ACTION OF SOME BRAID GROUPS ON HODGE ALGEBRAS

Stephen P. Humphries

Department of Mathematics Brigham Young University Provo, UT 84602, U.S.A. email steve@math.byu.edu

Abstract. In this paper we show that the braid groups B_{n} and the symmetric automorphism groups H(n) of the free group F_{n}, n = 3, 4 act in a non-linear way on an algebra with straightening law (ordinal Hodge algebra). We indicate various properties of these rings.

§1. Introduction.

In this paper we show that the braid groups B_{n} and the symmetric automorphism groups H(n) of the free group F_{n}, n = 3, 4 act in a non-linear way on an algebra with straightening law (ordinal Hodge algebra). We indicate various properties of these rings, for example that they are Cohen-Macaulay; we also calculate their growth series, which are rational functions, and their rings of invariants. These algebras will be subalgebras of the following polynomial rings. For R a commutative ring with identity and characteristic zero, usually taken to be ℤ, ℚ or ℂ, we let R_{(n)}[a_{ij}] denote the polynomial algebra

R[a_{12}, a_{13}, …, a_{1n}, a_{21}, a_{23}, …, a_{2n}, …, a_{n1}, …, a_{n,n−1}]

where the a_{ij} are commuting indeterminates. Let the group H(n) =< t_{ij}|1 ≤ i ≠ j ≤ n > act on R_{(n)}[a_{ij}] as follows: for λ ∈ ℤ, t_{kj}^{λ}(a_{rs}) = b_{rs}, where

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

Here t_{kj} = t_{kj}^{1}. For i = 1, …, n we will put t_{ii} = id for convenience. If we let λ ∈ R in (1.1), then we get elements t_{kj}^{λ} which generate a subgroup H̃(n) =< t_{kj}^{λ}|1 ≤ i ≠ j ≤ n, λ ∈ R > of Aut(R_{(n)}[a_{ij}]). Thus H̃(n) contains H(n) as a

Typeset by AMS-TEX

1991 Mathematics Subject Classification. 20F36, 13F50.

Key words and phrases. Braid groups, Hodge algebra.

STEPHEN P. HUMPHRIES

‘discrete’ subgroup. Let the symmetric group Σ_{n} act on R_{(n)}[a_{ij}] by permutation of subscripts and let Ĥ(n) =< H(n), Σ_{n} >. Then there is a split exact sequence

1 → H(n) → Ĥ(n) → Σ_{n} → 1.

In fact Ĥ(n) is the group of symmetric automorphisms of the free group F_{n} of rank n [Mc]. The action of H(n), Ĥ(n) and H̃(n) on R_{(n)}[a_{ij}] extends to an action of H(n), Ĥ(n) and H̃(n) on R_{(n+r)}[a_{ij}], r > 0 in the obvious way. This action of H(n) on R_{(n+r)}[a_{ij}] is only faithful if r > 0, otherwise the kernel is the free normal subgroup Inn(n) generated by the ‘inner’ automorphisms {J_{k} = t_{k1}t_{k2} … t_{kn}|k = 1, …, n} [Hu1, Theorem 2.5]. We note as in [Hu3] that there is a natural ring involution ∗ on R_{(n+r)}[a_{ij}] which commutes with the action of Ĥ(n); this involution is determined by its action on the generators a_{ij} which is as follows: a_{ij}^{∗} = −a_{ji}.

Note that 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]. A presentation 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

§2. Algebras with straightening law

Let A be an R-algebra and Π ⊂ A a finite subset with a partial order ≤. Then A is a graded algebra with straightening law (on Π, over R) (shortened to ASL most of the time) if we have:

(1) A = ⨁_{i≥0} A_{i} is a graded R-algebra such that A_{0} = R, the poset Π consists of homogeneous elements of positive degree which generate A as an Ralgebra.

(2) The products ψ_{1} … ψ_{m}, m ∈ ℕ, ψ_{i} ∈ Π, such that ψ_{1} ≤ ψ_{2} ≤ ⋯ ≤ ψ_{m} are linearly independent. They are called the standard monomials.

