A conjecture of Mallows and Sloane with the universal denominator of Hilbert series

: A conjecture of Mallows and Sloane conveys the dominance of Hilbert series for ﬁ nding basic invariants of ﬁ nite linear groups if the Hilbert series of the invariant ring is of a certain explicit canonical form. However, the conjecture does not hold in general by a well-known counterexample of Stanley. In this article, we give a constraint on lower bounds for the degrees of homogeneous system of parameters of rings of invariants of ﬁ nite linear groups depending on the universal denominator of Hilbert series de ﬁ ned by Derksen. We consider the conjecture with the universal denominator on abelian groups and provide some criteria guaranteeing the existence of homogeneous system of parameters of certain degrees. In this case, Stanley ’ s counterexample could be avoided, and the homogeneous system of parameters is optimal.


Introduction
Let V be an m-dimensional vector space over an algebraically closed field , ( ) S V* the symmetric algebra of the dual space V *, and x x ,…, m 1 a basis for V *.Then ( ) S V* is isomorphic to the polynomial ring [ ] denotes a finite linear group.The action of G on V induces an action on V * which extends to an action by algebra automorphisms on [ ] V .The ring of invariants The equivalent conditions for C-M property can be found in Benson [ A theorem of Hilbert, embellished by Serre, implies that ( ) , is a rational function of t (see, e.g., Atiyah and Macdonald [4,Theorem 11.1]).Serre's theorem has a counterpart in the non-commutative case, but Anick [5,6] found a very famous counterexample by showing how finitely presented Hopf algebras may be constructed which have irrational Hilbert series.Hilbert series is rational also for relatively free algebras of associative PIalgebras (algebras with polynomial identities) (see Belov [7,8]).More information about more general structure such as T-spaces can be found in the study by Belov [9,10]. Let denote the Hilbert series of [ ] V G .Then the Hilbert series can be written down imme- diately from the degrees of the basic invariants by (1): where deg( ).On the other hand, in the non-modular case, we have Molien's formula (see Molien [11]) to calculate the Hilbert series of the invariant ring, From this, we can make good guesses about the degrees of the basic invariants.Mallows and Sloane [12, Section 2] made the following conjecture: , such that (1) holds.
However, Conjecture 1.1 does not hold in general.Stanley [13] (or see Sloane [14, p. 101]) gives a counterexample to the conjecture.Consider the abelian group G of order 8 generated by the matrices Molien's formula yields .
Indeed, there is no h.s.o.p. with degree sequence ( ) 2, 2, 2 .An interpretation is given by a result of Kemper [15,Theorem 2] by the fact that the Krull dimension dim . The optimal h.s.o.p. is = canonical form (2), and there has been a lot of interest in determining the degrees ( for which there exists a regular sequence in the invariant ring with deg( Note that in a Cohen-Macaulay ring, an h.s.o.p. is a maximal regular sequence.Dixmier [17] made a conjecture concerning this question for binary forms, and it has attracted some attention [18][19][20].And a few authors have taken up this question for the natural action of the symmetric group on polynomial algebras [21][22][23][24].Derksen [25] introduced the universal denominators of Hilbert series.He gave formulas for the universal denominator of rings of invariants and showed that the universal denominator is actually equal to Dixmier's formula [17] for the denominator of the Hilbert series of invariants of binary forms.Moreover, there are many interesting properties in itself. By modular invariant theory, we understand the study of invariants of finite groups over fields whose characteristic may divide the order of the group.It is mainstream for the last few decades in the study of invariant theory of finite groups.Wehlau [26] and the references there in consider the generators of the ring of invariants for decomposable representations of C p with the field of characteristic p.For a Galois field q , the general linear group GL ( ) n, q is a finite group.The invariant rings of many interesting subgroups of GL ( ) n, q have been determined: the general and special linear groups (this goes back to L. Dickson, see, for instance, Smith [27,Section 8.1]), the groups B n and U n of upper triangular matrices and unipotent upper triangular matrices in GL ( ) n, q (see Smith [27,Proposition 5.5.6]), the finite symplectic groups (this goes back to D. Carlisle and P. Kropholler, see Benson [2,Section 8.3]), and the finite unitary groups [28].
