The Laurent coefficients of the Hilbert series of a Gorenstein algebra
By a theorem of R. Stanley, a graded Cohen-Macaulay domain is Gorenstein if and only if its Hilbert series satisfies the functional equation
where is the Krull dimension and is the a-invariant of . We reformulate this functional equation in terms of an infinite system of linear constraints on the Laurent coefficients of at . The main idea consists of examining the graded algebra of formal power series in the variable that fulfill the condition . As a byproduct, we derive quadratic and cubic relations for the Bernoulli numbers. The cubic relations have a natural interpretation in terms of coefficients of the Euler polynomials. For the special case of degree , these results have been investigated previously by the authors and involved merely even Euler polynomials. A link to the work of H. W. Gould and L. Carlitz on power sums of symmetric number triangles is established.
Key words and phrases:Gorenstein algebras, Euler polynomials, Bernoulli numbers
2010 Mathematics Subject Classification:Primary 05A15; Secondary 11B68, 13H10, 13A50.
- 1 Introduction
- 2 The for invariants of finite groups
- 3 Construction of the algebra
- 4 Proof of Theorem 1.1
- 5 Relation to Euler polynomials and Bernoulli numbers
- 6 Identities for the Bernoulli numbers
- 7 Power sum formulas revisited
- 8 Coefficient triangles
Let be a field. By a positively graded -algebra we mean a graded -algebra such that and for all we have . To a positively graded -algebra one associates its Hilbert series , i.e., the generating function
counting the dimensions of the homogeneous components . If is finitely generated, then the Hilbert series is actually a rational function . Moreover, the pole order in the Laurent expansion
The number is the so-called a-invariant of and can be understood as , see [18, Definition 5.1.5]. It is well-known that
see [14, Equation (3.32)]. We will call this quantity the degree, as it will play this role for a -graded algebra in this work. In the literature (e.g. in ), is frequently referred to as the degree, and we hope this will not lead to any confusion. The degree has a natural interpretation in terms of the canonical module of (see e.g. [2, 3]) as the degree shift in , where the shifted module of a graded module is defined via . In this context, is Gorenstein if and only if the canonical module is one-dimensional.
The starting point and initial motivation of this work is a reformulation of the functional equation (1.3) in terms of an infinite system of linear constraints on the Laurent coefficients . In a previous paper , we have treated the case of degree equal zero, i.e. . In the general case, the actual shape of the relations depends on the sign and parity of the degree .
Let be a formal Laurent series around of pole order at most . Then satisfies the functional equation (1.3) if and only if the following conditions are fulfilled, where we stipulate for .
If the degree is even, then for each :
If the degree is odd, then for each :
When , then for each :
regardless of the parity of . Moreover, if is odd, then for each :
Note that the relations among the Laurent coefficients are only unique up to scalar factors. Alternate choices of the scaling will arise in the course of Section 4 and be considered as well in Section 8.
The relations given in Theorem 1.1 in the case were first observed in [11, Section 8] in the context of experimental computations of the Laurent coefficients of Hilbert series of a certain class of rings related to symplectic quotients by the circle. An understanding of these relations suggested that these rings are all Gorenstein of degree (sometimes called graded Gorenstein or strongly Gorenstein), which was then verified in [10, Section 4]. The initial purpose of Theorem 1.1 is to demonstrate that analogous relations occur for Gorenstein rings of arbitrary degree . Hence, one application of Theorem 1.1 is to contexts in which the are more directly computable or otherwise more accessible than the rational expression of the Hilbert series. The relations for small values of can then be used to prove that a ring is not Gorenstein without computing a more complete description of the ring or its Hilbert series. We illustrate an example of such a context in Section 2 for invariants of finite groups.
In addition, the connection between the relations of Theorem 1.1 and the functional equation (1.3) has interesting consequences for some combinatorial number sequences and other combinatorial constructions. After introducing and describing the structure of the -graded algebra in Section 3, the principal tool used in the proof of Theorem 1.1 that is then given in Section 4, we turn our attention to these consequences. In Section 5, we reformulate the constraints of Theorem 1.1 to express the odd coefficients in terms of the even coefficients and vice versa, see Theorems 5.3 and 5.9. These reformulations are most succinctly stated using the coefficients of Euler polynomials as well as the Bernoulli numbers. Hence, the algebra structure of yields a unified proof of a large collection of quadratic and cubic identities for the Bernoulli numbers in Section 6, see Theorems 6.1 and 6.2.
