MODULAR COVARIANTS OF CYCLIC GROUPS OF ORDER p JONATHAN ELMER Abstract. Let G be a cyclic group of order p and let V; W be |G-modules. G ∗ G We study the modules of covariants |[V ] (W ) = (S(V ) ⊗ W ) . For V indecomposable with dimension 2, and W an arbitrary indecomposable mod- G G ule, we show |[V ] (W ) is a free |[V ] -module (recovering a result of Broer G and Chuai [1]) and we give an explicit set of covariants generating |[V ] (W ) G freely over |[V ] . For V indecomposable with dimension 3 and W an arbi- G trary indecomposable module, we show that |[V ] (W ) is a Cohen-Macaulay G |[V ] -module (again recovering a result of Broer and Chuai) and we give an G explicit set of covariants which generate |[V ] (W ) freely over a homogeneous G set of parameters for |[V ] . 1. Introduction Let G be a group, | a field, and V and W |G-modules. Then G acts on the set of functions V ! W according to the formula g · φ(v) = gφ(g−1v) for all g 2 G and v 2 V . Classically, a covariant is a G-equivariant polynomial map V ! W . An invariant is the name given to a covariant V ! | where | denotes the trivial indecomposable |G-module. If the field | is infinite, then the set of polynomial maps V ! W can be identified with S(V ∗) ⊗ W , where the action on the tensor product is diagonal and the action on S(V ∗) is the natural extension of the action on V ∗ by algebra automorphisms. Then the natural pairing S(V ∗) × S(V ∗) ! S(V ∗) is compati- ble with the action of G, and makes the invariants S(V ∗)G a |-algebra, and the covariants (S(V ∗) ⊗ W )G a S(V ∗)G-module. If G is finite and the characteristic of | does not divide jGj, then Schur's lemma implies that every covariant restricts to an isomorphism of some direct summand of S(V ∗) onto W . Thus, covariants can be viewed as \copies" of W inside S(V ∗). Otherwise, the situation is more complicated. The algebra of polynomial maps V ! | is usually written as |[V ]. In this article we will write |[V ]G for the algebra of G-invariants on V , and |[V ]G(W ) for the module of V − W covariants. We are interested in the structure of |[V ]G(W ) as a |[V ]G-module. Throughout, G denotes a finite group. This question has been considered by a number of authors over the years. For example, Chevalley and Sheppard-Todd [2], [10] showed that if the characteristic of | does not divide jGj and G acts as a reflection group on V , then |[V ]G is a polynomial algebra and |[V ]G(W ) is free. More generally, Eagon and Hochster [7] showed that if the characteristic of | does not divide jGj then |[V ]G(W ) is a Cohen-Macaulay module (and |[V ]G a Cohen-Macaulay ring in particular). In the modular case, Hartmann [5] and Hartmann-Shepler [6] gave necessary and sufficient Date: June 28, 2018. 2010 Mathematics Subject Classification. 13A50. Key words and phrases. modular invariant theory, covariants, free module, cohen-macaulay, hilbert series. 1 2 JONATHAN ELMER conditions for a set of covariants to generate |[V ]G(W ) as a free |[V ]G-module, provided that |[V ]G is polynomial and W =∼ V ∗. Broer and Chuai [1] remove the restrictions on both W and |[V ]G. 2. Preliminaries The purpose of this article is to investigate modular covariants for cyclic groups of order p. We are particularly interested in what happens when |[V ]G(W ) is not a free |[V ]G-module. Accordingly, from this point onwards we let G be a cyclic group of order p and | a field of characteristic p. We fix a generator σ of G. Recall that, up to isomorphism, there are exactly p indecomposable |G-modules V1;V2;:::;Vp, where the dimension of Vi is i. The isomorphism class of Vi is usually represented by a module of column vectors on which σ acts as left-multiplication by a single ∼ ∼ Jordan block of size i. Suppose V = Vm and W = Wn. We will choose bases v1; v2; : : : ; vm and w1; w2; : : : ; wn of V and W respectively for which the action of G is given by σw1 = w1 σw2 = w2 − w1 σw3 = w2 − w2 + w1 . σwn = wn − wn−1 + wn−2 − ::: ± w1 with analgous formula for the action on V (thus, the action of σ−1 is given by left-multiplication by a upper-triangular Jordan