4; 1) = \frac{3t^{12} − 12t^{10} + 18t^{8} − 13t^{6} + 6t^{4} − 3t^{2}}{(1 − t^{2})^{6}} =

3t^{2} + 12t^{4} + 40t^{6} + 102t^{8} + 219t^{10} + 419t^{12} + 738t^{14} + 1221t^{16}+

1923t^{18} + 2910t^{20} + 4260t^{22} + 6064t^{24} + 8427t^{26} + 11469t^{28} + …
R(5; 1) =

\frac{8t^{20} − 54t^{18} + 144t^{16} − 178t^{14} + 60t^{12} + 102t^{10} − 137t^{8} + 70t^{6} − 18t^{4} + 4t^{2}}{(1 − t^{2})^{10}}

= 4t^{2} + 22t^{4} + 110t^{6} + 453t^{8} + 1482t^{10} + 4083t^{12} + 9926t6^{14} + 21959t^{16}+

45122t^{18} + 87347t^{20} + 160922t^{22} + 284310t^{24} + …
R(6; 1) = \frac{45t^{30} − 520t^{28} + 2705t^{26} − 8340t^{24} + 16875t^{22} − 23430t^{20} + 22710t^{18}}{(1 − t^{2})^{15}} +

\frac{−15480t^{16} + 7695t^{14} − 3300t^{12} + 1586t^{10} − 745t^{8} + 235t^{6} − 40t^{4} + 5t^{2}}{(1 − t^{2})^{15}}

= 5t^{2} + 35t^{4} + 235t^{6} + 1380t^{8} + 6711t^{10} + 26630t^{12} + 89695t^{14}+

266305t^{16} + 716460t^{18} + 1780978t^{20} + 4147915t^{22} + 9144485t^{24} + ….

Theorem 7.5. The Poincaré series Q(n; i) and S(n; i) are rational functions of t where the denominator is a product of cyclotomic polynomials. In fact

Q(n; i) = \frac{P (n; i, i) + R(n; i)}{2} and S(n; i) = \frac{P (n; i, i) − R(n; i)}{2}.

In the dual direction we have the following result.

Theorem 7.6. For fixed k the functions p_{k}(n; i, j), q_{k}(n), r_{k}(n) and s_{k}(n) are all polynomials in n of degree k − 1.

This means that for fixed k we have ‘dual’ Poincaré series

∑_{n=0}^{∞} p_{k}(n; i, j)t^{n},  ∑_{n=0}^{∞} r_{k}(n)t^{n} etc.

which are also rational functions.

Examples 7.7. We give some of the polynomials for the rings A(n; i, i) and C(n; i) since those for B(n; i) and D(n; i) are related to these as indicated in Theorem 7.5.

p_{2}(n; i, i) = n − 1;
p_{3}(n; i, i) = (n − 1)(n − 2);
p_{4}(n; i, i) = \frac{(n − 1)(2n^{2} − 7n + 8)}{2};
p_{5}(n; i, i) = (n − 1)(n − 2)(n^{2} − 3n + 3);
r_{2}(n) = n − 1;
r_{4}(n) = \frac{(n − 1)(3n − 4)}{2};
r_{6}(n) = \frac{(n − 1)(16n^{2} − 59n + 60)}{6};
r_{8}(n) = \frac{(n − 1)(125n^{3} − 873n^{2} + 2134n − 1752)}{24};
r_{10}(n) = \frac{(n − 1)(648n^{4} − 7267n^{3} + 3138n^{2} − 60422n + 43140)}{60}.

From (1.1) we see that A(n + r; i, j) is invariant under the action of H̃(n) for all r ≥ 0 and that A(n; i, j) is invariant under the action of H̃(n)^{∗} = H̃(n)/Inn(n).

Proposition 7.8. The action of H(n) on A(n + 1; i, j) is faithful and the action of H(n)^{∗} on A(n; i, j) is faithful.