In Section 7, we consider another application to combinatorics, connecting the functional equation (1.3) to the power sum identities for symmetric number arrays (e.g. Pascal’s triangle) developed by H. W. Gould  and L. Carlitz . Specifically, by demonstrating that a generating function for these power sums is an element of , we re-derive these power sum identities as corollaries of the case of Theorem 1.1 and its reformulations in Section 5. To complete the paper, we illustrate the coefficient triangles that appear in Theorem 1.1 by displaying specific examples and consider the rescaling of the rows in Section 8.
Note that some of the auxiliary lemmas in this paper can be derived as special cases of known identities. For instance, Lemma 4.5 can be seen to be a consequence of [1, Equation (15.2.5)]. However, for the benefit of the reader, we provide elementary proofs.
Finally, let us observe that there exist Gorenstein algebras for each degree . For example, a hypersurface of degree in affine space is Gorenstein of degree , while invariant rings of unimodular finite group representations with no pseudoreflections are Gorenstein of degree by [19, 20], see Section 2. On the other hand, a polynomial ring in variables of degree has .
HCH and CS would like to thank Baylor University, and CS would like to thank the Universidade Federal do Rio de Janeiro, for their hospitality during work on this project. We would also like to thank Emily Cowie, whose work on a separate project  yielded a large family of examples used to discover some of the results contained here.
2. The for invariants of finite groups
In this section, we give an example of an application of Theorem 1.1 to computational invariant theory. We refer the reader to [2, Sections 2.5–6], [6, Section 2.6], or [17, Section 2.2] for background on the topic considerer here. For simplicity, we work over .
Let be a vector space over of dimension , and let be a finite subgroup of . By Molien’s formula, the Hilbert series of the ring of -invariant polynomials is given by
It is a well-known consequence of this result that the first two coefficients of the Laurent expansion of at are equal to
where is the number of pseudoreflections in , elements of whose fixed set in has codimension . By the same method used to determine these coefficients, we now demonstrate that the for can be computed using only those elements of that fix a subset of codimension at most .
For , let denote the eigenvalues of , where we assume that any eigenvalues with value occur last on this list. Choosing for each a basis for with respect to which is diagonal, Equation (2.1) becomes
Let denote the number of such that . For , let , , and . As the term in Equation (2.2) corresponding to an element has a pole order equal to at , the coefficient of the Laurent series depends only on the elements of . Specifically, we can express
Then as , the sum over is simply equal to , yielding . Similarly,
where the last sum has a pole at of order at most . In particular,
Continuing in this way,
To apply Theorem 1.1, given a finite group , the value of is determined from and using Equation (1.4). Note that in this context, is the number of pseudoreflections in . Furthermore, Equation (1.4) is reflected by Theorem 1.1 as Equation (1.5) with for even and as Equation (1.6) with for odd. Then, staying for the moment with the case even, the constraint in Theorem 1.1 corresponding to gives a necessary condition for to be Gorenstein that involves only the eigenvalues of elements of . Similarly, the constraint corresponding to arbitrary gives a necessary Gorenstein condition for involving only . Note that if contains no pseudoreflections, then is Gorenstein if and only if by a Theorem of Watanabe [20, Theorem 1]; hence, the Gorenstein property of can be established much more easily in this case.
As an explicit example, let be a primitive th root of unity and consider the subgroup of of order generated by and . Clearly, , and as contains the single pseudoreflection , we have . It follows that if were to be Gorenstein, we must have so that by Equation (1.6) with . However, as described above, one may easily compute using only the elements of , , , and that and so that and hence is not Gorenstein. In this simple example, we can conclude that the Gorenstein property fails with a computation involving only the elements of , and in particular without computing the invariant ring or its Hilbert series completely. In larger, less contrived examples, may be much smaller relative to the size of .
3. Construction of the algebra
For the remainder of this paper, let denote one of the fields , , or .