block). Now let x1; : : : ; xm be the dual basis to v1; v2; : : : ; vm; the action of G on these variables is then given by σx1 = x1 + x2; σx2 = x2 + x3; σx3 = x3 + x4: . σxm = xm: Note that |[V ] = |[x1; x2; : : : ; xm], and a general element of |[V ](W ) = |[V ]⊗W is given by φ = f1w1 + f2w2 + ::: + fnwn where each fi 2 |[V ]. We define the support of φ by Supp(φ) = fi : fi 6= 0g: The operator ∆ = σ − 1 2 |G will play a major role in our exposition. ∆ is a σ-twisted derivation on |[V ]; that is, it satisfies the formula (1) ∆(fg) = f∆(g) + ∆(f)σ(g) for all f; g 2 |[V ]. Further, using induction and the fact that σ and ∆ commute, one can show ∆ satisfies a Leibniz-type rule k X n (2) ∆k(fg) = ∆i(f)σn−i(∆n−i(g)): i i=0 MODULAR COVARIANTS OF CYCLIC GROUPS OF ORDER p 3 For any f 2 |[V ] we define the weight of f: wt(f) = minfi > 0 : ∆i(f) = 0g: Notice that ∆wt(f)−1 2 ker(∆) = |[V ]G for all f 2 |[V ]. Note also the following crucial observation: for all i = 1; : : : ; n − 1 we have (3) ∆(wi+1) + σ(wi) = 0 and ∆(w1) = 0. From this we obtain a simple characterisation of covariants: Proposition 1. Let G φ = f1w1 + f2w2 + ::: + fnwn 2 |[V ] (W ): i−1 Then there exists f 2 |[V ] with weight ≤ n such that fi = ∆ (f) for all i = 1; : : : ; n. Proof. We have n ! X 0 =∆ fiwi i=1 n X = (fi∆(wi) + ∆(fi)σ(wi)) i=1 n−1 X = (∆(fi) − fi+1)σ(wi)) + ∆(fn)σ(wn) i=1 where we used (3) in the final step. Now note that σ(wi)) = wi + (terms in wi+1; wi+2; : : : ; wn) for all i = 1; : : : ; n. Thus, equating coefficients of wi, for i = 1; : : : ; n gives ∆(f1) = f2; ∆(f2) = f3;:::; ∆(fn−1) = fn; ∆(fn) = 0: i−1 n Putting f = f1 gives fi = ∆ (f) for all i = 1; : : : ; n and 0 = ∆ (f) as required. Note that the support of any covariant is therefore of the form f1; 2; : : : ; ig for some i ≤ n. Where no confusion will arise, we write Supp(φ) = i if φ is a covariant and Supp(φ) = f1; 2; : : : ; ig. Introduce notation n Kn := ker(∆ ) and n In := im(∆ ): These are |[V ]G-modules lying inside |[V ]. Now we can define a map G Θ: Kn ! |[V ] (W ) n X i−1 (4) Θ(f) = ∆ (f)wi: i=1 Clearly Θ is injective, and Proposition 1 implies it is also surjective. Therefore Θ is a degree-preserving bijective map. 4 JONATHAN ELMER 3. Hilbert series Let | be a field and let S = ⊕i≥0Si be a positively graded |-vector space. The dimension of each graded component of S is encoded in its Hilbert Series X i H(S; t) = dim(Si)t : i≥0 G The observations following Proposition 1 imply that H(|[V ] (W ); t) = H(Kn; t). In this section we will outline a method for computing H(Kn; t). Each homogeneous component |[V ]i of |[V ] is a |G-module. As such, each one decomposes as a direct sum of modules isomorphic to Vk for some values of k. Write µk(|[V ]i) for the multiplicity of Vk in |[V ]i and define the series X i Hk(|[V ]) = µk(|[V ]i)t : i≥0 The series Hk(|[Vm]) were studied by Hughes and Kemper in [8]. They can G G also be used to compute the Hilbert series of |[Vm] ; since dim(Vk ) = 1 for all k = 1; : : : ; p we have p G X (5) H(|[Vm] ; t) = Hk(|[Vm]; t): k=1 Now observe that n n ≥ k dim(ker(∆nj )) = Vk k otherwise. Therefore n−1 p X X H(Kn; t) = kHk(|[V ]; t) + nHk(|[V ]; t): k=1 k=n We can also write this as as a series not depending on p: n−1 G X (6) H(Kn; t) = nH(|[V ] ; t) − ( (n − k)Hk(|[V ]; t)): k=1 using equation (5). G G We will need the Hilbert Series of In = |[V ] \ In in a later section. For all k = 1; : : : ; p we have 1 n > k dim(∆n(V ))G = k 0 otherwise. Therefore p G X H(In ; t) = Hk(|[V ]; t); k=n+1 which we can write independently of p as n G G X (7) H(In ; t) = H(|[V ] ; t) − ( Hk(|[V ]; t)): k=1 Note that G (8) H(Kn−1; t) + H(In−1; t) = H(Kn; t): We will make use of this in the final section. MODULAR COVARIANTS OF CYCLIC GROUPS OF ORDER p 5 4. Decomposition Theorems In this section we will compute the series Hk(|[V2]; t) and Hk(|[V3]; t) for all k = 1; : : : ; p − 1. Hughes and Kemper [8, Theorem 3.4] give the formula m−1 X γ − γ−1 1 − γp(m−1)tp Y (9) H ( [V ]; t) = γ−k (1 − γm−1−2jt)−1; k | m 2p 1 − tp γ2M2p j=0 where M2p represents the set of 2pth roots of unity in C.