Our goal is to determine the subgroups of the general linear group GL ( ) n, over an algebraically closed field such that Conjecture 1.1 holds with the restriction that the denominator of ( 2) is the universal denominator.
In Section 2, we give a constraint on the degrees of h.s.o.p. of rings of invariants determined by the universal denominator of Hilbert series (see Derksen and Kemper [16,Sect. 3.5.3]for other useful constraints).It provides another explanation for Stanley's counterexample and generalizes a result for symmetric groups (see Galetto et al. [24,Proposition 2.2]).We also give another counterexample of the conjecture with the universal denominator.With the constraint, the desired h.s.o.p. is the optimal h.s.o.p.Kemper [15] gave an algorithm to calculate the optimal h.s.o.p..In this article, we only use combinatorial methods.
Derksen [25] showed that the universal denominator of rings of invariants of finite groups is the least common multiple of { ( . The Shephard-Todd-Chevalley theorem (see Neusel and Smith [1, Theorem 7.1.4])says that [ ] V G is a polynomial algebra if and only if G is generated by pseudoreflections.An easy application of the Hilbert-Serre theorem implies that the degrees of the optimal h.s.o.p. of any group generated by pseudoreflections are determined by the universal denominator.If we consider the conjecture with the universal denominator, all pseudoreflection groups satisfies the conjecture, and Stanley's counterexample can be avoided.
In Section 3, we follow Stanley's counterexample and restrict our attention to abelian groups for which there exists an h.s.o.p. of certain degrees determined by the universal denominator.Since the universal denominator behaves nicely with respect to tensor products, the study of abelian groups can be reduced to the study of cyclic groups.We prove that the conjecture holds for minimal faithful representations of n (see Definition 3.3) and give a criterion.On this basis, we prove that if is an abelian group, the conjecture holds.We also consider a general three-dimensional representation of cyclic groups n with primitive nth roots of unity in the diagonal at the end of this article.
2 A constraint on the degrees of h.s.o.p.
We begin this section by recalling a criterion for h.s.o.p. in the invariant ring.
be homogeneous invariants of positive degree with . Then the following statements are equivalent: and ¯is an algebraic closure of ; A conjecture of Mallows and Sloane with universal denominator  3 The universal denominator considered in Derksen [25] is defined for a finitely generated r -graded module M over an r -graded ring R of finite type.We write the brief description of the universal denominator in the following.Definition 2.2.For a finitely generated multi-graded module M over a multi-graded ring R of finite type, the universal denominator of the Hilbert series of M is the least common multiple of the denominators of the Hilbert series of every multi-graded submodule N of M , denoted by udenom( ) M t , .
In this article, we consider the polynomial ring with the standard grading, and G is also an -graded ring.Derksen gave two descriptions of the universal denominator of a finitely generated multi-graded R-module M and gave the formula for the universal denominator of finite groups.We write it here with where m d is the dimension of the support of is the greatest common divisor of all where ζ is a dth root of unity.In conclusion, the universal denominator of Motivated by the theorem, we get a constraint on the degrees of h.s.o.p. in the invariant ring.

