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

Yang Zhang; Jizhu Nan; Yongsheng Ma

doi:10.1515/math-2024-0001

Article Open Access

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

Yang Zhang , Jizhu Nan and Yongsheng Ma

Published/Copyright: March 18, 2024

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$Open Mathematics$

From the journal Open Mathematics Volume 22 Issue 1

Abstract

A conjecture of Mallows and Sloane conveys the dominance of Hilbert series for finding basic invariants of finite 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 finite linear groups depending on the universal denominator of Hilbert series defined 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.

Keywords: Stanley’s counterexample; Molien’s theorem; optimal homogeneous system of parameters

MSC 2010: 13D40; 13A50

1 Introduction

Let V be an m -dimensional vector space over an algebraically closed field F , S ( V * ) the symmetric algebra of the dual space V * , and x 1 , … , x m a basis for V * . Then S ( V * ) is isomorphic to the polynomial ring F [ V ] = F [ x 1 , … , x m ] . G ⊂ GL ( V ) 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 F [ V ] . The ring of invariants

F [ V ] G = { f ∈ F [ V ] ∣ σ ⋅ f = f , ∀ σ ∈ G }

is a finitely generated graded connected F -subalgebra of F [ V ] (see, for example, Neusel and Smith [1, Theorem 2.1.4]).

A set f 1 , … , f m ∈ F [ V ] G of homogeneous polynomials is called a homogeneous system of parameters if f 1 , … , f m are algebraically independent and F [ V ] G is a finitely generated module over F [ f 1 , … , f m ] . We denote it by h.s.o.p. By Noether’s normalization lemma, we know that there always exists an h.s.o.p. in F [ V ] G (see Benson [2, Theorem 2.2.7]), and it is by no means unique, neither are its degrees d 1 = deg ( f 1 ) , … , d m = deg ( f m ) .

If char F does not divide the group order ∣ G ∣ , which is called the non-modular case, then F [ V ] G is Cohen-Macaulay (see Hochster and Eagon [3, Proposition 13]). It implies that for any given h.s.o.p. f 1 , … , f m ∈ F [ V ] G , there exist homogeneous free module generators g 1 , … , g l such that there is a decomposition

(1) F [ V ] G = ⊕ i = 1 l F [ f 1 , … , f m ] g i .

The equivalent conditions for C-M property can be found in Benson [2, Theorem 4.3.5].

Let A = ⊕ n = 0 ∞ A n be a finitely generated graded commutative F -algebra. Let M = ⊕ d = 0 ∞ M d be a finitely generated graded A -module such that A n M d ⊂ M n + d . Then it follows that dim F M d < ∞ . The Hilbert function H ( M , ⋅ ) : N → N of M is defined by H ( M , d ) = dim F M d , and the Hilbert series (sometimes called the Poincaré series) of M is the formal Laurent series

H ( M , t ) = ∑ d = 0 ∞ H ( M , d ) t d = ∑ d = 0 ∞ dim F ( M d ) t d .

A theorem of Hilbert, embellished by Serre, implies that H ( M , t ) 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 PI-algebras (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 H ( F [ V ] G , t ) denote the Hilbert series of F [ V ] G . Then the Hilbert series can be written down immediately from the degrees of the basic invariants by (1):

(2) H ( F [ V ] G , t ) = ∑ k ⩾ 0 dim F ( F [ V ] k G ) t k = 1 + t D 2 + ⋯ + t D l ( 1 − t d 1 ) ( 1 − t d 2 ) ⋯ ( 1 − t d m ) ,

where deg( f j ) = d j , deg( g i ) = D i ( D 1 = 0 ).

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,

H ( F [ V ] G , t ) = 1 ∣ G ∣ ∑ σ ∈ G 1 det ( 1 − σ t ) .

From this, we can make good guesses about the degrees of the basic invariants. Mallows and Sloane [12, Section 2] made the following conjecture:

Conjecture 1.1

Whenever H ( F [ V ] G , t ) 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 1 , … , f m of degrees d 1 , … , d m , and l homogeneous invariants g 1 , … , g l of degrees D 1 , … , D l , 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

1 1 i and − 1 − 1 1 ∈ GL 3 ( C )

Molien’s formula yields

H ( C [ V ] G , t ) = 1 ( 1 − t 2 ) 3 .

But C [ V ] G = C [ x 1 2 , x 2 2 , x 3 4 ] ( 1 ⊕ x 1 x 2 ) , which cannot be expressed in the form C [ f 1 , f 2 , f 3 ] , f i ∈ C [ V ] 2 G . 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 ( C [ V ] G ∕ ( C [ V ] 2 G ) ) = 1 . The optimal h.s.o.p. is f 1 = x 1 2 , f 2 = x 2 2 , f 3 = x 3 4 (see Derksen and Kemper [16, Example 3.5.6(b)]).

In view of the existence of h.s.o.p. of certain degrees in the invariant rings, the conjecture is equivalent to considering whether there exists an h.s.o.p. of degrees d 1 , … , d m if the Hilbert series is of a certain explicit canonical form (2), and there has been a lot of interest in determining the degrees ( d 1 , d 2 , … , d m ) for which there exists a regular sequence in the invariant ring with deg ( f i ) = d i . 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 [18–20]. And a few authors have taken up this question for the natural action of the symmetric group on polynomial algebras [21–24]. 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 F q , the general linear group GL ( n , F q ) is a finite group. The invariant rings of many interesting subgroups of GL ( n , F 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 , F 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 , F ) over an algebraically closed field F 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 { det ( i d − σ t ) ∣ σ ∈ G } . The Shephard-Todd-Chevalley theorem (see Neusel and Smith [1, Theorem 7.1.4]) says that F [ 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 Z n (see Definition 3.3) and give a criterion. On this basis, we prove that if G ≤ GL ( 2 , F ) is an abelian group, the conjecture holds. We also consider a general three-dimensional representation of cyclic groups Z n with primitive n th 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.

Proposition 2.1

[16, Proposition 3.5.1] Let f 1 , … , f m ∈ F [ V ] + G be homogeneous invariants of positive degree with m = dim F ( V ) . Then the following statements are equivalent:

f 1 , … , f m form an h.s.o.p.;
V F ¯ ( f 1 , … , f m ) = 0 , where V F ¯ ( f 1 , … , f m ) is defined as { v ∈ F ¯ ⊗ F V ∣ f i ( v ) = 0 f o r i = 1 , … , m } and F ¯ is an algebraic closure of F ;
the Krull dimension dim ( F [ V ] ∕ ( f 1 , … , f m ) ) is zero;
dim ( F [ V ] ∕ ( f 1 , … , f i ) ) = m − i for i = 1 , … , m .

The universal denominator considered in Derksen [25] is defined for a finitely generated Z r -graded module M over an N 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 F [ V ] = F [ x 1 , … , x m ] with the standard grading, and F [ V ] = ⊕ d ∈ N F [ V ] d . Naturally, the ring of invariants F [ V ] G = ⊕ d ∈ N F [ V ] d G is also an N -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 M = F [ V ] G equipped with the N -grading for finite groups.

Note that for d ∈ N , ( 1 − t d ) = ∏ i ∣ d ϕ i ( t ) . ϕ i ( t ) ∈ Z [ t ] denotes the i th cyclotomic polynomial, whose zeros are exactly the primitive i th roots of unity.

Definition 2.3

Let I [ d ] be the ideal of F [ V ] G generated by all F [ V ] e G for which d dose not divide e .

