Normal Reflection Subgroups of Complex Reflection Groups

We study normal reflection subgroups of complex reflection groups. Our approach leads to a refinement of a theorem of Orlik and Solomon to the effect that the generating function for fixed-space dimension over a reflection group is a product of linear factors involving generalized exponents. Our refinement gives a uniform proof and generalization of a recent theorem of the second author.

1. Introduction 1.1. Lie Groups. Hopf proved that the cohomology of a real connected compact Lie group G is a free exterior algebra on r = rank(G) generators of odd degree [Hop64]. Its Poincaré series is therefore given by (1 + q 2ei+1 ).
Chevalley presented these e i for the exceptional simple Lie algebras in his 1950 address at the International Congress of Mathematicians [Che50], and Coxeter recognized them from previous work with real reflection groups [Cox51]. This observation has led to deep relationships between the cohomology of G and the invariant theory of the corresponding Weyl group W = N G (T )/T , where T is a maximal torus in G [Ree95, RS19]-notably, where V = Lie(T ) is the reflection representation of W , S(V * ) is the algebra of polynomial functions on V , and I + W is the ideal generated by the W -invariant polynomials in S(V * ) with no constant term. For more details, we refer the reader to the wonderful survey [BG94].
1.2. Complex Reflection Groups. It turns out that the e i in Equation (1) can be computed from the generating function for the dimension of the fixed space fix(w) := dim(ker(1 − w)) for w ∈ W , via the remarkable formula: (2) w∈W q fix(w) = r i=1 (q + e i ). G is a free exterior algebra over the ring S(V * ) G of G-invariant polynomials, which gives a factorization of the Poincaré series of the G-invariant differential forms Computing the trace of the projection 1 |G| g∈G g to the subspace of G-invariants on S(V * ) ⊗ V * , specializing to u = q(1 − x) − 1, and taking the limit as x → 1 gives the Shephard-Todd result in Equation (2).
1.3. Galois twists and cohomology. More generally, the fake degree of an mdimensional simple G-module M is the polynomial encoding the degrees in which M occurs in the coinvariant algebra S(V * )/I + G ≃ C G : The fake degree of a reducible G-module is defined to be the sum of the fake degrees of its simple direct summands. The integers e i (M ) in Equation (4) are called the M -exponents of G.
Letting ζ G denote a primitive |G|-th root of unity, for σ ∈ Gal(Q(ζ G )/Q) the Galois twist V σ is the representation of G obtained by applying σ to its matrix entries. In [OS80], Orlik and Solomon gave a beautiful generalization of Equations (2) and (3) that takes into account these Galois twists (see Section 2.3).
where the λ i (g) are the eigenvalues of g ∈ G acting on V .
When σ : ζ G → ζ G is complex conjugation, Orlik and Solomon [OS80,Thm. 4.8] further connected their Theorem 1.1 to the cohomology of the complement of the corresponding hyperplane arrangement-in this case V σ ≃ V * as a G-representation, and the co-exponents e i (V * ) are the degrees of the generators of the cohomology ring of the complement of the hyperplane arrangement.
1.4. Normal Reflection Subgroups of Complex Reflection Groups. Let G ⊂ GL(V ) be a complex reflection group. We say that N G is a normal reflection subgroup of G if it is a normal subgroup of G that is generated by reflections. The main theorem of this paper, Theorem 1.4, gives a new refinement of Theorem 1.1 to accommodate a normal reflection subgroup. The following result is a special case of [BBR02], where they consider the more general notion of "bon sous-groupe distingué" in lieu of our normal reflection subgroup N of G.
Theorem 1.2. Let G ⊂ GL(V ) be a complex reflection group and let N G be a normal reflection subgroup. Then G/N = H acts as a reflection group on the vector space V /N = E.
The "bon sous-groupes distingués" of [BBR02] are precisely those normal subgroups for which the associated quotient group is a reflection group acting on the tangent space at 0 of V /N , which is a strictly weaker condition than being a normal reflection subgroup. Our proof of Theorem 1.2 in Section 3 follows the ideas of [BBR02], but specialized to our more restricted setting where the normal subgroup under consideration is actually a normal reflection subgroup. In this more restricted setting, we are able to prove the new results Theorems 1.3 and 1.4 stated below.
The technical definition of the G-module U N σ that mediates the statement of the following result is given in Definition 2.9. Since we are dealing with multiple reflection groups acting on multiple spaces, we will begin labeling exponents and degrees by their corresponding groups.