Proof. Let α ∈ H(n)^{∗}, α ≠ id. Then by Corollary 2.4 of [Hu1] we see that for 1 ≤ r ≠ s ≤ n + 1, α(a_{rs}) = y_{rs} has a unique monomial summand z_{rs} of maximal degree and further that, since α ≠ id, there are 1 ≤ u ≠ v ≤ n+1 such that z_{uv} and z_{vu} have degree greater than 1. Now if u, v ≠ i, j, then a_{iu}a_{uv}a_{vj} ∈ A(n + 1; i, j) and so α(a_{iu}a_{uv}a_{vj}) = y_{iu}y_{uv}y_{vj} has a unique monomial summand z_{iu}z_{uv}z_{vj} of degree greater than 3. Thus α(a_{iu}a_{uv}a_{vj}) ≠ a_{iu}a_{uv}a_{vj} showing that H(n)^{∗} acts faithfully on A(n + 1; i, j). The remaining cases are similar. □

Again using (1.1) one can show that the other graded rings that we study are also invariant under the action of H̃(n):

Proposition 7.9. The action of H(n) on B(n+r; i), C(n+r; i) and D(n+r; i) is faithful if r > 0 and the action of H(n)∗ on B(n; i), C(n; i) and D(n; i) is faithful.

Proof. This is similar to the proof of Proposition 7.8. □

§8 Proof of Theorem 7.3

Let K_{n} denote the complete directed graph on the vertices {1, 2, …, n} and with edges {(i, j)|1 ≤ i ≠ j ≤ n}. Let G_{k}(n; i, j) be the set of all monomials of degree k in A_{k}(n; i, j).

Let E = E_{n} be the edge set of K_{n}; then a weighted graph is a function w : E → ℤ^{≥0}. The edge e ∈ E will be said to have weight w(e). If w ∈ G_{k}(n; i, j) and 1 ≤ r ≠ s ≤ n, then w_{rs} will denote the exponent of a_{rs} in w and we let |w| = ∑_{r≠s} w_{rs}. Thus each w ∈ G_{k}(n; i, j) determines a weighted graph. We let γ(w) be the weighted graph where edge (r, s) has weight w_{rs}. We write |γ(w)| = |w| which we call the degree of γ(w). Let P = P_{n} be the set of non-negatively-weighted graphs on n vertices, and let P (k) = P_{n}(k) be those of weight k. We will consider P as a commutative monoid where one just adds the weights of the edges of any two elements of P to obtain a third; the identity is the weighted graph with all weights 0. There is a natural map Ψ : P → P where if γ ∈ P has weight γ_{ij} on the edge (i, j), then Ψ(γ) has weight γ_{ij} − min(γ_{ij}, γ_{ji}) on (i, j). This amounts to ‘canceling’ all the cycles of length 2 in γ. We will also define the restriction

Ψ_{k} = Ψ|_{P (k)} : P (k) → P.

In counting generators for C(n; i) we will repeatedly use the fact that the number of weighted graphs of total weight k having m edges is equal to the number of non-negative solutions of the equation

x_{1} + x_{2} + ⋯ + x_{m} = k.

This is well-known to be the binomial coefficient \binom{k+m−1}{m−1}. We will also have occasion to use the following result, the proof of which is by induction on k.

Lemma 8.1. If h_{0}(t) = \frac{1}{1−t} − 1 and h_{i}(t) = t\frac{d}{dt}(h_{i−1}(t)) for i > 0, then the power series expansion for h_{k}(t) is ∑_{m=1}^{∞} m^{k}t^{m}. Also h_{k}(t) is a rational function whose denominator is (1 − t)^{k+1}. In particular any power series of the form

∑_{m=1}^{∞} ∑_{k=1}^{N} a_{k}m^{k}t^{m}

where a_{k} ∈ ℂ[t] for k = 1, …, N, is a rational function whose denominator is a product of cyclotomic polynomials dividing (1 − t)^{N+1}. □

Now note that if w is a monomial in C_{k}(n; i), then w_{rs} = w_{sr} for 1 ≤ r ≠ s ≤ n. Thus Ψ_{k}(w) = 0 and w has even weight. In particular R(n; i) is really a power series in t^{2}. We will thus assume that k is even. Given γ ∈ P we define supp (γ), the support of γ to be the weighted graph with weight 1 on each edge of γ with non-zero weight and with weight 0 on all other edges. Note that γ = supp(γ) + γ′ for some graph γ′ ∈ P. We also define S^{∗} to be the finite set of supports of the monomial generators of C(n, i). Thus each weighted graph in S^{∗} has an edge containing the vertex i and is a connected union of distinct 2-cycles in K_{n}. Further, if w is a monomial generator of C_{k}(n; i), then