G d
Since [ ] V G is an integral domain and a finitely generated -algebra , by the dimension theorem, , and is given as follows: (b) In this example, we show a special case of Proposition 2.6 when G is the symmetric group S m .Suppose

Galetto et al. [24, Proposition 2.2] consider the degrees of elements of an homogeneous system of parameters in [ ]
V S m and shows that In Section 4 of the study [25], Derksen shows that suppose that V is a vector space on which the finite group G acts linearly over a field of characteristic 0, then the universal denominator of ( [ ] ) H V t , G is expressed as follows: where denotes the dth cyclotomic polynomial, whose zeros are exactly the primitive dth roots of unity.Derksen [25,Example 4.6] also considers the action of the symmetric group S m on = V m , where is an algebraically closed field of characteristic 0. Suppose ∈ g S m has cycle structure ( ) and are the lengths of the cycles of the permutation g.The permutation action of g on V can be viewed as a block matrix and the ith block of the matrix is a permutation representation of the cyclic group ki .Since is algebraically closed, then the block matrix can be in Jordan canonical form with all k i th roots of unity in the diagonal.If ζ is a primitive dth root of unity, then ζ is an eigenvalue of g if and only if d divides k i .Then we have , and max dim .
A conjecture of Mallows and Sloane with universal denominator  5 Since divided by d, then we have , where e j is the jth elementary symmetric function of degree j.The Hilbert series of [ ] V S m is expressed as follows: It is clear that the elementary symmetric polynomials form an optional hsop with degree sequence ( ) m 1, 2, …, .Proposition 2.6 implies that there is a lower bound for the degree of an h.s.o.p., and it is easy to compute the degree by Theorem 2.5( ) c if the degree of the universal denominator meets the degree of an h.s.o.p..However, it is not easy in general.Indeed, the universal denominator of the Hilbert series of the invariant ring is not even of the form ( )( ) ( ) . We illustrate the phenomenon through the following examples.
, where G is the group of order 12 generated by where ζ is a primitive 12th root of unity.Then where ζ is a primitive 12th root of unity.The Hilbert series with universal denominator is 2 is spanned by x x 2 3 as a vector space, and [ ] are regular sequences.For graded Cohen-Macaulay rings, an h.s.o.p. is equivalent to a maximal regular sequence (see [16,Proposition 2.6.3]).Suppose there is an invariant h of degree 12 such that h, x x is an h.s.o.p..It implies that h, x 3 , x 3 is an h.s.o.p., which contradicts the algebraic independence.Therefore, there does not exist an h.s.o.p. with a degree sequence ( ) 12, 3, 2 .
In the next section, we intend to consider the abelian groups for which there exists an h.s.o.p. with the degree sequence determined by the universal denominator of the Hilbert series of the invariant ring.

Mallows and Sloane's conjecture with the universal denominator
Mallows and Sloane [12] said that the conjecture is true for finite unitary groups generated by reflections.
In fact, they are exactly natural examples whose optimal h.s.o.p. are determined by the universal denominators since their invariant rings are all polynomial algebras, see Example 2.7 for symmetric groups.
be the polynomial ring, graded in such a way that deg( . If M is a finitely generated R-module, then by Hilbert's syzygy theorem (see Eisenbud [29,Theorem 1.13]), there exists a resolution where F i is a finitely generated free R-module for all i, and the sequence is exact.The Hilbert series of M can be computed by the exact sequence with Hilbert-Serre Theorem (see [4,Theorem 11.1]).
where ( ) [ ] ∈ f t t .Then the denominator of ( ) . By Definition 2.2, udenom( ) ( )( ) ( ) . In the rest of this section, we consider the conjecture with the universal denominator for abelian groups.The study of abelian groups can be reduced to cyclic groups by the structure theorem of finitely generated abelian groups and the following lemma (or see [25,Lemma 1.12] for finitely generated multi-graded modules).
be finite groups, where V V , 1 2 are finite-dimensional vector spaces over a field where char does not divide the orders of the groups.Then Proof.Note that and Then the result follows from Theorem 2.5 (a) and (c).□ If G is a finite abelian subgroup of ( ) m GL , , where char does not divide the order of G, then G is diagonalizable.So we will always assume that every element of G is a diagonal matrix [13,30].
Throughout this section, n denotes the cyclic group of order n for ∈ n *. ζ denotes a primitive nth root of unity.[ ] m denotes the number set { } m 1, …, for ∈ m *. ( ) o σ denotes the order of the element σ in a group G. denotes an algebraically closed field.

The universal denominator of cyclic groups
Let ρ: . We consider a commutative diagram: where ( ) j n is a copy of n , and φ is a monomorphism of groups given by ( ) . Then η is also a monomorphism of groups given by , η is a faithful representation of n .By Theorem 2.5(c), where ka j is a non-negative integer such that Then we have Therefore, by Theorem 2.5 (c), where for some degree sequence ( ) for some degree sequence ( ) . Therefore, ( ) ( ) , which is obviously a factor of n. □ Clearly, In conclusion, we have s n udenom , is of the form 1 1 for for 0 1.

G d d d d s s m 1
Note that we sometimes denote γ s , τ s by γ ρ s , , τ ρ s , , respectively, since they depend on some given representation ρ.