In the special case σ = 1, it is well-known that U N σ ≃ E as G-modules (see Definition 2.9 and Lemma 4.1), so that Theorem 1.3 coincides with [AW20a, Theorem 1.3] in this case. As we explain in Remark 4.2, in this special case where σ = 1 the equalities in Theorem 1.3 are compatible with the relations d i = e i + 1 between classical exponents and degrees for the three reflection groups involved.
An essential tool in our proof of Theorem 1.3 is Proposition 3.3, which gives a graded G-module isomorphism C G ≃ C H ⊗ C N relating the spaces of harmonic polynomials for N and H to that for G, which is an interesting and useful result in its own right.
Our Theorem 1.4 below generalizes the Orlik-Solomon formula from Theorem 1.1 to take into account the additional combinatorial data arising from a normal reflection subgroup. The technical definition of the G-module U N σ is again given in Definition 2.9.
Theorem 1.4. Let G ⊂ GL(V ) be a complex reflection group of rank r and let N G be a normal reflection subgroup. Let E = V /N and σ ∈ Gal(Q(ζ G )/Q). Then for a suitable choice of indexing we have where the λ i (g) are the eigenvalues of g ∈ G acting on V .
In view of Theorem 1.3, specializing to t = 1 recovers Theorem 1.1. Moreover, since U N σ ≃ E as G-modules when σ = 1 (see again Definition 2.9 and Lemma 4.1), Theorem 1.4 coincides with [AW20a, Theorem 1.5] in this case, which when similarly specialized to t = 1 recovers Equation (2). As explained in Remark 4.12, one can also recover Theorem 1.1 for the reflection group N from Theorem 1.4 by applying 1 r! ∂ r ∂t r on both sides. In the special case σ = 1, one can recover Equation (2) for the reflection group H by specializing Theorem 1.4 to q = 1 and dividing by |N | on both sides, but this same specialization does not seem to be directly related to Theorem 1.1 for H in general for arbitrary σ ∈ Gal(Q(ζ G )/Q).
Our proof of Theorem 1.4 follows a similar strategy to the one employed in [OS80]: we compute the Poincaré series for (S(V * )⊗ (U N σ ) * ) G in two equivalent and standard ways, and then obtain Theorem 1.4 from a well-chosen specialization. However, a delicate technical issue arises in that our specialization does not provide the correct contribution term-by-term in the left-hand side of Theorem 1.4. We overcome this technical difficulty by applying the results of [BLM06], where the authors develop a "twisted invariant theory" for cosets N g of a reflection group N ⊂ GL(V ) for g ∈ GL(V ) an element of the normalizer of N in GL(V ). Our proof of Theorem 1.4 applies the results of [BLM06] to the special situation where the cosets N g all come from g ∈ G, a reflection group containing N as a normal reflection subgroup, to show that our specialization argument does provide the correct contribution coset-by-coset.
In summary, we have applied results and insights from [BBR02,BLM06] in the development of new results in the invariant theory for complex reflection groups G taking into account the additional combinatorial data arising from normal reflection subgroups N G and their corresponding reflection group quotients H = G/N . The setting of [BBR02] considers more general N (their "bon sous-groups distingués"), whereas the setting of [BLM06] considers more general cosets N g (for arbitrary g in the normalizer of N ). The general study of normal reflection subgroups initiated in this paper is both natural, since it lies in the intersection [BBR02]∩ [BLM06], as well as productive, as evidenced for example by Theorems 1.3 and 1.4, the graded G-module isomorphism C G ≃ C H ⊗ C N of Proposition 3.3, and the ancillary results in Section 3 relating the amenability of different modules with respect to the groups G, N , and H.
1.5. Organization. We recall standard results about complex reflection groups in Section 2. In Section 3, we introduce normal reflection subgroups and prove ancillary results, relating spaces of harmonic polynomials and amenability with respect to different reflection groups. We prove the main results stated in the introduction, Theorems 1.3 and 1.4, in Section 4. In Section 5 we recall the caseby-case results of [Wil19] and discuss how they are obtained in a case-free way by the methods of the present paper. In Section 6 we provide a complete classification of the normal reflection subgroups of the irreducible complex reflection groups. Finally, in Section 7 we give several examples that illustrate our general results.

Invariant Theory of Reflection Groups
Let V be a complex vector space of dimension r. A reflection is an element of GL(V ) of finite order that fixes some hyperplane pointwise. A complex reflection group G is a finite subgroup of GL(V ) that is generated by reflections. A complex reflection group G is called irreducible if V is a simple G-module; V is then called the reflection representation of G. A (normal) reflection subgroup of G is a (normal) subgroup that is generated by reflections. In what follows, a G-module will always be a complex representation of G.