γ(w) = supp(γ(w)) + γ′

where γ′ is a union of 2-cycles in supp(γ(w)) and γ′ has weight k − |supp(γ(w))|. Thus we will identify the monomial generators of C_{k}(n; i) with such weighted graphs. Moreover if γ′′ ∈ S^{∗} has weight less than k and if γ′ is any weighted graph of weight k − |γ′′| which is a union of 2-cycles contained in γ′′, then γ′′ + γ′ is also a monomial generator of C(n; i) of weight k. It follows that the number of monomial generators of C(n; i) having weight k and support γ′′ is equal to

\binom{(\frac{k}{2} − \frac{|γ′′|}{2}) + \frac{|γ′′|}{2} − 1}{\frac{|γ′′|}{2} − 1} = \binom{\frac{k}{2} − 1}{\frac{|γ′′|}{2} − 1}.

Thus

dim (C_{k}(n; i)) = ∑_{γ′′∈S^{∗}} \binom{\frac{k}{2} − 1}{\frac{|γ′′|}{2} − 1}.

Now for fixed γ′′ ∈ S^{∗} the binomial coefficient \binom{\frac{k}{2} −1}{\frac{|γ′′}{2} −1} is a polynomial in k of degree

\frac{|γ′′|}{2} − 1 and so by lemma 8.1

∑_{k≥|γ′′|} \binom{\frac{k}{2} − 1}{\frac{|γ′′|}{2} − 1}t^{k}

is a rational function whose denominator is a divisor of (1 − t^{2})^{\frac{γ′′}{2}} . Since |γ′′| ≤ n(n − 1) Theorem 7.3 follows. □

§9 Proof of Theorem 7.1

We first reformulate the problem in terms of weighted graphs again. If 1 ≤ i ≤ n, then we let

I_{w}(i) = ∑_{j=1}^{n} w_{ij} and T_{w}(i) = ∑_{j=1}^{n} w_{ji}.

The following result is clear.

Lemma 9.1. For a monomial w ∈ A(n; i, j) and γ(w) as defined above we have (i) γ(w) is connected (ii) if i = j, then at each vertex k of K_{n} we have I_{w}(k) = T_{w}(k); (iii) if i ≠ j, then at each vertex k ≠ i, j of K_{n} we have I_{w}(k) = T_{w}(k), while

I_{w}(i) = T_{w}(i) + 1 and I_{w}(j) = T_{w}(j) − 1. Moreover any weighted graph satisfying the above three conditions defines a unique ij-monomial in A(n; i, j). □

Thus we will need to count all such weighted graphs, which we will call ij-graphs. We will prove the case i = j, the other case being only slightly more involved. As in the proof of 7.3 we first partition using supports:

p_{k}(n; i, i) = ∑_{γ′′∈S^{∗}} r_{k}(γ′′)

where r_{k}(γ′′) is the cardinality of the set R_{k}(γ′′) of weight k ii-graphs with support equal to γ′′ ∈ S^{∗}. Next we partition each R_{k}(γ′′) using the function Ψ. First order the set of oriented cycles (without base point) in K_{n} as follows: order them by length and then for cycles of the same length choose some (arbitrary) total ordering. Thus distinct cycles in K_{n} have different orders; i.e. we have a total order on the set of oriented cycles in K_{n}. We use this ordering to represent each graph Ψ_{k}(γ) as a sum of cycles in a unique way. Given γ_{0} = Ψ_{k}(γ) there is a unique cycle γ_{1} of maximal order in γ_{0}. Thus we have γ_{0} = γ_{1} + γ′_{0} where |γ′_{0}| < |γ_{0}|. Repeating this process for γ′_{0} gives the representation referred to. We will write this representation as

Ψ_{k}(γ) ≈ n_{1}γ_{1} + n_{2}γ_{2} + ⋯ + n_{s}γ_{s}

where γ_{1} > γ_{2} > ⋯ > γ_{s} and n_{1}, n_{2}, …, n_{s} ≠ 0. A set {γ_{1}, γ_{2}, …, γ_{p}} of cycles in K_{n} will be called oriented if for each 1 ≤ r, s ≤ n the edges (r, s) and (s, r) do not both belong to the weighted graph γ_{1} + γ_{2} + ⋯ + γ_{p}.

