Degree of reductivity of a modular representation

Degree of reductivity of a modular representation

Abstract.

For a finite dimensional representation of a group over a field , the degree of reductivity is the smallest degree such that every nonzero fixed point can be separated from zero by a homogeneous invariant of degree at most . We compute explicitly for several classes of modular groups and representations. We also demonstrate that the maximal size of a cyclic subgroup is a sharp lower bound for this number in the case of modular abelian -groups.

2000 Mathematics Subject Classification:
13A50
Second author is supported by a grant from Tübitak:112T113

Introduction

Separating points from zero by invariants is a classical problem in invariant theory. While for infinite groups it is quite a problem to describe those points where this is (not) possible (leading to the definition of Hilbert’s Nullcone), the finite group case is easier. We fix the setup before going into details. Unless otherwise stated, denotes an algebraically closed field of characteristic . We consider a finite dimensional representation of a finite group over . We call a -module. The action of on induces an action of on via for and . Any homogeneous system of parameters (hsop) of the invariant ring has as its common zero set, hence every nonzero point can be separated from zero by one of the . Moreover, Dade’s algorithm [6, section 3.3.1] produces an hsop in degree , hence every nonzero point can be separated from zero by an invariant of degree at most . Therefore, for a given nozero point , the number

is bounded above by the group order , and hence so is the supremum of the taken over all . There has been a recent interest in this number, see [5, 7, 8]. In [7], another related number is introduced, which is defined to be zero if and otherwise as the supremum of all taken over all nonzero fixed points . We propose the name degree of reductivity for . Note that a group is called reductive, if for every and every , there exists a homogeneous positive degree invariant such that , hence the suggested name. It was shown in [7], that , the supremum of the taken over all , equals the size of a Sylow- subgroup of . The goal of this paper is to give more precise information on and compute it explicitly for several classes of modular groups (i.e., is divisible by ) and representations. In Section 1 we show that for a cyclic -group and every faithful -module , we have . In that situation we compute for every as well. The most important stepstone that we lay to our main results is a restriction of the degrees of certain monomials that appear in invariant polynomials. We think that this restriction can also be useful for further studies targeting the generation of the invariant ring. In Section 2 we consider an abelian -group and show that the maximal size of a cyclic subgroup of is a lower bound for for every faithful -module . We also work out the Klein four group and compute the - and -values for all its representations. It turns out that our lower bound is sharp for a large number of these representations. In the final section we deal with groups whose order is divisible by only once and put a squeeze on the -values of the representations of these groups.

For a general reference for invariant theory we refer the reader to [1, 4, 6, 13].

1. Modular cyclic groups

Let be the cyclic group of order . Fix a generator of . It is well known that there are exactly indecomposable -modules over , and each indecomposable module is afforded by acting via a Jordan block of dimension with ones on the diagonal. Let be an arbitrary -module over . Write

where each is spanned as a vector space by . Then the action of is given by for and . Note that the fixed point space is -linearly spanned by . The dual is isomorphic to . Let denote the corresponding dual basis, then we have

and the action of is given by for and for . We call the for terminal variables. Set . Notice that if and . Since for , and is an additive map, we have the following, see also the discussion in [12, before Lemma 1.4].

Lemma 1.

Let and be a monomial that appears in . If a monomial appears in , then there is another monomial that appears in such that appears in as well.

We say that a monomial lies above if appears in . We will use the well-known Lucas-Theorem on binomial coefficients modulo a prime in our computations (see [9] for a short proof):

Lemma 2 (Lucas-Theorem).

Let be integers with base--expansions and , where for . Then .

The following lemma is the main technical stepstone for the rest of the paper.

Lemma 3.

For , define

Let be a monomial consisting only of terminal variables, that appears in an invariant polynomial with nonzero coefficient. Then divides for all .

Proof.

As the case is trivial, we will assume from now. Let be an invariant polynomial in which appears with a nonzero coefficient, and . Without loss of generality, we assume and . Set . For simplicity we denote with . Then , and the claim is . We proceed by induction on and at each step we verify the claim for all such that . Assume and . By way of contradiction, we assume . Then we can write , where and are non-negative integers with . We have and . Since appears in with coefficient one, it follows that the coefficient of in is . Therefore appears in . As only consists of terminal variables, it can be seen easily that it is the only monomial lying above , which is a contradiction by Lemma 1.