Definition 2.4

Let ζ ∈ F * and let σ ∈ G . Then we define

V ζ σ = { v ∈ V ∣ σ ⋅ v = ζ ⋅ v } .

Theorem 2.5

[25, Theorem 1.10, Theorem 4.2] Let G be a finite group and F be an algebraically closed field whose characteristic does not divide the order of G.

udenom ( F [ V ] G , t ) = ∏ d ∈ N ϕ d ( t ) m d
where m d is the dimension of the support of ( F [ V ] G ) [ d ] ≔ F [ V ] G ∕ I [ d ] F [ V ] G .
The universal denominator udenom ( F [ V ] G , t ) is the greatest common divisor of all
( 1 − t d 1 ) ⋯ ( 1 − t d s )
for which there exist homogeneous f 1 , … , f s ∈ F [ V ] G of degrees d 1 , … , d s , respectively, such that F [ V ] G is a finitely generated F [ f 1 , … , f s ] module.
The dimension of the zero set of the ideal I [ d ] , which is equal to m d is
max { dim V ζ σ ∣ σ ∈ G } ,
where ζ is a dth root of unity. In conclusion, the universal denominator of H ( F [ V ] G , t ) is
lcm { det ( i d − σ t ) ∣ σ ∈ G } .

Motivated by the theorem, we get a constraint on the degrees of h.s.o.p. in the invariant ring.

Proposition 2.6

Let f 1 , … , f m be an h.s.o.p. of F [ V ] G with degree sequence ( d 1 , … , d m ) . We define β d ≔ # { 1 ⩽ j ⩽ m ∣ d d i v i d e s d j } for d ∈ N . m d is defined as in Theorem 2.5. Then

β d ⩾ m d .

Proof

Let T be a set consisting of f j such that d ∤ d j . Then ( T ) ⊂ I [ d ] ⊂ F [ V ] G , and hence,

dim ( F [ V ] G ∕ ( T ) ) ⩾ dim ( F [ V ] G ∕ I [ d ] ) .

Since F [ V ] G ⊆ F [ V ] is integral, by Krull’s principal ideal theorem (see, for example, Eisenbud [29, Theorem 10.1]) or Proposition 2.1 and [15, Proposition 5],

dim ( F [ V ] G ∕ ( T ) ) = dim ( F [ V ] ∕ ( T ) ) = dim ( F [ V ] ) − # T = β d .

Since F [ V ] G is an integral domain and a finitely generated F -algebra , by the dimension theorem,

m d = dim Supp ( ( F [ V ] G ) [ d ] ) = dim V Spec ( F [ V ] G ) ( I [ d ] ) = dim ( F [ V ] G ∕ ( I [ d ] ) ) .

In conclusion, we have β d ⩾ m d .□

Example 2.7

(a) Recall Stanley’s counterexample mentioned in Section 1. By Definition 2.4 and Theorem 2.5(c), we have m 4 = 1 , m 2 = 3 , m 1 = 3 , and m d = 0 for other d ∈ N . The universal denominator of F [ V ] Z 2 × Z 4 is given as follows:

∏ d ∈ N ϕ d ( t ) m d = ϕ 4 ( t ) ϕ 2 ( t ) 3 ϕ 1 ( t ) 3 = ( 1 − t 2 ) 2 ( 1 − t 4 ) .

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 ) , and the degrees of the universal denominator exactly meet those of the optimal h.s.o.p. mentioned in Section 1.

(b) In this example, we show a special case of Proposition 2.6 when G is the symmetric group S m . Suppose f 1 , f 2 , … , f m form an homogeneous system of parameters of F [ V ] G , where deg ( f j ) = d j and dim F ( V ) = m . For d ∈ N , define β d the number of j for 1 ⩽ j ⩽ m satisfying that d divides d j , i.e.,

β d ≔ # { 1 ⩽ j ⩽ m ∣ d divides d j } for d ∈ N .

Galetto et al. [24, Proposition 2.2] consider the degrees of elements of an homogeneous system of parameters in F [ V ] S m and shows that

β d ⩾ ⌊ m ∕ d ⌋ for 1 ⩽ d ⩽ m .

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 F of characteristic 0, then the universal denominator of H ( F [ V ] G , t ) is expressed as follows:

udenom ( F [ V ] G , t ) = ∏ d ∈ N ϕ d ( t ) m d = lcm { det ( I − t g ) ∣ g ∈ G } ,

where

m d = max { dim V ζ g ∣ g ∈ G } , V ζ g = { v ∈ V ∣ g ⋅ v = ζ ⋅ v } .

ϕ d ( t ) ∈ Z [ t ] denotes the d th cyclotomic polynomial, whose zeros are exactly the primitive d th roots of unity.

Derksen [25, Example 4.6] also considers the action of the symmetric group S m on V = F m , where F is an algebraically closed field of characteristic 0. Suppose g ∈ S m has cycle structure ( k 1 , k 2 , … , k r ) with k 1 ⩾ k 2 ⩾ ⋯ ⩾ k r ⩾ 1 and k 1 + k 2 + ⋯ + k r = m , i.e., k 1 , k 2 , … , k r 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 i th block of the matrix is a permutation representation of the cyclic group Z k i . Since F 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 d th root of unity, then ζ is an eigenvalue of g if and only if d divides k i . Then we have

dim V ζ g = # { i ∣ d divides k i } , and max g ∈ S m { dim V ζ g } = ⌊ m ∕ d ⌋ .

The maximum is reached if g has ⌊ m ∕ d ⌋ d-cycles. The result β d ⩾ m d is obtained for G = S m .

Since

1 − t n = ∏ d ∣ n ϕ d ( t ) ,

and there are ⌊ m d ⌋ integers { d , 2 d , … , ⌊ m d ⌋ d } ⊂ { 1 , 2 , … , m } divided by d , then we have

udenom ( F [ V ] S m , t ) = ∏ d = 0 ∞ ϕ d ( t ) ⌊ m ∕ d ⌋ = ( 1 − t ) ( 1 − t 2 ) ⋯ ( 1 − t m ) .

It is well known that F [ V ] S m = F [ e 1 , e 2 , … , e m ] , where e j is the j th elementary symmetric function of degree j . The Hilbert series of F [ V ] S m is expressed as follows:

H ( F [ V ] S m , t ) = 1 ( 1 − t ) ( 1 − t 2 ) ⋯ ( 1 − t m ) .

It is clear that the elementary symmetric polynomials form an optional hsop with degree sequence ( 1 , 2 , … , m ) .

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 ( 1 − t d 1 ) ( 1 − t d 2 ) ⋯ ( 1 − t d m ) . We illustrate the phenomenon through the following examples.

Example 2.8

( a ) Consider the invariant ring C [ V ] G , where G is the group of order 12 generated by

ζ 2 ζ 3 ∈ GL 2 ( C ) ,

where ζ is a primitive 12th root of unity. Then

udenom ( C [ V ] G , t ) = ϕ 6 ( t ) ϕ 4 ( t ) ϕ 3 ( t ) ϕ 2 ( t ) ϕ 1 ( t ) 2 = ( 1 − t 6 ) ( 1 − t ) ( 1 + t 2 ) .

By Proposition 2.6, f 1 = x 1 6 , f 2 = x 2 4 is an optimal h.s.o.p. of C [ V ] G , and ( 1 − t 6 ) ( 1 − t 4 ) = udenom ( C [ V ] G , t ) ϕ 2 ( t ) .

