Non-induced modular representations of cyclic groups

Abstract We compute the ring of non-induced representations for a cyclic group, Cn, over an arbitrary field and show that it has rank φ(n), where φ is Euler’s totient function—independent of the characteristic of the field. Along the way, we obtain a “pick-a-number” trick; expressing an integer n as a sum of products of p-adic digits of related integers.


Introduction
Given a finite group G and a field k of characteristic p ≥ 0, we may study the representation theory of G over k via the representation ring R kG , whose elements are (isomorphism classes of) kG modules, {V}.
Here, and throughout, the representations of G over k will be identified with kG-modules.The addition and multiplication operations on the representation ring correspond to taking the direct sum and tensor product of modules.That is; {V} + {U} = {V ⊕ U} and {V} • {U} = {V ⊗ k U}.
Observe that the representation ring differs from the Grothendieck ring, as in the Grothendieck ring we have {V} + {U} = {W} whenever there is a short exact sequence 0 → V → W → U → 0. We shall abuse notation when working in the representation ring and write V for the isomorphism class {V}.
Recall that if H is a subgroup of G, and V is a kG-module, then we may consider V as a kH-module, simply by only considering the action of the elements in kH.This module will be denoted by V ↓ G H , or often just V ↓ H , and is called the restriction of V to H.It is clear that if K ≤ H ≤ G, then is the kG-module induced from H to G. Here, the G action is given by left multiplication on the first factor.We call this process induction and say that V ↑ G H is an induced module.Induction is also transitive, which is to say The quotient of the representation ring by this ideal gives a measure of the kG-modules which are not induced from kH-modules.We shall now turn our attention to the kG-modules which are not induced from any proper subgroup.That is, we study the quotient which we shall refer to as the ring of non-induced representations.This was considered for cyclic groups by Srihari in [Sri21], where the following is proved: Theorem 2. Let G be a cyclic group of order n and k an algebraically closed field of characteristic 0. Then , where Φ n is the nth cyclotomic polynomial.In particular, rank R kG where ϕ(n) is the number of integers less than n coprime to n.
The main goal of this paper is to prove an analogous result over fields of positive characteristic.We first shall observe that the proof of Theorem 2 provided in [Sri21] actually shows: Corollary 3. Let G be a cyclic group of order n and k an algebraically closed field of characteristic p such that p n. Then R kG .
In particular the rank of the ring of non-induced representations is ϕ(n).

Cyclic p-groups
Throughout this section, let q = p α and denote by G = C q the cyclic group of order q generated by element g.Let k be a field of characteristic p.

The representation ring
Lemma 4. [AF78, Proposition 1.1]For k and G as above, i) there is a ring isomorphism kG ∼ − → k [X] (X q ), defined by sending g to 1 + X, ii) under this isomorphism, a complete set of (pairwise non-isomorphic) kG-modules are given by iii) the trivial module, V 1 , is the unique irreducible kG-module.
To understand R kG , we first need to understand the structure of multiplication.In other words we need to understand the decomposition of the tensor product V r ⊗ V s into indecomposable parts.The decomposition rule was known to Littlewood, and has been discussed by a number of authors.The following multiplication table is given in the lecture notes of Almkvist and Fossum [AF78] and is derived by Green [Gre62]: Proposition 5.For each k < α and s ≤ p k+1 we have the following decompositions.If s ≤ p k , then This suggests the construction in [AF78] which defines, for 0 To complete the multiplication rule, Renuad [Ren79] gives a reduction theorem which allows us to express V r ⊗ V s in terms of the tensor product of smaller modules: This allows the tensor product to be reduced to the tensor product of smaller modules, which can be calculated via repeated applications of Theorem 6, and finally Proposition 5, or similar multiplication rules appearing in [Ren79].
To aid in exposing the structure of R kG , we adjoin elements With this set up Alkvist and Fossum are able to completely determine the structure of R kG by identifying R kG with a quotient of a polynomial ring.Before we can state their result, we will first state some identities involving the χ i and then define some families of polynomials.
Lemma 7. Let i ≤ j and 0 < s < p, then we have and Proof.The first fact can be verified by considering the expansion of (µ i + µ −1 i ) s , while the second is obtained from applications of Proposition 5. Now, still following [AF78], we define some families of polynomials.These are easier to state if we allow ourselves to use the language of quantum numbers which are briefly introduced here.Definition 8.The quantum number, [n] We shall write [n] x for the nth quantum number evaluated at X = x.
The first quantum numbers are Notice that the coefficients are such that [n] 2 = n.We can also write In this formulation, Consider the polynomials in Z[X 0 , . . ., X α−1 ], We are now ready to state a structure theorem for R kG .