(3) For all incomparable ψ, ν ∈ Π the product ψν has a representation

ψν = ∑ a_{μ}μ, a_{μ} ∈ R, μ a standard monomial

where the μ’s occuring in the above sum with non-zero coefficients each contain a factor ζ ∈ Π such that ζ ≤ ψ, ζ ≤ ν.

The standard monomials form a basis for A as an R-module. For a proof of this fact and general information about ASL’s see [BV]. ASL’s are called ordinal Hodge algebras in [DEP].

Let c_{ijk...rs} represent the cycle a_{ij}a_{jk} … a_{rs}a_{si}. Then the cycles generate a subalgebra of R_{(n)}[a_{ij}] denoted Y_{n}. A cycle c_{ijk...rs} will be called simple if i, j, k, …, r, s are all distinct. Let Π_{n} be the set of simple cycles. It is easy to see that every cycle in R_{(n)}[a_{ij}] is a product of simple cycles in Π_{n}.

Lemma 2.1. For r ≥ 0 the algebra Y_{n+r} is invariant under the action of Ĥ(n) and B_{n}. The kernel of the action of Ĥ(n) on Y_{n+r} is trivial if r > 0 and is equal to Inn(n) if r = 0. The kernel of the action of B_{n} on Y_{n+r} is trivial if r > 0 and is equal to the centre Z(B_{n}) if r = 0.

Proof. This uses equation (1.1) and is similar to the proofs of Propositions 7.8 and 7.9 of [Hu3]. □

For n = 3, 4 we make Y_{n} into an ASL on Π_{n} by giving the simple cycles the following partial orders: for n=3 three ways of doing this are indicated in Figures 1.1, 1.2 and 1.3.

In Figures 1 {a, b} = {c_{123}, c_{132}} and {x, y, z} = {c_{12}, c_{23}, c_{31}}. We note the similarity of Figure 1.2 with the diagram giving the partial order corresponding to the determinantal ring of a 2 × 2 matrix on page 39 of [BV].

It is easy to check that this is an ASL: since c_{123}c_{132} = c_{12}c_{23}c_{31} we get condition (3) in the definition; further, any monomial in Y_{3} is easily seen to have either the form c_{123}^{e}c_{12}^{f}c_{23}^{g}c_{31}^{h} or the form c_{132}^{e}c_{12}^{f}c_{23}^{g}c_{31}^{h}, thus giving (2).

For n = 4 we use the partial order indicated in Figure 2.

The top diagram in Figure 2 depicts a cellulated tetrahedron (viewed from above) with faces, edges and vertices labeled by the simple cycles. The bottom diagram in Figure 2 shows the cell structure for the base of the tetrahedron. The 2-cells are labeled by the cycles c_{ij} in a natural way. In this order we also have

c_{12} < c_{13} < c_{14} < c_{23} < c_{24} < c_{34} < c_{rst}

for all r, s, t ∈ {1, 2, 3, 4} where c_{rst} is a vertex of the above tetrahedron. If we ignore the sign changes, then the involution ∗ on R_{n}[a_{ij}] gives rise to a map of the cells of this tetrahedron which is just the dual map of the tetrahedron.

Theorem 2.2. With the above order on simple cycles Y_{4} is an ASL.

Proof. First one checks that products of incomparable terms for this partial order have the correct form, for example:

c_{ijk}c_{ikj} = c_{ij}c_{jk}c_{ki} for all i, j, k = 1, 2, 3, 4;
c_{123}c_{1324} = c_{124}c_{13}c_{23};
c_{1234}c_{1342} = c_{134}c_{234}c_{12} etc.