The conjecture for minimal faithful representations of cyclic groups
In this subsection, we focus on representations of n for which there is no primitive nth root of unity in the diagonal.In particular, we define the minimal faithful representations of n , which are inspired by pseudoreflections.They are the most important family of representations of n which satisfy the conjecture, and we prove this in Theorem 3.6.Moreover, we provide a criterion for the representations.
Let ∈ n .n has prime factorization , o ζ a j .These are pseudo-reflections.
a m 1 is a polynomial algebra, and udenom . Now, we consider the minimal faithful representations, which weaken the condition that ( ) o ζ a j 's are pairwise coprime.
, we call ρ an m-dimensional minimal faithful representation of n or minimal for short.
Remark 3.4.For the representation ρ in Definition 3.3, the faithfulness is clear since the order of diag( . Moreover, we should explain that the minimality of ρ is up to faithfulness.It means that if we remove any element, for example, , then ρ is not faithful any more.Therefore, we call ρ an m-dimensional minimal faithful representation of n .Remark 3.5.Let n have prime factorization for ⩽ ≠ ′ ⩽ j j m 1 .Note that the pseudo-reflection representations of n are minimal.Two-dimensional faithful representations of n without primitive n-th roots of unity in the diagonal are minimal.for ⩽ ≠ ′ ⩽ j j m 1 . When j = 1, without loss of generality, let we have the following equivalent statements: . The first equivalence follows from Theorem 3.6.The second, third, and fourth follow from equation (4).□ Here, let , …, a a j j r 1 .Lemma 3.8.With the same conditions as in Theorem 3.6, let J be a non-empty subset of [ ] m .If udenom We proceed by induction on θ.When = θ 0, it is trivial by Theorem 3.6.

Suppose there exists
This contradicts the fact that i 0 is the minimal positive integer such that ≡ ia ia , if there is at least one primitive nth root of unity in the diagonal, it is equivalent to the faithful representation d , and the Hilbert series (sometimes called the Poincaré series) of M is the formal Laurent series

Conjecture 1 . 1 .
Whenever ( [ ] ) H V t ,G can be put in the form of (2) by cancelling common factors and/or by multiplying numerator and denominator by the same polynomial, then a matching set of basic invariants can be found, consisting of m algebraically independent homogeneous invariants f f ,

ζ σ Theorem 2 . 5 .
[25, Theorem 1.10, Theorem 4.2]  Let G be a finite group and be an algebraically closed field whose characteristic does not divide the order of G.
the ring of invariants [ ] V n is generated by the monomials ( )

1 2 3 .
If there exist h.s.o.p. of certain degrees for η k and η l , respectively, then so does it for η in the following cases: By Noether's normalization lemma, we know that there always exists an h.s.o.p. in [ ] V G (see Benson [2, Theorem 2.2.7]), and it is by no means unique, neither are its degrees finitely generated graded connected -subalgebra of [ ] V (see, for example, Neusel and Smith [1, The by Krull's principal ideal theorem (see, for example, Eisenbud [29, Theorem 10.1]) or Proposition 2.1 and [15, Proposition 5], Example 2.7.(a) Recall Stanley's counterexample mentioned in Section 1.By Definition 2.4 and Theorem 2.5(c), we have , and the degrees of the universal denominator exactly meet those of the optimal h.s.o.p. mentioned in Section 1.
d By Proposition 2.6, for any h.s.o.p. of the invariant ring, it contains at least one homogeneous polynomial of degree divisible by 4. It implies that there are no h.s.o.p. with the degree sequence ( ) 2, 2, 2 We give another counterexample of the conjecture of Mallows and Sloane with the universal denomi- nator.Consider the invariant ring [ ] V G, where G is the group of order 12 generated by By Proposition 2.6, there exists an h.s.o.p. with the degree sequence determined by the universal denominator of the Hilbert series of the invariant ring if and only if 1The conjecture for representations of n with primitive nth roots of unity in the diagonalIn this subsection, we confirm the conjecture with the universal denominator is true for two-dimensional faithful representations of n , and Stanley's counterexample could be avoided in this case.Moreover, we find the universal denominator is always of the form ( might be less than n.So it is convenient to classify the representations into two cases depending on whether there is a primitive nth root of unity in the diagonal.
d , 1 2 .Assume both ζ a 1 and ζ a 2 are not primitive nth root of unity.It is a minimal representation, and it is a special case in Theorem 3.6 for = m 2 with the h.s.o.p.
For any m-dimensional faithful representation of n with primitive n-th roots of unity in the diagonal, it is equivalent to the form it is not clear whether there exists an h.s.o.p. in certain degrees.We present the following result.