( b ) We give another counterexample of the conjecture of Mallows and Sloane with the universal denominator. Consider the invariant ring C [ V ] G , where G is the group of order 12 generated by

ζ ζ 2 ζ 10 ∈ GL 3 ( C ) ,

where ζ is a primitive 12th root of unity. The Hilbert series with universal denominator is

H ( C [ V ] G , t ) = f ( t ) udenom ( C [ V ] G , t ) = t 12 + t 8 + t 7 + 2 t 6 + 1 ( 1 − t 12 ) ( 1 − t 3 ) ( 1 − t 2 ) .

C [ V ] 2 G is spanned by x 2 x 3 as a vector space, and C [ V ] 3 G is spanned by x 1 2 x 3 . It is easy to see that, if f 1 = g 1 h 1 with deg ( g 1 ) ⩾ 1 and deg ( h 1 ) ⩾ 1 , then f 1 , f 2 , … , f m is a regular sequence if and only if both g 1 , f 2 , … , f m and h 1 , f 2 , … , f m 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 1 2 x 3 , x 2 x 3 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.

3 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.

Indeed, let R = F [ x 1 , x 2 , … , x m ] be the polynomial ring, graded in such a way that deg ( x i ) = d i > 0 . If M is a finitely generated R -module, then by Hilbert’s syzygy theorem (see Eisenbud [29, Theorem 1.13]), there exists a resolution

0 → F m → F m − 1 → ⋯ → F 1 → F 0 → M → 0 ,

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]).

H ( M , t ) = ∑ i = 0 m ( − 1 ) i H ( F i , t ) = f ( t ) ( 1 − t d 1 ) ( 1 − t d 2 ) ⋯ ( 1 − t d m )

where f ( t ) ∈ Z [ t ] . Then the denominator of H ( M , t ) divides ( 1 − t d 1 ) ( 1 − t d 2 ) ⋯ ( 1 − t d m ) . By Definition 2.2, udenom ( R , t ) = ( 1 − t d 1 ) ( 1 − t d 2 ) ⋯ ( 1 − t d m ) .

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).

Lemma 3.1

Let G 1 ≤ GL ( V 1 ) and G 2 ≤ GL ( V 2 ) be finite groups, where V 1 , V 2 are finite-dimensional vector spaces over a field F where char F does not divide the orders of the groups. Then

udenom ( F [ V 1 ⊕ V 2 ] G 1 × G 2 , t ) = udenom ( F [ V 1 ] G 1 , t ) ⋅ udenom ( F [ V 2 ] G 2 , t ) .

Proof

Note that

H ( F [ V 1 ⊕ V 2 ] G 1 × G 2 , t ) = H ( F [ V 1 ] G 1 ⊗ F [ V 2 ] G 2 , t ) = H ( F [ V 1 ] G 1 , t ) ⋅ H ( F [ V 2 ] G 2 , t )

and

max { dim ( V 1 ⊕ V 2 ) ζ σ ∣ σ ∈ G 1 × G 2 } = max { dim ( V 1 ) ζ σ ∣ σ ∈ G 1 } + max { dim ( V 2 ) ζ σ ∣ σ ∈ G 2 } .

Then the result follows from Theorem 2.5 (a) and (c).□

If G is a finite abelian subgroup of GL ( m , F ) , where char F 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, Z n denotes the cyclic group of order n for n ∈ N * . ζ denotes a primitive n th root of unity. [ m ] denotes the number set { 1 , … , m } for m ∈ N * . o ( σ ) denotes the order of the element σ in a group G . F denotes an algebraically closed field.

3.1 The universal denominator of cyclic groups

Let ρ : Z n ↪ GL ( m , F ) , 1 ¯ ↦ diag ( ζ a 1 , … , ζ a m ) be a faithful representation of Z n , where 0 ⩽ a j ⩽ n − 1 for j ∈ [ m ] . Then the ring of invariants F [ V ] Z n is generated by the monomials

x I = x 1 i 1 ⋯ x m i m suchthat ∑ j = 1 m i j a j ≡ 0 ( mod n ) ,

with i 1 + ⋯ + i m ⩽ n . We consider a commutative diagram:

where Z n ( j ) is a copy of Z n , and φ is a monomorphism of groups given by ( a 1 ¯ , … , a m ¯ ) ↦ diag ( ζ a 1 , … , ζ a m ) . Then η is also a monomorphism of groups given by 1 ¯ ↦ ( a 1 ¯ , … , a m ¯ ) , i.e., η is a faithful representation of Z n . By Theorem 2.5(c),

(3) udenom ( F [ V ] Z n , t ) = lcm { ( 1 − ζ k a 1 ¯ t ) ⋯ ( 1 − ζ k a m ¯ t ) ∣ k ∈ [ n ] } ,

where k a j ¯ is a non-negative integer such that k a j ¯ ≡ k a j ¯ (mod n ).

Let s ∈ N . For k ¯ ∈ Z n = { 1 ¯ , 2 ¯ … , n ¯ = 0 ¯ } , we define

V s k ≔ V ζ s k ¯ = { v ∈ V ∣ k ¯ ⋅ v = ζ s v } ,

Γ s k ≔ { j ∈ [ m ] ∣ k a j ¯ = s } .

Then we have

# Γ s k = dim V s k .

Therefore, by Theorem 2.5 (c),

udenom ( F [ V ] Z n , t ) = ∏ d ∈ N ϕ d ( t ) m d = ∏ s = 0 n − 1 ( 1 − ζ s t ) γ s ,

where

γ s ≔ max { dim V s k ∣ k ∈ [ n ] } = max { # Γ s k ∣ k ∈ [ n ] } ,

in other words, γ s is the maximum number of s occurring in ( k a 1 ¯ , … , k a m ¯ ) for 1 ⩽ k ⩽ n . Clearly,

γ s = m d if d = o ( ζ s ) = n g c d ( n , s ) .

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 β d = m d for some h.s.o.p.. It is necessary to consider the conditions such that udenom ( F [ V ] G , t ) is of the form ( 1 − t d 1 ) ⋯ ( 1 − t d m ) for some degree sequence ( d 1 , … , d m ) .

Lemma 3.2

Let ρ : Z n ↪ GL ( m , F ) , 1 ¯ ↦ diag ( ζ a 1 , … , ζ a m ) be a faithful representation of Z n , 0 ⩽ a j ⩽ n − 1 for j ∈ [ m ] . Suppose udenom ( F [ V ] Z n , t ) is the form ∏ j = 1 m ( 1 − t d j ) where d j ∈ N * and char F ∤ d j for j ∈ [ m ] . Then the d j ’s are factors of n.

Proof

Suppose that udenom ( F [ V ] Z n , t ) = ∏ i = 1 m ( 1 − t d i ) ∈ F [ t ] for some degree sequence ( d 1 , … , d m ) . Then ζ d i − 1 is a zero, where ζ d i is a primitive d i th root of unity. But the zeros of udenom ( F [ V ] Z n , t ) are of the form ( ζ k a j ¯ ) − 1 for k , j ∈ [ m ] . Then ζ d i = ζ k a j ¯ for some k , j ∈ [ m ] . Therefore, d i = o ( ζ d i ) = o ( ζ k a j ¯ ) = n g c d ( n , k a j ¯ ) , which is obviously a factor of n .□

Let ( d 1 , … , d m ) ∈ ( N * ) m be a degree sequence such that d j divides n for j ∈ [ m ] . Then

( 1 − t d 1 ) ⋯ ( 1 − t d m ) = ∏ d ∈ N ϕ d ( t ) u d = ∏ s = 0 n − 1 ( 1 − ζ s t ) τ s ,

where

u d ≔ # { j ∈ [ m ] ∣ d divides d j } , τ s ≔ # { j ∈ [ m ] ∣ o ( ζ s ) divides d j } .