Next assume that and let be arbitrary. Note that the induction hypothesis is that the assertion holds for every pair with and . Consider the base--expansion of where . Let denote the smallest integer such that . We claim that , which is equivalent to . By way of contradiction assume . Define . Then the base--expansion of is . As in the basis case, we see that the coefficient of in is . By the Lucas-Theorem, . So appears in . By Lemma 1 there exists another monomial in that lies above . We have for some . Since and divides it follows that

Let denote the subgroup of generated by . Note that and consider as an -module. From it follows that decomposes into indecomposable -modules such that become terminal variables with respect to the -action. Note that by assumption, and as , we have . Also the -module generated by has dimension . Therefore the monomial appearing in consists only of terminal variables with respect to the -action and is a terminal variable whose index would appear in the set corresponding to the considered -action. Therefore, the induction hypothesis (with and ) applied to yields . As we have assumed , it follows that divides , which is a contradiction to (*) above. ∎

With this lemma we can precisely compute the degree required to separate a nonzero fixed point from zero.

Theorem 4.

Let be a nonzero fixed point, where . Let denote the set of all such that , and denote the maximal integer such that for all . Then .

In particular, if is a faithful -module, then .

Proof.

Any homogeneous invariant polynomial of positive degree that is nonzero on must contain a monomial with a nonzero coefficient in the variables of the set . With as defined above, by the previous lemma the exponents of the in are divisible by for all . Hence , so it remains to prove the reverse inequality. The maximality condition on implies the existence of a such that . Then the Jordan block representing the action of on has order , and so the orbit product is an invariant homogeneous polynomial of degree . Furthermore, for every and the corresponding element in the orbit, we have , where we used . Hence, , which shows .

For the final statement, note that if is a faithful -module, then there is a satisfying . Now for , in the notation above we have and , so the first part yields . It follows as claimed. ∎

We now consider the general modular cyclic group , where is a non-negative integer with . Let and be the subgroups of of order and , respectively. Fix a generator of and a generator of . For every and an -th root of unity , there is an -dimensional -module with basis such that for , and for . It is well-known that the form the complete list of indecomposable -modules, see [10, Lemma 3.1] for a proof.

Notice that the indecomposable module is faithful if and only if and is a primitive -th root of unity. Let denote the corresponding basis for . We have an isomorphism , where the action of on is given by an upper diagonal Jordan block. Note that if , we have , and so .

Proposition 5.

Let and be a faithful indecomposable -module, i.e. and is a primitive -th root of unity. Then .

Proof.

Let be a homogeneous invariant of positive degree such that . Then contains the monomial with a nonzero coefficient. Considered as a -module, is isomorphic to the indecomposable -module . Since is particularly -invariant, we get from Lemma 3 that divides . As is also -invariant, and acts just by multiplication with on every variable, it follows that is -invariant, hence we have . As is a primitive -th root of unity, it follows that divides . Since and are coprime we get that divides . Therefore . The reverse inequality always holds by Dade’s hsop algorithm. ∎

Now let be arbitrary and be an arbitrary th root of unity. Define such that and let denote the order of as an element of the multiplicative group (then ). Then can be considered as a faithful -module, hence the result above yields . As the -value of a direct sum of modules is the maximum of the -values of the summands (see fore example [7, Proposition 3.3]), the proposition above allows to compute for every -module. This precisises the result of [7, Corollary 4.2], which states . As an interesting example, take again a primitive th root of unity and consider the -module . Note that though is a faithful -module, we get from the above

which is strictly smaller than if and .

2. Modular abelian -groups

Before we focus on abelian -groups, we start with a more general lemma.

Lemma 6.

Let be a -group, a faithful -module and let be of order such that (the center of ). Then .

Proof.

Let denote the subgroup of generated by . We follow the notation of the previous section and consider the decomposition of as a -module. Since is also faithful as -module, we have . We can choose a suitable basis of such that acts on this basis via sums of Jordan blocks of dimensions . Set . Let denote the image of the map on . Since , we have that commutes with every , hence is a -module. We also have because if . On the other hand, is spanned by for . But , so we get that and in particular for . Hence is spanned -linearly by . Moreover, since every modular action of a -group on a nonzero module has a non-trivial fixed point, we have

Choose any nonzero vector . As is in the span of , every homogeneous polynomial of positive degree that is nonzero on must contain a monomial with nonzero coefficient in the variables . Since is also -invariant, Lemma 3 applies and we get that the exponents of these variables in this monomial are all divisible by . It follows as desired. ∎

In the following two examples, is an algebraically closed field of characteristic .