Given two simple cycles c_{U}, c_{V} ∈ Π_{n} we now define a notion of relative orientation of this pair of simple cycles: Consider the complete graph K_{n} with vertices labeled 1, …, n. We will say that cycles c_{U} and c_{V} are oriented the same if the oriented edge paths in K_{n} defined by U and V do not contain a common edge that is oriented differently in each case. Here, for example, the closed oriented edge path corresponding to U would be

i_{1} → i_{2} → ⋯ → i_{k−1} → i_{k} → i_{1}

if U = i_{1}i_{2} … i_{k}.

It is easy to see that the highest level monomials in Figure 2 are the c_{ijk} that are at the centres of the triangles of the tetrahedron in Figure 2. All the c_{rstu} are at the next highest level and the c_{ijk} that are at the vertices of the tetrahedron are at the level just below the c_{rstu}. In Figure 3 we give a different view of the partial order on Π_{4}. This shows the order of the elements more clearly than Figure 2 and will be particularly useful in the next section.

Now note that any monomial μ in Y_{4} can be written uniquely as μ = μ_{1}μ_{2} where μ_{2} is a product of powers of the c_{ij} and μ_{1} is a product of powers of the c_{ijk} and c_{ijkl}, all of which are (pairwise) oriented the same. Note that pairs of cycles in each of the ‘levels’ in Figure 3 are all oriented differently; for example c_{123} and c_{134} do not agree in orientation along the edge from 1 to 3 and that in this case we have c_{123}c_{134} = c_{13}c_{1234}. Now one checks that elements ψ_{1}, ψ_{2} ∈ Π_{4} having the same orientation are comparable in the above partial order i.e. we either have ψ_{1} < ψ_{2} or ψ_{1} > ψ_{2}; conversely if ψ_{1}, ψ_{2} ∈ Π_{4} are comparable, then they have the same orientation. Thus for any monomial μ = μ_{1}μ_{2} as above the element μ_{2} occurs as a standard monomial in the given order.

We now show that the standard monomials are linearly independent. Clearly the standard monomials are actually monomials in Y_{4}. We say that a monomial μ in Y_{n} is oriented consistently if all the simple cycles c_{S} which divide μ are (pairwise) oriented the same. Note that if we write μ = μ_{1}μ_{2} as above, then μ_{1} is oriented consistently.

Lemma 2.3. If μ ∈ Y_{4} is any monomial which is oriented consistently, then μ can be written uniquely as a product c_{ijk}^{α}c_{rstu}^{β}c_{abc}^{γ}. where c_{ijk} > c_{rstu} > c_{abc}.

Proof. One first checks that any two similarly oriented c_{ijk} (one of which must necessarily be in the top (first) level of Figure 3, the other in the third level) share a unique edge and leave a unique edge unoriented; these two edges are disjoint (do not share a vertex). A choice of orientation for the edge with undetermined orientation determines a unique simple cycle c_{rstu} which is oriented the same as each of the two similarly oriented c_{ijk}, there thus being only two such c_{rstu} which are compatible with the two c_{ijk}. We now check three cases.

a) Here μ is divisible by two of the c_{ijk}, say c_{ijk} and c_{i′j′k′}. Then we have the situation described above. It follows that μ is not divisible by any of the other c_{xyz} and that there are only two possible choices of c_{rstu} that divide μ. If we have one of these, say c_{rstu}, then the other is prohibited from being a divisor of μ, since μ is oriented consistently. One easily checks that c_{ijk}^{α}c_{i′j′k′}^{β}c_{rstu}^{γ} = c_{ijk}^{α′}c_{i′j′k′}^{β′}c_{rstu}^{γ′} if and only if α = α′, β = β′ and γ = γ′. This does this case.

b) Suppose that μ is divisible by only one of the c_{xyz}, say c_{ijk}. Then there are only three of the c_{rstu} that have the same orientation as c_{ijk} and any pair of these are not oriented the same. Thus μ has the form c_{ijk}^{α}c_{rstu}^{β} for a unique c_{rstu} (unless

β = 0). Clearly c_{ijk}^{α}c_{rstu}^{β} = c_{ijk}^{α′}c_{rstu}^{β′} if and only if α = α′ and β = β′. This does this case.