Clearly,

τ s = u d if d = o ( ζ s ) = n g c d ( n , s ) .

In conclusion, we have

udenom ( F [ V ] G , t ) is of the form ( 1 − t d 1 ) ⋯ ( 1 − t d m ) ⇔ m d = u d for d ∈ N ⇔ γ s = τ s for 0 ⩽ s ⩽ n − 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 .

3.2 The conjecture for minimal faithful representations of cyclic groups

In this subsection, we focus on representations of Z n for which there is no primitive n th root of unity in the diagonal. In particular, we define the minimal faithful representations of Z n , which are inspired by pseudo-reflections. They are the most important family of representations of Z n which satisfy the conjecture, and we prove this in Theorem 3.6. Moreover, we provide a criterion for the representations.

Let n ∈ N . n has prime factorization p 1 r 1 ⋯ p λ r λ ( λ > 1 ) . Let ρ : Z n → GL ( m , F ) , 1 ¯ ↦ diag ( ζ a 1 , … , ζ a m ) be a representation of Z n such that [ o ( ζ a 1 ) , … , o ( ζ a m ) ] = n and o ( ζ a j ) = n g c d ( n , a j ) < n . This yields m ⩾ 2 . Let V = F m . If o ( ζ a j ) ’s are pairwise coprime, then 2 ⩽ m ⩽ λ and ( V , Z n ) = ⊕ j = 1 m ( F , Z o ( ζ a j ) ) . These are pseudo-reflections. F [ V ] Z n = F [ x 1 o ( ζ a 1 ) , … , x m o ( ζ a m ) ] is a polynomial algebra, and udenom ( F [ V ] ρ ( Z n ) , t ) = ( 1 − t o ( ζ a 1 ) ) ⋯ ( 1 − t o ( ζ a m ) ) .

Now, we consider the minimal faithful representations, which weaken the condition that o ( ζ a j ) ’s are pairwise coprime.

Definition 3.3

Let ρ : Z n → GL ( m , F ) , 1 ¯ ↦ diag ( ζ a 1 , … , ζ a m ) , be a representation of Z n such that [ o ( ζ a 1 ) , … , o ( ζ a m ) ] = n and o ( ζ a j ) = n g c d ( n , a j ) < n . If [ o ( ζ a 1 ) , … , o ( ζ a j ) ^ , … , o ( ζ a m ) ] < n for any 1 ⩽ j ⩽ m , we call ρ an m -dimensional minimal faithful representation of Z n or m i n i m a l for short.

Remark 3.4

For the representation ρ in Definition 3.3, the faithfulness is clear since the order of diag ( ζ a 1 , … , ζ a m ) is n by [ o ( ζ a 1 ) , … , o ( ζ a m ) ] = n . Moreover, we should explain that the minimality of ρ is up to faithfulness. It means that if we remove any element, for example, ζ a j , j ∈ { 1 , 2 , … , m } , in the diagonal of diag ( ζ a 1 , … , ζ a m ) such that [ o ( ζ a 1 ) , … , o ( ζ a j ) ^ , … , o ( ζ a m ) ] < n , i.e., the order of diag ( ζ a 1 , … , ζ a j ^ , … , ζ a m ) < n , then ρ is not faithful any more. Therefore, we call ρ an m -dimensional minimal faithful representation of Z n .

Remark 3.5

Let n have prime factorization p 1 r 1 ⋯ p λ r λ . If ρ is minimal, then λ ⩾ 2 and 2 ⩽ m ⩽ λ . In Definition 3.3, it is easy to see that the condition [ o ( ζ a 1 ) , … , o ( ζ a j ) ^ , … , o ( ζ a m ) ] < n for any j ∈ [ m ] is equivalent to o ( ζ a j ) ∤ o ( ζ a j ′ ) for 1 ⩽ j ≠ j ′ ⩽ m .

Note that the pseudo-reflection representations of Z n are minimal. Two-dimensional faithful representations of Z n without primitive n -th roots of unity in the diagonal are minimal.

Theorem 3.6

Let n ∈ N and n have prime factorization p 1 r 1 ⋯ p λ r λ ( λ ⩾ 2 ) . Let ρ : Z n ↪ GL ( m , F ) , 1 ¯ ↦ diag ( ζ a 1 , … , ζ a m ) be an m-dimensional minimal faithful representation of Z n . If udenom ( F [ V ] Z n , t ) is of the form ∏ j = 1 m ( 1 − t d j ) and char F ∤ d j for j ∈ [ m ] , then there exists an h.s.o.p. with degree sequence ( d 1 , … , d m ) .

Proof

Suppose that udenom ( F [ V ] G , t ) = ( 1 − t d 1 ) ⋯ ( 1 − t d m ) . For any j ∈ [ m ] , since ( 1 − ζ a j t ) divides udenom ( F [ V ] G , t ) , we have that ζ a j is a d i th root of unity for some i ∈ [ m ] , i.e. o ( ζ a j ) ∣ d i for some i ∈ [ m ] . On the other hand, for any k ∈ [ o ( G ) ] , where o ( G ) denotes the order of G , since ( 1 − ζ k a 1 t ) ⋯ ( 1 − ζ k a m t ) divides ( 1 − t o ( ζ a 1 ) ) ⋯ ( 1 − t o ( ζ a m ) ) , by equation (3), udenom ( F [ V ] G , t ) divides ( 1 − t o ( ζ a 1 ) ) ⋯ ( 1 − t o ( ζ a m ) ) , and hence, ( 1 − t d i ) divides ( 1 − t o ( ζ a 1 ) ) ⋯ ( 1 − t o ( ζ a m ) ) for all i ∈ [ m ] . Therefore, for any i ∈ [ m ] , ζ d i is an o ( ζ a j ) th root of unity for some j ∈ [ m ] , i.e. d i ∣ o ( ζ a j ) for some j ∈ [ m ] .

We will show that o ( ζ a j ) = d j by induction on j . According to Remark 3.5, the minimal condition is equivalent to o ( ζ a j ) ∤ o ( ζ a j ′ ) for 1 ⩽ j ≠ j ′ ⩽ m . When j = 1, without loss of generality, let o ( ζ a 1 ) ∣ d 1 . Since o ( ζ a 1 ) ∤ o ( ζ a j ) for j ∈ [ m ] \ { 1 } , then o ( ζ a 1 ) = d 1 . Suppose o ( ζ a j ) = d j for j ∈ [ m − 1 ] . Then o ( ζ a m ) ∣ d m , which yields d m = o ( ζ a m ) . Therefore, udenom ( F [ V ] Z n , t ) = ∏ j = 1 m ( 1 − t o ( ζ a j ) ) and x 1 o ( ζ a 1 ) , … , x m o ( ζ a m ) is the desired h.s.o.p..□

The theorem gives a sufficient condition of the existence of an h.s.o.p. with the degree sequence ( d 1 , … , d m ) determined by the universal denominator for a minimal faithful representation of Z n . We present a criterion for minimal faithful representations of Z n such that udenom ( F [ V ] G , t ) is of the form ( 1 − t d 1 ) ⋯ ( 1 − t d m ) . It is necessary to do this because there is an exception in Example 2.8(a).