Induced representations
The subgroups of G are all cyclic p groups generated by some power of g.Consider H = g p α−β , the subgroup of G of order p β .The group algebra kH ⊆ kG is identified, under the isomorphism in Lemma 4, with k[X p α−β ]/(X p α ).Its indecomposable representations are again each of the form Inducing to obtain a kG-module we get: Lemma 10.We have an isomorphism of kG-modules Proof.Note kG ⊗ kH W r is cyclic as a kG-module, generated by e ⊗ 1 and has dimension rp α−β .
Thus Ind G H R kH consists of all kG-modules V i such that i is divisible by p β .Hence, since G has a unique maximal subgroup, and induction is transitive, ∑ H<G Ind G H R kH consists of all the kG-modules V i such that i is divisible by p.We have thus shown: Proposition 11.Let G = C q the cyclic group of order q = p α and let k be a field of characteristic p. Then To describe the ring of non-induced representations, , via the isomorphism in Theorem 9 we would aim to write the kG-modules V i where p | i in terms of the χ i .Instead, we shall change basis and give an alternative description of the ideal ∑ H<G Ind G H R kH in terms of modules which are easier to describe by polynomials in the χ i .

Change of basis
Our new basis for R kG shall use the language of quantum numbers.Observe, from Proposition 5, that tensoring with V 2 satisfies a similar relation to the quantum numbers. Moreover, This relationship motivates defining a new basis.
Definition 14.Let r < q = p α and write r − Figure 1: The change of basis matrix, expressing V i in terms of U j over characteristic p = 3.Here i increases downwards along rows and j to the right along columns and a cell is filled if the coefficient is 1 and empty if it is 0. Notice, for example that V i = U i whenever i = ap k with 1 ≤ a < p.This picture appears also in [STWZ21, Figure 2], where it shows the Weyl-Cartan matrix counting the decomposition multiplicities of simple sl 2 -modules in Weyl sl 2 -modules over characteristic p.
Of course, each U j ∈ R kG can be written in terms of the indecomposable modules V i using repeated applications of Theorem 9 and Proposition 5.The largest indecomposable appearing in this expression for U r comes from the term ∏ α−1 i=0 χ i r i .This largest module appearing in this term corresponds to the largest term in ∏ α−1 i=0 V p i +1 r i .Using the reduction theorem, Theorem 9, again we see that largest module appearing in our expression for U r is V r .In particular, the set {U j : 0 < j < q} is a basis for R kG and the change-of-basis matrix is lower triangular (see Fig. 1).
We have a ring homomorphism from R kG to Z simply by taking dimensions.As χ i is the difference of the indecomposable modules V p i +1 and V p i −1 , this homomorphism sends χ i to 2. In particular, image of U r under the dimension homomorphism can be realised by evaluating the polynomials at χ i = 2.As observed earlier, the quantum polynomials are such that [r] 2 = r, thus the "dimension" of U r is ∏ α−1 i=0 (r i + 1).
Example 15.Let p = 5, α = 3.We then have that where the third equality follows from the identity χ 2 1 = V 2p+1 + V 2p−1 + 2, and the final equality follows from Proposition 5, which shows V 11 In fact, we are able to give a closed form for the V r in terms of the U j .See Fig. 1 for a visual representation of this proposition.
Proposition 16.The exact form for the V i is as follows.
Proof.(Sketch) This can be shown by induction on r, with the base case being when r is a p-power, and thus V r = U r .If the equation holds for s < r, and r = p β + r for some β such that r < p β+1 , then the inductive step follows by considering Proposition 22.Let G be a cyclic group and k a field of characteristic p. Suppose G is of order n = mq, where q = p α and p m. Let {V i : 0 < i < m} be the complete set of indecomposable kC m -modules and let {V i : 0 < i < q} be the complete set of indecomposable kC q -modules.Then ∑ H<G Ind G H R kH = V i ⊗ k V j : 0 < i < m, 0 < j < q, p | j or i | m , which is the ideal generated by ∑ H<C m Ind C m H R kH and ∑ H<C q Ind C q H R kH .
In particular, rank R kG Note that the ranks of both the ring of non-induced representations and the ideal of induced representations are independent of the characteristic of the field, even though these sets may differ from field to field.
induction gives rise to a well defined (additive) group homomorphism of the representation rings, which we denote Ind G H : R kH → R kG .