c) If μ is divisible by none of the c_{xyz}, then only one of the c_{rstu} can be a divisor since no two c_{rstu} are oriented the same. Thus μ = c_{rstu}^{α} for some unique c_{rstu}. This does this case and completes the proof of Lemma 2.3. □

This completes the proof of Theorem 2.2. □

Remark 2.4. We note that one can check that it is not possible to find a partial order on the elements of Π_{5} such that two such elements are comparable if and only if they are oriented the same. In fact one cannot even find such a partial order on the subset {c_{ijk}|1 ≤ i < j < k ≤ 5}.

Remark 2.5. Let X = (x_{ij}) be an n × n matrix of indeterminates and N = {1, …, n}. For each pair ((i_{1}, …, i_{k}), (j_{1}, …, j_{k})) of sequences of elements of N we let

(i_{1}, …, i_{k})|(j_{1}, …, j_{k})

denote the minor of X corresponding to the submatrix with rows (i_{1}, …, i_{k}) and with columns (j_{1}, …, j_{k}). One of the most important examples of an ASL is given by the polynomial ring R[x_{ij}] with a certain natural ordering of the symbols (i_{1}, …, i_{k})|(j_{1}, …, j_{k}) [BV, DEP]. Let I be the ideal of R[x_{ij}] generated by the elements x_{ii}, i = 1, …, n. Then R[a_{ij}] is isomorphic to R[x_{ij}]/I and the action of H(n) on R[a_{ij}] gives the following action on the minors:

t_{rs}^{u}((i_{1}, …, i_{k})|(j_{1}, …, j_{k})) =
(i_{1}, …, i_{k})|(j_{1}, …, j_{k}) − ux_{sr}(i_{1}, …, i_{k})^{(rs)}|(j_{1}, …, j_{k})

+ ux_{rs}(i_{1}, …, i_{k})|(j_{1}, …, j_{k})^{(rs)} − u^{2}x_{rs}x_{sr}(i_{1}, …, i_{k})^{(rs)}|(j_{1}, …, j_{k})^{(rs)}

where (i_{1}, …, i_{k})^{(rs)} is the sequence (i_{1}, …, i_{k}) with s replaced by r etc. This gives an action of the groups B_{n}, H(n) and Ĥ(n) on a quotient of an ASL for all values of n.

Remark 2.5. We note that by [Hu1, section 5], if we anti-symmetrise by letting a_{ij} = −a_{ji} for all 1 ≤ i < j ≤ n, then we obtain an action of B_{n}, H(n) etc. on the ring R[a_{ij}|1 ≤ i < j ≤ n whose kernel is the same as that of the original action.

§3. Properties of the ASL’s

If P is a poset and a, b ∈ P, then we write a ≪ b if a < b and there is no c ∈ P with a < c < b. A poset P is called wonderful if in the poset P ∪ {∞, −∞} obtained from P by adjoining a greatest and a least element we have: for y_{1}, y_{2} < z and x ≪ y_{1}, y_{2} there is an element y ≤ z such that y_{1}, y_{2} ≪ y.

Lemma 3.1. The posets Π_{n}, n = 3, 4 are both wonderful posets.

Proof. For n=3 it is easy to see that any of the partial orders in Figure 1 give a wonderful poset. It is less easy to see this for n=4. The amount of checking needed for the proof can however be reduced if one notes the following:

1) One need really only concern oneself with the partial order on the elements in the top four levels of Figure 3.

2) The symmetric group Σ_{4} acts on the c_{U} by permutation of subscripts in such a way that the elements of the alternating group A_{n} fix the top three level sets shown in Figure 3 and any element not in A_{4} interchanges levels 1 and 3, while keeping level 2 fixed.

3) The action of Σ_{4} described in 2) acts transitively on the set of c_{ijk} and A_{4} acts transitively on the set of c_{rstu}.