We start with m = 2 .

Proposition 3.7

Let ρ : Z n ↪ GL ( 2 , F ) , 1 ¯ ↦ diag ( ζ a 1 , ζ a 2 ) be a faithful representation of Z n such that o ( ζ a j ) < n and char F ∤ o ( ζ a j ) for j = 1 , 2 . Let d be the greatest common divisor of o ( ζ a 1 ) and o ( ζ a 2 ) . We define i 0 ∈ [ n ] as the minimal positive integer i such that i a 1 ≡ i a 2 (mod n ). Then i 0 = n d . Moreover, there is an h.s.o.p. of a certain degree determined by udenom ( F [ V ] Z n , t ) if and only if d ∣ ( a 1 − a 2 ) .

Proof

Let ⟨ ζ a j ⟩ denotes the subgroup of Z n generated by ζ a j . Since

(4) ⟨ ζ a 1 ⟩ ∩ ⟨ ζ a 2 ⟩ = ζ n o ( ζ a 1 ) ∩ ζ n o ( ζ a 2 ) = ζ lcm n o ( ζ a 1 ) , n o ( ζ a 2 ) = ζ n g c d ( o ( ζ a 1 ) , o ( ζ a 2 ) ) = ζ n d ,

we have the following equivalent statements:

udenom ( F [ V ] Z n , t ) is of the form ( 1 − t d 1 ) ( 1 − t d 2 ) ⇔ udenom ( F [ V ] Z n , t ) = ( 1 − t o ( ζ a 1 ) ) ( 1 − t o ( ζ a 2 ) ) ⇔ γ ρ , s = 2 , n d ∣ s ; 0 , n o ( ζ a 1 ) ∤ s or n o ( ζ a 2 ) ∤ s ; 1 , otherwise. ⇔ i 0 = n d ⇔ n d a 1 ≡ n d a 2 ( mod n ) ⇔ d ∣ ( a 1 − a 2 ) .

The first equivalence follows from Theorem 3.6. The second, third, and fourth follow from equation (4).□

Here, let J = { j 1 , … , j r } ⊂ [ m ] , n J = l c m ( o ( ζ a j 1 ) , … , o ( ζ a j r ) ) . If ρ : Z n ↪ GL ( m , F ) , 1 ¯ ↦ diag ( ζ a 1 , … , ζ a m ) is minimal, so is ρ J : Z n J ↪ GL ( r , F ) , 1 ¯ ↦ diag ( ζ a j 1 , … , ζ a j r ) .

Lemma 3.8

With the same conditions as in Theorem 3.6, let J be a non-empty subset of [ m ] . If udenom ( F [ V ] ρ ( Z n ) , t ) is the form ∏ j = 1 m ( 1 − t d j ) and char F ∤ d j for j ∈ [ m ] , then udenom ( F [ V ] ρ J ( Z n J ) , t ) = ∏ j ∈ J ( 1 − t o ( ζ a j ) ) .

Proof

Let J ′ = [ m ] \ J , θ = ∣ J ′ ∣ . We proceed by induction on θ . When θ = 0 , it is trivial by Theorem 3.6. For θ = m − r − 1 > 0 , ∣ J ∣ = r + 1 . Let J = { j 1 , … , j r + 1 } ⊂ [ m ] . ρ J : Z n J ↪ GL ( r + 1 , F ) is defined by 1 ¯ ↦ diag ( ζ a j 1 , … , ζ a j r + 1 ) . Suppose that udenom ( F [ V ] ρ J ( Z n J ) , t ) = ( 1 − t o ( ζ a j 1 ) ) ⋯ ( 1 − t o ( ζ a j r + 1 ) ) , i.e., γ ρ J , s = τ ρ J , s for s ∈ [ n ] . Then we consider θ = m − r . We have ∣ J ∣ = r , ρ J ′ : Z n J ↪ GL ( r , F ) , 1 ¯ ↦ diag ( ζ a j 1 , … , ζ a j λ ^ , … , ζ a j r + 1 ) for some λ ∈ [ r + 1 ] . Assume that udenom ( F [ V ] ρ J ′ ( Z n J ) , t ) = ∏ s = 1 n ( 1 − ζ s t ) ν s . Then

ν s = γ ρ J , s − 1 = τ ρ J , s − 1 , if n o ( ζ a j λ ) ∣ s ; γ ρ J , s = τ ρ J , s , otherwise .

Hence, udenom ( F [ V ] ρ J ′ ( Z n ) , t ) = LCM f ρ ( t ) 1 − t o ( ζ a j λ ) = ( 1 − t o ( ζ a j 1 ) ) ⋯ ( 1 − t o ( ζ a j λ ) ) ^ ⋯ ( 1 − t o ( ζ a j r + 1 ) ) . This completes the proof.□

Theorem 3.9

Let n ∈ N and n have prime factorization p 1 r 1 ⋯ p λ r λ ( λ ⩾ 2 ) . Let ρ : Z n ↪ GL ( m , F ) , and 1 ¯ ↦ diag ( ζ a 1 , … , ζ a m ) be an m-dimensional minimal faithful representation of Z n . For { j 1 , … , j r } ⊂ [ m ] , let d j 1 , … , j r = g c d ( o ( ζ a j 1 ) , … , o ( ζ a j r ) ) . Then the following statements are equivalent:

udenom ( F [ V ] Z n , t ) is of the form ∏ j = 1 m ( 1 − t d j ) and char F ∤ d j for j ∈ [ m ] ;
for any non-empty subset { j 1 , … , j r } ⊂ [ m ] , q a j 1 ≡ ⋯ ≡ q a j r ( mod n ) , where q is a positive integer less than or equal to n and n J d j 1 , … , j r ∣ q , n J = lcm ( o ( ζ a j 1 ) , … , o ( ζ a j r ) ) ;
d j 1 , j 2 ∣ ( a j 1 − a j 2 ) for any pair j 1 , j 2 ∈ [ m ] .

Proof

( a ) ⇔ ( b ) Suppose udenom ( F [ V ] Z n , t ) = ∏ j = 1 m ( 1 − t d j ) . For any non-empty subset J = { j 1 , … , j r } ∈ [ m ] , by Theorem 3.6 and Lemma 3.8, we have udenom ( F [ V ] ρ J ( Z n J ) , t ) = ∏ i = 1 r ( 1 − t o ( ζ a j i ) ) . Since < ζ a j 1 > ∩ ⋯ ∩ < ζ a j r > = < ζ n J d j 1 , … , j r > , we have γ ρ J , s = τ ρ J , s = r for 1 ⩽ s ⩽ n and n J d j 1 , … , j r ∣ s . Therefore, s a j 1 ≡ ⋯ ≡ s a j r ( mod n ) , where s is a positive integer less than or equal to n and n J d j 1 , … , j r ∣ s .

Conversely, suppose ∏ j = 1 m ( 1 − t o ( ζ a j ) ) = ∏ s = 0 n − 1 ( 1 − ζ s t ) τ s . Since ⟨ ζ a j 1 ⟩ ∩ ⋯ ∩ ⟨ ζ a j r ⟩ = ⟨ ζ n J d j 1 , … , j r ⟩ , condition ( b ) implies γ ρ , s = # { j ∈ [ m ] ∣ o ( ζ s ) divides o ( ζ a j ) } , which is τ s . Therefore, udenom ( F [ V ] ρ ( Z n ) , t ) = ∏ j = 1 m ( 1 − t o ( ζ a j ) ) .