Example 7.

Consider the dihedral group of order with relations , and . Then . Hence the lemma applies, and for every faithful -module we have .

Example 8.

Consider the quaternion group of order . There is an element of order such that . From the lemma it follows that for every faithful -module , we have .

Example 9.

Consider the non-abelian group

of order (where we write ). Note that . The element is of order , and it can be checked easily that . From the lemma it follows that for every faithful -module , we have .

Recall that that for a group , the exponent of is the least common multiple of the orders of its elements. In particular for an abelian group, the exponent is the maximal order of an element. As a corollary of the above lemma, we get:

Theorem 10.

Let be a non-trivial -group. Then for every faithful -module we have

If is an abelian -group, we particularly have

Proof.

First note that for -groups, its center is non-trivial, so particularly we have . Now chose an element of maximal order . Then Lemma 6 applies and yields . Finally, if is an abelian -group, we have . ∎

For a recent related study of the invariants of abelian -groups we refer the reader to [3]. We also remark that the inequality in Theorem 10 is sharp, see Theorem 15 below.

The Klein four group

Let denote the Klein four group with generators and , and an algebraically closed field of characteristic . The goal of this section is to compute the - and -value of every -module (in all cases, both numbers are equal here). We first give the -value for each indecomposable representation of the Klein four group. The complete list of indecomposable representations is for example given in [2, Theorem 4.3.3]. There, the indecomposable representations are classified in five types (i)-(v), and we will use the same enumeration. For the notation of the modules, we follow [10] but note that there types (iv) and (v) are interchanged. The first type (i) is just the regular representation , and here we have by [7, Theorem 1.1 and Proposition 2.4].

The type (ii) representations are parameterized by a positive integer and . Then is defined as the -dimensional representation spanned by such that the action is given by , for and , for and . Let be the elements of corresponding to . Then we have , for and , and for .

Lemma 11.

In the notation as above, we have that equals if , and it equals if .

Proof.

If , the corresponding matrix group is of order , and the result follows easily. If it follows from [6, Theorem 3.7.5] that is generated by and the norm , as those two invariants form an hsop and the product of their degrees equals the group order . Now the claim follows easily. ∎

Proposition 12.

In the notation as above, we have for all and .

Proof.

We have , from Dade’s hsop algorithm, hence it is enough to show . Consider the point . Any homogeneous invariant of positive degree separating from zero must contain . Lemma 13 implies , so , finishing the proof. ∎

Set for . Since for every polynomial , the assertion of Lemma 1 holds for for . We say that a monomial lies above the monomial with respect to if appears in .

Lemma 13.

Assume that with . Then does not appear in a polynomial in for .

Proof.

Assume that appears in . Since spans a two dimensional indecomposable summand as a -module, Lemma 3 applies and we get that is divisible by . Assume on the contrary that . Then . So appears in . Since is the only other monomial in that lies above with respect to we get that appears in as well. Moreover since the coefficient of in and is one, it follows that the coefficients of and in are equal. Call this nonzero coefficient . Then the coefficient of in is . Since and are the only monomials in that lie above with respect to , we get that if , giving a contradiction. Next assume that . Then, since appears in and we get that appears in . This gives a contradiction by Lemma 1 again because is the only monomial that lies above with respect to . Finally, we note that the cases and correspond to the same matrix group and so their invariants are the same. ∎

The type (iii) representations are -dimensional representations () which are obtained from just by interchanging the actions of and . In particular, and have the same invariant ring, so we get as a corollary from Lemma 11 and Proposition 12 that and for all .

The type (iv) representations for are -dimensional representations. (Note that in [10], these representations are listed as type (v).) They are linearly spanned by , where for and , for , , , and for . Let be the elements of corresponding to . Then we have for and , and for .

Proposition 14.

We have for all .

Proof.

Again by Dade’s hsop-algorithm, we have . Consider the point , and let be homogeneous of minimal positive degree such that . Then must appear in with a nonzero coefficient. Since spans a two dimensional indecomposable summand as a -module and is also -invariant, Lemma 3 applies and we get that is divisible by . By [12, Proposition 5.8.3], does not appear in a -invariant polynomial. It follows , so we are done. ∎

The type (v) representations for are -dimensional representations. (Note that in [10], these representations are given type (iv).) They are afforded by and , where denotes a matrix whose entries are all zero. In [12, section 4] (with notation ), an hsop consisting of invariants of degree at most is given for . As the -value is clearly not one, it follows