4) The stabiliser of one of the c_{ijk} is a cyclic subgroup of A_{4} of order 3 and so the action of A_{4} described in 2) acts 2-transitively on the set of c_{ijk} in each of the first and third levels.

We now check all the cases. Suppose that x = c_{34} in the definition of a wonderful poset. Then by the 2-transitivity mentioned above we can (without loss of generality) take y_{1} = c_{234}, y_{2} = c_{124}. Now the only z with y_{1}, y_{2} < z are z = c_{1234}, c_{123}, c_{134}, ∞. Thus if we take y = c_{1234}, then the condition y_{1}, y_{2} ≪ y is satisfied. (We note that if x = c_{ij} < c_{34}, then there is nothing to check.)

Next suppose that x = c_{ijk} is in the third level. Then we may take x = c_{234} since A_{4} acts transitively on the third level cycles. We note that x is stabilised by the 3-cycle (123). Now the subgroup < (123) > of A_{4} acts 2-transitively on the y_{i} with x ≪ y_{i} and so we may take y_{1} = c_{1234}, y_{2} = c_{1342}. Now the only z with y_{1}, y_{2} < z are z = c_{134}, ∞. Thus if we take y = c_{134}, then the condition y_{1}, y_{2} ≪ y is satisfied.

Next suppose that x = c_{rstu} is in the second level. Then we may take x = c_{1234} since A_{4} acts transitively on the second level cycles. Now we can only take y_{1} = c_{123}, y_{2} = c_{134} (or vice-versa). Clearly the only z with y_{1}, y_{2} < z is z = ∞. Thus if we take y = ∞, then the condition y_{1}, y_{2} ≪ y is satisfied.

If x is in the top level, then there is nothing to check. This completes the proof of Lemma 3.1. □

In fact one can check (using the same ideas) that the posets Π_{n}, n = 3, 4 are both distributive lattices. From Lemma 3.1 we immediately have

Corollary 3.2. If R is a Cohen-Macaulay ring, then the rings Y_{n}, n = 3, 4 are both Cohen-Macaulay.

Proof. This follows from Lemma 3.1 and Theorem 5.14 of [BV]. □

The rank of Ξ ⊂ Π is equal to the length of a maximal chain in Ξ.

Corollary 3.3. Let R be a noetherian ring. Then, using any of the partial orders in Figure 1, the ring Y_{3} has dimension 4+dim(R). The ring Y_{4} has dimension 9+dim(R).

Proof. This follows from Theorem 5.10 of [BV] since the rank of Π_{3} is clearly 4 and the rank of Π_{4} is 9. □

Basic facts about Cohen-Macaulay rings, including the vanishing of certain Ext’s can be found in [M].

Let A be an arbitrary ASL on Π. An ideal in Π is a subset Ω ⊂ Π such that ξ ∈ Ω and v ≤ ξ implies v ∈ Ω. Let Ω ⊂ Π be an ideal in Π. By [BV, Proposition 5.1 (b)] A/AΩ is an ASL on Π \ Ω. If we take A = Y_{n}, n = 3, 4 and Ω_{n} = {c_{ij}|1 ≤ i, j ≤ n}, then Z_{n} = Y_{n}/Y_{n}Ω_{n} is an ASL on Ψ_{n} = Π_{n} \ Ω_{n}. From Figure 3 we see that Z_{n} is a symmetric ASL: it is also an ASL with respect to the reverse order on Π \ Ω. Now Ψ_{n} has rank 1 if n=3 and rank 3 if n=4.

We now consider the Poincaré series of these algebras. For convenience let R be a field. For fixed 1 ≤ i ≤ n, A(n; i) is the R-subalgebra of R_{(n)}[a_{ij}] generated by all monomials in Y_{n} that contain the index i (in their set of subscripts). Note that A(n; i) has a natural grading by degree:

A(n; i) = ⨁_{k=0}^{∞} A_{k}(n; i).

Here each A_{k}(n; i) is a vector space of dimension p_{k}(n; i). In [Hu3] we calculated the Poincaré series of A(n; i):

P (n; i) = ∑_{k=0}^{∞} p_{k}t^{k},

where p_{k} = p_{k}(n; i).