( b ) ⇔ ( c ) By Proposition 3.7, for any pair j i 1 , j i 2 ∈ { j 1 , … , j r } , we have d j i 1 , j i 2 ∣ ( a j i 1 − a j i 2 ) ⇔ q a j i 1 ≡ q a j i 2 (mod n ), where q is a positive integer less than or equal to n and n J ′ d j i 1 , j i 2 ∣ q , n J ′ = l c m ( o ( ζ a j i 1 ) , o ( ζ a j i 2 ) ) . Since d j 1 , … , j r ∣ d j i 1 , j i 2 and n J ′ ∣ n J , we have n J ′ d j i 1 , j i 2 ∣ n J d j 1 , … , j r , and hence, q a j i 1 ≡ q a j i 2 (mod n ) for q ⩽ n and n J d j 1 , … , j r ∣ q . Therefore, q a j 1 ≡ ⋯ ≡ q a j r ( mod n ) for q ⩽ n and n J d j 1 , … , j r ∣ q .□

Example 3.10

Consider a minimal faithful representation ρ : Z 60 ↪ GL ( 2 , F ) , 1 ¯ ↦ diag ( ζ 3 , ζ 8 ) , where ζ is a primitive 60th root of unity. We list ( 3 i ¯ , 8 i ¯ ) for 1 ⩽ i ⩽ 60 :

( 3 , 8 ) , ( 6 , 16 ) , ( 9 , 24 ) , ( 12 , 32 ) , ( 15 , 40 ) , ( 18 , 48 ) , ( 21 , 56 ) , ( 24 , 4 ) , ( 27 , 12 ) , ( 30 , 20 ) , ( 33 , 28 ) , ( 36 , 36 ) ̲ , ( 39 , 44 ) , ( 42 , 52 ) , ( 45 , 0 ) , ( 48 , 8 ) , ( 51 , 16 ) , ( 54 , 24 ) , ( 57 , 32 ) , ( 0 , 40 ) , ( 3 , 48 ) , ( 6 , 56 ) , ( 9 , 4 ) , ( 12 , 12 ) ̲ , ( 15 , 20 ) , ( 18 , 28 ) , ( 21 , 36 ) , ( 24 , 44 ) , ( 27 , 52 ) , ( 30 , 0 ) , ( 33 , 8 ) , ( 36 , 16 ) , ( 39 , 24 ) , ( 42 , 32 ) , ( 45 , 40 ) , ( 48 , 48 ) ̲ , ( 51 , 56 ) , ( 54 , 4 ) , ( 57 , 12 ) , ( 0 , 20 ) , ( 3 , 28 ) , ( 6 , 36 ) , ( 9 , 44 ) , ( 12 , 52 ) , ( 15 , 0 ) , ( 18 , 8 ) , ( 21 , 16 ) , ( 24 , 24 ) ̲ , ( 27 , 32 ) , ( 30 , 40 ) , ( 33 , 48 ) , ( 36 , 56 ) , ( 39 , 4 ) , ( 42 , 12 ) , ( 45 , 20 ) , ( 48 , 28 ) , ( 51 , 36 ) , ( 54 , 44 ) , ( 57 , 52 ) , ( 0 , 0 ) ̲ .

We have i 0 = 12 , d = g c d ( o ( ζ 3 ) , o ( ζ 8 ) ) = g c d ( 20 , 15 ) = 5 and clearly i 0 = 60 d and 5 ∣ ( 3 − 8 ) . Moreover, by counting γ ρ , s , the maximum number of s occurring in ( 3 i ¯ , 8 i ¯ ) for 1 ⩽ i ⩽ 60 , we have udenom ( F [ V ] Z 60 , t ) = ∏ s = 0 59 ( 1 − ζ s ) γ ρ , s = ( 1 − t 20 ) ( 1 − t 15 ) .

3.3 The conjecture for representations of Z n with primitive n th roots of unity in the diagonal

In this subsection, we confirm the conjecture with the universal denominator is true for two-dimensional faithful representations of Z n , and Stanley’s counterexample could be avoided in this case. Moreover, we find the universal denominator is always of the form ( 1 − t d 1 ) ( 1 − t d 2 ) for m = 2 unlike the exception in Example 2.8(a), which is an example of minimal representations. We also consider a general case for m = 3 .

Note that, by formula (2), we have:

H ( F [ V ] G , t ) = 1 + t D 2 + ⋯ + t D k ( 1 − t d 1 ) ⋯ ( 1 − t d m ) = ( 1 + t D 2 + ⋯ + t D k ) ( 1 + t d 1 + ( t d 1 ) 2 + ⋯ ) ⋯ ( 1 + t d m + ( t d m ) 2 + ⋯ ) = ( 1 + α 1 t d 1 + ⋯ + α m t d m ) + ⋯ .

Obviously α j ⩾ 1 . Then we have

(5) dim F F [ V ] d j G ⩾ 1 .

Proposition 3.11

Let ρ : Z n ↪ GL ( 2 , F ) , 1 ¯ ↦ diag ( ζ a 1 , ζ a 2 ) be a faithful representation of Z n , where 0 ⩽ a j ⩽ n − 1 for j = 1 , 2 . Suppose the denominator of formula (2) is udenom ( F [ V ] Z n , t ) , and d j ∈ N * , char F ∤ d j for j = 1 , 2 . Then there exists an h.s.o.p. for F [ V ] Z n with the degree sequence ( d 1 , d 2 ) .

Proof

Let n be a positive integer. n has a unique factorization p 1 r 1 p 2 r 2 ⋯ p l r l , where p 1 , p 2 , … , p l are different prime numbers. If l = 1 , then there is at least one primitive n th root of unity in { ζ a 1 , ζ a 2 } . If l ⩾ 2 , then both o ( ζ a 1 ) and o ( ζ a 2 ) might be less than n . So it is convenient to classify the representations into two cases depending on whether there is a primitive n th root of unity in the diagonal. Suppose that udenom ( F [ V ] Z n , t ) = ( 1 − t d 1 ) ( 1 − t d 2 ) for some degree sequence ( d 1 , d 2 ) .

If there is at least one primitive n th root of unity in the diagonal, then ( 1 − t n ) divides udenom ( F [ V ] Z n , t ) . By Lemma 3.2, without loss of generality, let d 1 = n . By equation (5), there are invariants of degree d 1 , d 2 . Since invariant generators of cyclic groups are monomials, let f 1 = x 1 n + x 2 n , f 2 = x 1 k x 2 d 2 − k , then f 1 , f 2 are invariants and V ( f 1 , f 2 ) = 0 . By Proposition 2.1, f 1 , f 2 is the desired h.s.o.p. with the degree sequence ( d 1 , d 2 ) .

Assume both ζ a 1 and ζ a 2 are not primitive n th 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. x 1 o ( ζ a 1 ) , x 2 o ( ζ a 2 ) .□

Remark 3.12

The condition that H ( F [ V ] Z n , t ) is the form (2) is important for the first case in the proposition. For example, let

Z n = ζ 0 0 ζ − 1 ≤ GL ( 2 , C ) .

where n ⩾ 3 is odd. udenom ( F [ V ] Z n , t ) = ( 1 − t n ) ( 1 − t ) . However, there is no invariant of degree 1. Actually,

H ( F [ V ] Z n , t ) = ∑ i = 0 n − 1 ( − t ) i ( 1 − t n ) ( 1 − t ) .