Lemma 9.2. If Ψ_{k}(γ) ≈ n_{1}γ_{1} + n_{2}γ_{2} + ⋯ + n_{s}γ_{s}, then {γ_{1}, γ_{2}, …, γ_{s}} is an oriented set of cycles in K_{n}. Further, if Ψ_{k}(γ′) ≈ n′_{1}γ′_{1} + n′_{2}γ′_{2} + ⋯ + n′_{s′}γ′_{s′}, and Ψ_{k}(γ) = Ψ_{k}(γ′), then s = s′ and γ_{i} = γ′_{i}, n_{i} = n′_{i} for 1 ≤ i ≤ s.

Proof. Since Ψ_{k} ‘kills’ all cycles of length 2 in K_{n}, the first statement is clear by construction. For the second statement we induct on N = n_{1} + n_{2} + ⋯ + n_{s}, the case N = 0 being clear. Now if N > 0, then there is a cycle in Ψ_{k}(γ) = Ψ_{k}(γ′) of maximal order. By construction this must be γ_{1} and it must also be γ′_{1}. Thus γ_{1} = γ′_{1}. We may now proceed by induction after subtracting one γ_{1} from both sides of the equation n_{1}γ_{1} + n_{2}γ_{2} + ⋯ + n_{s}γ_{s} = n′_{1}γ′_{1} + n′_{2} + ⋯ + n′_{s′}γ′_{s′}. □

For γ ∈ P we let F (γ) be the set of pairs {(r, s), (s, r)} of edges of γ with γ_{rs}, γ_{sr} ≠ 0 and we let f(γ) = card(F (γ)). Note that f(γ) ≤ \frac{n(n−1)}{2}. Now if Ψ_{k}(γ) ≈ n_{1}γ_{1} + n_{2}γ_{2} + ⋯ + n_{s}γ_{s}, m = k − (n_{1}|γ_{1}| + n_{2}|γ_{2}| + ⋯ + n_{s}|γ_{s}|) is even and m ≥ 2f(γ), then, putting f = f(γ), there are

( \begin{matrix}(\frac{m}{2} − f) + f − 1 \\ f − 1 \end{matrix} ) = ( \begin{matrix}\frac{m}{2} − 1 \\ f − 1 \end{matrix} )

graphs in Ψ_{k}^{−1}(Ψ_{k}(γ))∩R_{k}(γ′′). Now summing over all oriented sets {γ_{1}, γ_{2}, …, γ_{s}} of cycles in supp(γ′′) and over all n_{1}, n_{2}, …, n_{p} such that m = k −(n_{1}|γ_{1}|+n_{2}|γ_{2}|+ ⋯ + n_{p}|γ_{p}|) is even and non-negative will give card(R_{k}(γ′′)). Now note that for fixed f each \binom{\frac{m}{2} −1}{f−1} is a polynomial in \frac{m}{2} of degree f − 1. Theorem 7.1 will follow from the above, lemma 8.1 and

Lemma 9.3. The sum over all n_{1}, n_{2}, …, n_{s} of all the \binom{\frac{m}{2} −1}{f−1}’s where f is fixed and m = k −(n_{1}|γ_{1}|+n_{2}|γ_{2}|+⋯+n_{s}|γ_{s}|) is even and non-negative is a polynomial in k with integral coefficients of degree s + f − 1.