2.1. Chevalley-Shephard-Todd's Theorem. Let S(V * ) be the symmetric algebra on the dual vector space V * , and write S(V * ) G for its G-invariant subring. By a classical theorem of Shephard-Todd [ST54] and Chevalley [Che55], a finite subgroup G ⊂ GL(V ) is a complex reflection group if and only if S(V * ) G is a polynomial ring, and in this case S(V * ) G is generated by r algebraically independent homogeneous polynomials-the degrees d 1 ≤ · · · ≤ d r of these polynomials are invariants of G.
is a complex reflection group if and only if there exist r = dim(V ) homogeneous algebraically independent polynomials G 1 , . . . , G r such that S(V * ) G = C[G 1 , . . . , G r ]. In this case, Let I + G ⊂ S(V * ) denote the ideal generated by homogeneous G-invariant polynomials of positive degree. In [Che55], Chevalley proved that, as an ungraded G-module, S(V * )/I + G affords the regular representation of G. Since I + G is G-stable, we may choose a G-stable complement C G ⊂ S(V * ), so that S(V * ) ≃ I + G ⊕ C G as graded G-modules, and C G is a graded version of the regular representation of G. Chevalley also proved in [Che55] that S(V * ) ≃ S(V * ) G ⊗ C G as graded G-modules. A canonical choice for such a G-stable complement C G is the space of G-harmonic polynomials [LT09, Corollary 9.37], that is, polynomials in S(V * ) that are annihilated by all G-invariant polynomial differential operators with no constant term [LT09, Definition 9.35].
Remark 2.2. The space C G of G-harmonic polynomials is stabilized by the normalizer of G in GL(V ) [LT09,Proposition 12.2]. This fact will be essential in our treatment of normal reflection subgroups.
Our choice of notation C for the space of harmonic polynomials, instead of the more common and natural H used in the literature, is meant to avoid unfortunate phonetic confusion with the quotient group H = G/N that will play a prominent role in the rest of the paper.
2.2. Solomon's Theorem. We recall the following celebrated theorem of Solomon.
is a free exterior algebra over the ring of G-invariant polynomials: Computing the trace on S(V * )⊗ V * of the projection to the G-invariants 1 |G| g∈G g gives a formula for the Poincaré series as a sum over the group.
2.3. Orlik-Solomon's Theorem. The reflection representation V of G ⊂ GL(V ) can be realized over Q(ζ G ), where ζ G denotes a primitive |G|-th root of unity, in the sense that there is a choice of basis for V with respect to which G ⊂ GL r (Q(ζ G )).
For σ ∈ Gal Q(ζ G )/Q , the Galois twist V σ of V is the representation of G on the same underlying vector space V obtained by applying σ to the matrix entries of g ∈ GL r (Q(ζ G )). Alternatively and equivalently, one can define V σ by applyingσ to the matrix entries of g in terms of any basis of V , forσ any extension of σ to a field automorphism of C.
In [OS80], Orlik and Solomon gave the following generalization of Theorem 2.3.
where the degrees of the homogeneous generators of (U G σ ) * : Computing the Poincaré series in two ways as in Corollary 2.4 gives the following formula.
is called amenable. This amenability condition can be shown to be equivalent to the requirement that (S(V * ) ⊗ M * ) G be a free exterior algebra over S(V * ) G . Since, in particular, Galois twists V σ of the reflection representation V of G are amenable, the following theorem generalizes Theorem 2.5.
where (U G M ) * := (C G ⊗ M * ) G and the degrees of the homogeneous generators of (U G M ) * are e G i (M ), the M -exponents of G. From this, one can pursue the usual strategy of computing the Poincaré series of (S(V * ) ⊗ M * ) G in two different ways to obtain the following.
However, it is no longer clear how to specialize Corollary 2.8 in the same way as Corollaries 2.4 and 2.6 to obtain an analogue of Equation (2) and Theorem 1.1 in this generality.