There are negative integer coefficients in the numerator.

Remark 3.13

Consider Stanley’s counterexample given in Section 1.

H ( C [ V 1 ] Z 2 , t ) = f 1 ( t ) udenom ( C [ V 1 ] Z 2 , t ) = 1 + t 2 ( 1 − t 2 ) 2 .

H ( C [ V 2 ] Z 4 , t ) = f 2 ( t ) udenom ( C [ V 2 ] Z 4 , t ) = 1 ( 1 − t 4 ) .

By Lemma 3.1, we have

H ( C [ V 1 ⊕ V 2 ] Z 2 × Z 4 , t ) = f ( t ) udenom ( C [ V 1 ⊕ V 2 ] Z 2 × Z 4 , t ) = 1 + t 2 ( 1 − t 2 ) 2 ( 1 − t 4 ) .

x 1 2 , x 2 2 , x 3 4 is the desired h.s.o.p. with degree sequence ( 2 , 2 , 4 ) .

By the proposition, for finding the cyclic group Z n ≤ GL ( 2 , F ) which satisfies the conjecture with universal denominator, it is necessary (not sufficient) to determine whose universal denominator is the form ( 1 − t d 1 ) ( 1 − t d 2 ) .

Lemma 3.14

Let ρ : Z n ↪ GL ( 2 , F ) , 1 ¯ ↦ diag ( ζ a 1 , ζ a 2 ) be a faithful representation of Z n , where 0 ⩽ a j ⩽ n − 1 and char F ∤ o ( ζ a j ) for j = 1 , 2 . We define i 0 ∈ [ n ] as the minimal positive integer i such that i a 1 ≡ i a 2 ( mod n ) . Then i 0 ∣ n . Moreover, if i ⩽ n , then i a 1 ≡ i a 2 ( mod n ) if and only if i = q i 0 for 1 ⩽ q ⩽ n i 0 .

Proof

Proof by contradiction. Suppose n = q i 0 + r with 0 < r < i 0 . We have ( q i 0 + r ) a 1 ≡ 0 (mod n ) and ( q i 0 + r ) a 2 ≡ 0 (mod n ). Since i 0 a 1 ≡ i 0 a 2 (mod n ), we have r a 1 ≡ r a 2 (mod n ). This contradicts the fact that i 0 is the minimal integer such that i a 1 ≡ i a 2 ( mod n ) . Thus, r = 0 .

Suppose there exists i ∈ [ n ] such that i a 1 ≡ i a 2 (mod n ) and i ≠ q i 0 . Let q ′ ∈ { 1 , 2 , … , n i 0 } satisfy 0 < i − q ′ i 0 < i 0 . Since i 0 a 1 ≡ i 0 a 2 (mod n ), we have q ′ i 0 a 1 ≡ q ′ i 0 a 2 (mod n ). Combining with i a 1 ≡ i a 2 (mod n ), we have ( i − q ′ i 0 ) a 1 ≡ ( i − q ′ i 0 ) a 2 (mod n ). This contradicts the fact that i 0 is the minimal positive integer such that i a 1 ≡ i a 2 (mod n ). The converse is obvious.□

For Z n ≤ GL ( 2 , F ) , if there is at least one primitive n th root of unity in the diagonal, it is equivalent to the faithful representation

η k : Z n ↪ Z n × Z n , 1 ¯ ↦ ( 1 ¯ , k ¯ ) for some 0 ⩽ k ⩽ n − 1 .

Note that if k = 0, then F [ x 1 , x 2 ] Z n ≅ F [ x 1 ] Z n ⊗ F [ x 2 ] and the group acts trivially on F [ x 2 ] . It is equivalent to Z n ≤ GL ( 1 , F ) . Here, we always omit this case.

Proposition 3.15

Let η k : Z n ↪ Z n × Z n , 1 ¯ ↦ ( 1 ¯ , k ¯ ) be a representation of Z n , where 1 ⩽ k ⩽ n − 1 and char F ∤ n . Then i 0 = n g c d ( n , k − 1 ) (defined in Lemma 3.14). Moreover, udenom ( F [ V ] Z n , t ) = ( 1 − t n ) ( 1 − t g c d ( n , k − 1 ) ) .

Proof

Since i 0 ≡ k i 0 (mod n ), then i 0 ( k − 1 ) ≡ 0 (mod n ), and hence, i 0 = o ( k − 1 ¯ ) = n g c d ( n , k − 1 ) . By Lemma 3.14, we have

γ η k , s = 2 , if i 0 ∣ s or s = q i 0 for 1 ⩽ q ⩽ n i 0 ; 1 , if i 0 ∤ s or s ≠ q i 0 for 1 ⩽ q ⩽ n i 0 .

Then udenom ( F [ V ] Z n , t ) = ( 1 − t n ) ( 1 − t n i 0 ) = ( 1 − t n ) ( 1 − t g c d ( n , k − 1 ) ) .□

Example 3.16

Consider η 5 : Z 8 ↪ Z 8 × Z 8 , 1 ¯ ↦ ( 1 ¯ , 5 ¯ ) . We list ( i ¯ , 5 i ¯ ) for 1 ⩽ i ⩽ 8 :

( 1 , 5 ) , ( 2 , 2 ) ̲ , ( 3 , 7 ) , ( 4 , 4 ) ̲ , ( 5 , 1 ) , ( 6 , 6 ) ̲ , ( 7 , 3 ) , ( 0 , 0 ) ̲ .

Then we have i 0 = 2 and

γ η 5 , 2 = γ η 5 , 4 = γ η 5 , 6 = γ η 5 , 0 = 2 ; γ η 5 , 1 = γ η 5 , 3 = γ η 5 , 5 = γ η 5 , 7 = 1 .

Therefore, udenom ( F [ V ] Z 8 , t ) = ∏ s = 0 7 ( 1 − ζ s t ) γ η 5 , s = ( 1 − t 8 ) ( 1 − t 4 ) .

For any m -dimensional faithful representation of Z n with primitive n -th roots of unity in the diagonal, it is equivalent to the form

η : Z n ↪ ∏ j = 1 m Z n ( j ) , 1 ¯ ↦ ( 1 ¯ , k 1 ¯ , … , k m − 1 ¯ )

for some 0 ⩽ k i ⩽ n − 1 , 1 ⩽ i ⩽ m − 1 . In fact, we always assume that k i ⩾ 1 .

For m ⩾ 3 , it is not clear whether there exists an h.s.o.p. in certain degrees. We present the following result.

Proposition 3.17

Let η : Z n ↪ Z n × Z n × Z n , 1 ¯ ↦ ( 1 ¯ , k ¯ , l ¯ ) be a three-dimensional representation of Z n , where 1 ⩽ k , l ⩽ n − 1 . Suppose udenom ( F [ V ] η ( Z n ) , t ) is the form ( 1 − t d 1 ) ( 1 − t d 2 ) ( 1 − t d 3 ) , where char F ∤ d 1 , d 2 , d 3 and d 1 ⩾ d 2 ⩾ d 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:

n ∣ d 1 , o ( k ¯ ) ∣ d 2 and o ( l ¯ ) ∣ d 3 ;
n ∣ d 1 , g c d ( n , k − 1 ) ∣ d 2 and o ( l ¯ ) ∣ d 3 ;
n ∣ d 1 , l c m ( g c d ( n , k − 1 ) , g c d ( n , l − 1 ) ) ∣ d 2 and d 3 = a + b , where a , b are non-negative integers and a k + b l ≡ 0 ( mod n ) .