Proof. This is by induction on s ≥ 0. The case s = 0 follows by expanding \binom{\frac{m}{2} −1}{f−1} as a polynomial in \frac{m}{2} of degree f − 1 and then using the Euler-MacLaurin formula. The cases s ≥ 1 follow inductively by splitting the sum up as

∑_{m≥2 even} ∑_{n_{1}} ∑_{n_{2},...,n_{s}} \binom{\frac{m}{2} − 1}{f − 1}. □

By lemma 8.1 and the above proof the degree of the denominator in the Poincaré series P (n; i, i) is dependent upon the function f(n) defined to be

max{card({γ_{1}, γ_{2}, …, γ_{s}})|{γ_{1}, γ_{2}, …, γ_{s}} is an oriented set of cycles in K_{n}}.

The function f(n) is not well understood; see sections 4, 5 and 10 of [Mo] where the maximum number of cycles of a given length in a tournament matrix is considered (each oriented set of cycles {γ_{1}, γ_{2}, …, γ_{s}} in K_{n} gives a tournament matrix with a 1 in the ij position if and only if some γ_{h} has (i, j) as an edge).

§10 Proof of Theorem 7.5

Let T : A_{k}(n; i, i) → B_{k}(n; i) be defined by

T (x) = x + x^{∗}.

Then T is a group homomorphism whose image is in B(n; i) since ∗ has order 2. Now T is onto, since our ring of scalars contains ℚ. Let X = {x_{1}, …, x_{r}} be a minimal set of monomial generators for A_{k}(n; i, i). We can partition X as X = X_{1} ∪ X_{2} where X_{1} = {x_{i} ∈ X|x_{i}^{∗} = x_{i}} and X_{2} = X\X_{1}. Then X_{1} is a minimal set of monomial generators for C_{k}(n; i) and the elements of X_{2} come in pairs {x_{i}, x_{i}^{∗}}. This gives the formula for Q(n; i) in Theorem 7.5.

Now if x ∈ A_{k}(n; i, i), then we can write x as a sum of symmetric and antisymmetric parts:

x = \frac{x + x^{∗}}{2} + \frac{x − x^{∗}}{2} ,

from which the formula for S(n; i) in Theorem 7.5 follows. □

§11 Proof of Theorem 7.6

We will only sketch the proof this time since it is similar to those above. For a fixed k > 0 there are only a finite number (depending only on k) of (isomorphism classes of) connected, rooted, weighted graphs Γ on k vertices of the various types (A(n; i, j), B(n; i), C(n; i) or D(n; i)). For a fixed type we partition over these. Now for the cases A(n; i, i), B(n; i), C(n; i) and D(n; i) the number of rooted embeddings of Γ in K_{n} is

\frac{(n − 1)(n − 2) … (n − v(Γ) + 1)}{|Sym_{0}(Γ)|}

where v(Γ) is the number of vertices of Supp(Γ) and Sym_{0}(Γ) is the group of rooted symmetries of Γ. Note that this is a polynomial of degree at most k − 1 in n. Summing over the finite number of connected, rooted, weighted graphs Γ on k vertices gives 7.8 for the cases A(n; i, i), B(n; i), C(n; i) and D(n; i). For A(n; i, j) with i ≠ j the number of embeddings of Γ in K_{n} is

\frac{(n − 1)(n − 2) … (n − v(Γ) + 1)}{|Sym_{1}(Γ)|}

where Sym_{1}(Γ) is the group of symmetries of Γ which fix the vertices i, j. This proves 7.6. □

Note that 7.6 shows that if we let

(x)_{m} = x(x − 1)(x − 2) … (x − m + 1),

then for all k, n we have

r_{2k}(n) = ∑_{m=2}^{k+1} \frac{r_{km}}{(m − 1)!}(n)_{m}

where r_{km} is the number of rooted, labeled, connected weight k graphs with m vertices. For example r_{k2} = 1 and if m = k + 1, then we are considering labeled trees on k + 1 vertices, each edge having weight 1. A result of Cayley [G-J, p. 174] says that r_{k,k+1} = (k + 1)^{k}.

Remarks 11.1.

1. We note that there are generalizations of the results of section 7 to the situation where we allow some of the a_{rs} to be 0. The results and proofs are essentially the same as those given above.

2. The methods of proof of the various results of section 7 allow one to calculate the Poincaré series given as examples in section 7.

References

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[G-J] Goulden I.P., Jackson D.M., Combinatorial Enumeration (1983), Wiley Series in Discrete

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