We obtained the following results:

P (3; 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 (4; 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} + …

This allows us to calculate the Poincaré series V_{n}(t) of Y_{n}, n = 3, 4.

Proposition 3.4.

V_{3}(t) = \frac{−t^{8} + t^{7} + 2t^{6} − t^{5} − 2t^{4} − t^{3} + 3t^{2}}{(1 + t + t^{2})(1 + t)^{2}(1 − t)^{4}}

= 3t^{2} + 2t^{3} + 6t^{4} + 6t^{5} + 12t^{6} + 12t^{7} + 21t^{8} + 22t^{9} + 33t^{10} + 36t^{11} + 50t^{12}

+ 54t^{13} + 72t^{14} + 78t^{15} + 99t^{16} + 108t^{17} + 133t^{19} + 144t^{20} + …

V_{4}(t) = \frac{−4t^{20} + 8t^{19} + 6t^{18} − 12t^{17} − 11t^{16} − 2t^{15} + 25t^{14} + 10t^{13} − 12t^{12}}{(1 + t + t^{2})^{2}(1 + t)^{5}(1 + t^{2})(1 − t)^{9}}

+ \frac{−14t^{11} − 15t^{10} + 14t^{9} + 17t^{8} + 4t^{7} − 16t^{6} − 12t^{5} + 10t^{4} + 4t^{3} − 6t^{2}}{(1 + t + t^{2})^{2}(1 + t)^{5}(1 + t^{2})(1 − t)^{9}}

= 6t^{2} + 8t^{3} + 24t^{4} + 48t^{5} + 106t^{6} + 192t^{7} + 369t^{8} + 624t^{9} + 1080t^{10}

+ 1728t^{11} + 2787t^{12} + 4248t^{13} + 6498t^{14} + 9528t^{15} + 13962t^{16}

+ 19824t^{17} + 28066t^{18} + 38760t^{19} + …

Proof. Apart from the above we also have P (2, 1) = \frac{t^{2}}{1−t^{2}} . Let μ be a monomial in Y_{n}, n = 3, 4 and let I be its set of indices. Then we have the following mutually exclusive possibilities: either a) 1 ∈ I or b) 1 ∉ I and 2 ∈ I or c) 1, 2 ∉ I and 3 ∈ I. This then gives V_{3}(t) = P (3; 1) + P (2, 1) and V_{4}(t) = P (4, 1) + P (3; 1) + P (2, 1). The result follows from the expressions for P (n; 1), n = 2, 3, 4 given above. □

In [Hu3] we also found the invariant subring of Y_{n}, n = 3, 4 for the action of B_{n}: let T_{i} be the n×n-matrix (transvection) differing from the n×n identity matrix I_{n} only in row i, where the ij entry is a_{ij} if i ≠ j and 1 if i = j. Let the characteristic polynomial of T_{1}T_{2} … T_{n} be ∑_{k=0}^{n} d_{nk}t^{k}. Let M_{n} = T_{1} + T_{2} + ⋯ + T_{n} − nI_{n}.

Proposition 3.5. The invariant subring for the action of B_{3} on Y_{3} is the subring

R[d_{31}, d_{32}] = R[a_{12}a_{23}a_{31} + a_{13}a_{32}a_{21}, a_{12}a_{23}a_{31} + a_{12}a_{21} + a_{13}a_{31} + a_{23}a_{32}].

The invariant subring for the action of B_{4} on Y_{4} is the subring R[d_{41}, d_{42}, d_{43}].

For all n ≥ 2 the invariant subring for the action of H_{n} on Y_{n} is the subring R[det(M_{n})].

Proof. The first two statements follow from [Hu3, Theorem 1.1]. The last statement is just [Hu2, Theorem 6]. □

Acknowledgement

The author would like to thank S. DiPippo and W. E. Lang for suggesting a connection with Hodge algebras.

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