Proof

Since udenom ( F [ V ] η ( Z n ) , t ) is invariant whenever the image of η ( 1 ¯ ) is ( 1 ¯ , k ¯ , l ¯ ) or ( 1 ¯ , l ¯ , k ¯ ) , without loss of generality, we could set ( n d 2 ) ¯ ∈ < k ¯ > and ( n d 3 ) ¯ ∈ < l ¯ > , then d 2 ∣ o ( k ¯ ) and d 3 ∣ o ( l ¯ ) .

With the earlier discussion, we have d 1 = n , d 2 = o ( k ¯ ) and d 3 = o ( l ¯ ) with an h.s.o.p. x 1 n , x 2 o ( k ¯ ) , x 3 o ( l ¯ ) .
By Propositions 3.15 and 3.11, x 1 n + x 2 n , ( x 1 κ x 2 g c d ( n , k − 1 ) − κ ) d 2 g c d ( n , k − 1 ) , x 3 o ( l ¯ ) , where 0 ⩽ κ ⩽ g c d ( n , k − 1 ) , is an h.s.o.p. with degree sequence ( d 1 , d 2 , d 3 ) .
If l c m ( g c d ( n , k − 1 ) , g c d ( n , l − 1 ) ) ∣ d 2 , then g c d ( n , k − 1 ) ∣ d 2 and g c d ( n , l − 1 ) ∣ d 2 . By Propositions 3.15 and 3.11, there are invariants of degree d 2 of the form of ( x 1 κ 1 x 2 g c d ( n , k − 1 ) − κ 1 ) d 2 g c d ( n , k − 1 ) , ( x 1 κ 2 x 3 g c d ( n , l − 1 ) − κ 2 ) d 2 g c d ( n , l − 1 ) , where 0 ⩽ κ 1 ⩽ g c d ( n , k − 1 ) , 0 ⩽ κ 2 ⩽ g c d ( n , l − 1 ) . The last condition guarantees there is an invariant of degree d 3 of the form of x 2 a x 3 b . Then x 1 n + x 2 n + x 3 n , ( x 1 κ 1 x 2 g c d ( n , k − 1 ) − κ 1 ) d 2 g c d ( n , k − 1 ) + ( x 1 κ 2 x 3 g c d ( n , l − 1 ) − κ 2 ) d 2 g c d ( n , l − 1 ) , x 2 a x 3 b is an h.s.o.p. with degree sequence ( d 1 , d 2 , d 3 ) .□

Example 3.18

Consider the representations of Z 12 which satisfy the case ( c ) in Proposition 3.17. There exists an h.s.o.p. with degree sequence ( n , g c d ( n , k − 1 ) ) for η k when k = 1 , 4 , 5 , 7 , 9 , 10 , 11 . Let ζ be a primitive 12th root of unity. We list the hsop of F [ x 1 , x 2 ] Z 12 for all k as follows:

η 1 : Z 12 ↪ GL ( 2 , F ) , 1 ¯ ↦ ζ 0 0 ζ . udenom ( F [ x 1 , x 2 ] Z 12 , t ) = ( 1 − t 12 ) ( 1 − t 12 ) with an hsop x 1 12 , x 2 12 .
η 4 : Z 12 ↪ GL ( 2 , F ) , 1 ¯ ↦ ζ 0 0 ζ 4 . udenom ( F [ x 1 , x 2 ] Z 12 , t ) = ( 1 − t 12 ) ( 1 − t 3 ) with an hsop x 1 12 , x 2 3 .
η 5 : Z 12 ↪ GL ( 2 , F ) , 1 ¯ ↦ ζ 0 0 ζ 5 . udenom ( F [ x 1 , x 2 ] Z 12 , t ) = ( 1 − t 12 ) ( 1 − t 4 ) with an hsop x 1 12 + x 2 12 , x 1 2 x 2 2 .
η 7 : Z 12 ↪ GL ( 2 , F ) , 1 ¯ ↦ ζ 0 0 ζ 7 . udenom ( F [ x 1 , x 2 ] Z 12 , t ) = ( 1 − t 12 ) ( 1 − t 6 ) with an hsop x 1 12 + x 2 12 , x 1 5 x 2 .
η 9 : Z 12 ↪ GL ( 2 , F ) , 1 ¯ ↦ ζ 0 0 ζ 9 . udenom ( F [ x 1 , x 2 ] Z 12 , t ) = ( 1 − t 12 ) ( 1 − t 4 ) with an hsop x 1 12 , x 2 4 .
η 10 : Z 12 ↪ GL ( 2 , F ) , 1 ¯ ↦ ζ 0 0 ζ 10 . udenom ( F [ x 1 , x 2 ] Z 12 , t ) = ( 1 − t 12 ) ( 1 − t 3 ) with an hsop x 1 12 + x 2 12 , x 1 2 x 2 .
η 11 : Z 12 ↪ GL ( 2 , F ) , 1 ¯ ↦ ζ 0 0 ζ 11 . udenom ( F [ x 1 , x 2 ] Z 12 , t ) = ( 1 − t 12 ) ( 1 − t 2 ) with an hsop x 1 12 + x 2 12 , x 1 x 2 .

And there are only two 3-dimensional representations of Z 12 satisfying case (c). We list the hsop of F [ x 1 , x 2 , x 3 ] Z 12 as follows:

η : Z 12 ↪ GL ( 3 , F ) , 1 ¯ ↦ ζ 0 0 0 ζ 7 0 0 0 ζ 10 .
udenom ( F [ x 1 , x 2 , x 3 ] Z 12 , t ) = ( 1 − t 12 ) ( 1 − t 6 ) ( 1 − t 3 )
with an hsop x 1 12 + x 2 12 + x 3 12 , x 1 x 2 5 + x 1 4 x 3 2 , x 2 2 x 3 .
η : Z 12 ↪ GL ( 3 , F ) , 1 ¯ ↦ ζ 0 0 0 ζ 7 0 0 0 ζ 11 .
udenom ( F [ x 1 , x 2 , x 3 ] Z 12 , t ) = ( 1 − t 12 ) ( 1 − t 6 ) ( 1 − t 4 )
with an hsop x 1 12 + x 2 12 + x 3 12 , x 1 x 2 5 + x 1 3 x 3 3 , x 2 2 x 3 2 .

Acknowledgements

The authors would like to express their sincere gratitude to the referees and editors for a very careful reading of this article and for all their insightful comments, which led to a number of improvements. Special thanks to Dr. Yang Chen for a number of invaluable conversations at the genesis of this work and reminding the article “Degrees of Regular Sequences With a Symmetric Group Action” in the reference. Thanks also to Dr. Jiangbei Hu for his assistance with programming.

Funding information: This work was supported by the National Nature Science Foundation of China (Grant No. 12171194).
Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.
Conflict of interest: The authors state no conflicts of interest.
Data availability statement: Data sharing is not applicable to this article as no datasets were generated or analyzed during the current study.

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Received: 2022-11-08

Revised: 2023-11-20

Accepted: 2024-02-26

Published Online: 2024-03-18

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https://doi.org/10.1515/math-2024-0001

Keywords for this article

Stanley’s counterexample; Molien’s theorem; optimal homogeneous system of parameters

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