Let us assume that is a formal Laurent series over the field around of pole order at most . Clearly, the substitution is a formal power series in the variable , i.e. an element of . Conversely, a formal power series defines a formal Laurent series . Assuming that satisfies the functional equation (1.3), we derive
Multiplying by and substituting , we find that satisfies the functional equation
Let be the space of formal power series in the variable satisfying Equation (3.1). We introduce the -graded vector space .
Using the above computations, it is an easy task to verify the following statements, which we leave to the reader.
The substitution establishes an isomorphism between the space of formal Laurent series of pole order at most in the variable around satisfying Equation (1.3) and . With respect to the Cauchy product of formal power series, forms a -graded algebra.
In , the authors have investigated the algebra , i.e., the algebra of formal power series in the variable invariant under the Möbius transformation . We recall the following.
Theorem 3.3 ().
For a formal power series , the following conditions are equivalent:
is a formal composite with , i.e., there exists a such that .
For each , the relation holds.
In fact, in (ii.), instead of , we could have worked with any whose class in is nonzero. Here is the maximal ideal of . This was noted in [10, Remark 2.4], where we constructed examples of such series that are related to the Genocchi sequence. Another natural example slipped through our attention, namely:
Moreover, the reader can readily check that
is invertible. Hence, for any , it follows that . Using , we can write uniquely in the form
for . Conversely, every such series is an element of .
The -graded algebra is isomorphic to the algebra of Laurent polynomials with coefficients in in the variable of degree .
4. Proof of Theorem 1.1
Let , and let . In this section, we prove Theorem 1.1, that is equivalent to the relations described in Equations (1.5), (1.6), (1.7), and (1.8). In the case , Equations (1.5) and (1.8) involve only the coefficients for , while Equation (1.7) involves the for when is even and when is odd (in the case of even, there is no relation involving ). Hence, we will first assume in this case that for in Subsection 4.1 and deal with Equation (1.7) separately in Subsection 4.2.
We first consider the case of even with the assumption that the first coefficients vanish.
Let and let be an even integer. If , assume that for each . Then if and only if .
Note that when , the assumption that the first of the vanish ensures that is a power series. By Equation (3.1), if and only if , which by a simple computation using and the fact that is even is equivalent to , i.e. . ∎
Let and let be an even integer. If , assume that for each . Then the satisfy Equation (1.5) for each if and only if .
To proceed to the odd case, we start with the following useful characterization of solutions for the constraints in Theorem 1.1 when is positive and odd, i.e. for some . Note that Equation (4.1) simply rewrites Equation (1.6) replacing .
Let , and let be an integer. Then
for each if and only if there is a power series such that
where, for , we interpret as the indefinite integral (with vanishing constant of integration). If , then Equation (4.2) is equivalent to
for some .
For each , multiplying both sides by the scalar , we rewrite Equation (4.1) as
Define by and , and then for each , the satisfy Equation (4.4) if and only if
which, along with is equivalent to . By Theorem 3.3, and as has no constant nor linear terms by definition, this is equivalent to the existence of a such that
Multiplying by yields
Dividing both sides by yields Equation (4.2).
We next have the following, which demonstrates Theorem 1.1 in the case .
A power series is an element of if and only if the satisfy
for each .
As above, by Equation (3.1), if and only if
Let , and then there is a such that by Theorem 3.4. Define
and then we may express
where the integral is the formal integral of power series, interpreted term by term. Substituting into Equation (4.8) and applying the product rule, we have
To proceed, we will need to establish the following identity. We will use to denote the rising Pochhammer symbol and to denote the falling Pochhammer symbol.
Let be a positive integer and let . Then
We first use the binomial series to rewrite
Differentiating, we express the left-hand side of Equation (4.9) as
With this, we now prove the following, which demonstrates Theorem 1.1 when is odd, i.e. for .
As , we have by Lemma 4.3 that the satisfy Equation (4.1) if and only if there is a power series satisfying Equation (4.3). Using Theorem 3.3 and the generator given in Equation (3.2), given , there is a power series such that . That is, the satisfy Equation (4.1) if and only if
Applying Lemma 4.5 to each term and noting that the term vanishes, we continue