Definition 2.9. Let M be a G-module. We define the Orlik-Solomon space U G M to be the dual G-module to (U G M ) * := (C G ⊗ M * ) G . In the special case where M = V , we write

Normal Reflection Subgroups
The following theorem is a special case of results in [BBR02] (where they consider the more general notion of "bon sous-groupe distingué" in lieu of our normal reflection subgroup N of G). We emphasize that our proof follows the ideas in [BBR02], specialized to our more restricted setting. Proof. We claim that there exist homogeneous generators N 1 , . . . , N r of S(V * ) N such that E * = span C {N 1 , . . . , N r } is H-stable. By Theorem 2.1, the ring of Ninvariants S(V * ) N = C[Ñ 1 , . . . ,Ñ r ] for some homogeneous algebraically indepen-dentÑ i . Let I + ⊂ S(V * ) N be the ideal generated by homogeneous N -invariants of positive degree. Then both I + and I 2 + are H-stable homogeneous ideals, and therefore the algebraic tangent space I + /I 2 + to E = V /N at 0 inherits a graded action of H that is compatible with the (graded) quotient map π : I + ։ I + /I 2 + . Hence there exists a graded H-equivariant section ϕ : I + /I 2 + → I + . Letting N i = ϕ • π(Ñ i ) we see that N 1 , . . . , N r are still homogeneous algebraically independent generators for S(V * ) N with deg(N i ) = deg(Ñ i ) and E * := span C {N 1 , . . . , N r } is H-stable. Let x = {x 1 , . . . , x r } denote a dual basis for V and N = {N 1 , . . . , N r } denote an H-stable basis for E * as above. Since the action of H on the polynomial ring S(E * ) = S(V * ) N is obtained from the action of G on S(V * ), it preserves x-degrees as well as N-degrees. Therefore we may choose the fundamental G-invariants G i (x) ∈ S(V * ) G = (S(V * ) N ) H = S(E * ) H to be simultaneously x-homogeneous and N-homogeneous, so that H i (N) := G i (x) form a set of Nhomogeneous generators for the polynomial ring S(E * ) H . Since any algebraic relation f (H 1 , . . . , H r ) = 0 would result in an algebraic relation f (G 1 , . . . , G r ) = 0, the N-homogeneous H i (N) must be algebraically independent. By Theorem 2.1, H is a complex reflection group.
We see that in fact there exist algebraically independent (bi)homogeneous polyno- Having chosen the fundamental G-invariants G i to have degrees d G 1 ≤ · · · ≤ d G r , we implicitly index the N -degrees d N i and H-degrees d H i so that Equation (5) is satisfied.
Remark 3.2. Unlike in the real case [Gal05,BD10], H is not necessarily a reflection subgroup of G or even a subgroup of G. A counterexample is given by G 8 = G⊲N = G(4, 2, 2), so that G/N ≃ S 3 -but S 3 is not a subgroup of G 8 .
3.1. Harmonic polynomials. The space of N -harmonic polynomials C N ⊂ S(V * ) is G-stable [LT09, Proposition 12.2] and isomorphic to the regular representation of N [LT09, Corollary 9.37]. The space of H-harmonic polynomials C H ⊂ S(E * ) is bigraded, by x-degree as well as by N-degree, and therefore it admits a Cbasis of H-harmonic polynomials that are simultaneously x-homogeneous and Nhomogeneous. The following result elaborates on [BLM06, Corollary 8.4] in our present setting.
Proof. Putting together the isomorphisms S(V * ) ≃ S(E * ) ⊗ C N as graded Nmodules, S(E * ) ≃ S(E * ) H ⊗ C H as bigraded H-modules (equivalently, as bigraded G-modules of N -invariants), and S(V * ) G = S(E * ) H , we obtain the isomorphism S(V * ) ≃ S(V * ) G ⊗ C H ⊗ C N as graded G-modules. Letting π : S(V * ) → S(V * )/I + G denote the canonical projection, we see that C⊗C H ⊗C N must surject onto the image S(V * )/I + G ≃ C G , because S(V * ) G is generated as a C-algebra by the generators of the ideal I + G . But this surjection C H ⊗ C N → C G of graded G-modules must then be an isomorphism, because Since C H ⊂ S(E * ) consists of N -invariants, we have (C H ⊗ C N ) N = C H ⊗ (C N ) N = C H ⊗ C, and therefore (C G ) N ≃ C H as graded H-modules, as claimed.
Remark 3.4. It follows from the isomorphism S(V * ) ≃ S(V * ) G ⊗ C G as graded G-modules that the Poincaré series of C G can be written [Che55, Theorem B] Since |G| = d 1 · · · d r , it is natural to ask for a combinatorial interpretation of Hilb(C G ; q) as a weighted sum over the elements of G. When G is a real reflection group, G acts simply transitively on the connected components of its real hyperplane complement. Assigning some base connected component R e to the identity element e ∈ G gives a bijection between group elements and regions sending g ↔ R g , and we can define the statistic inv(g) to be the number of inversions of g ∈ Gthat is, the number of hyperplanes separating the connected component R g from R e . Then the Poincaré series of C G has the well-known interpretation In principle, our Proposition 3.3 gives a new method for producing such combinatorial interpretations: given interpretations of Hilb(C H ; q) and Hilb(C N ; q) and coset representatives {h} for G/N , we can write Remark 3.5. It follows from Proposition 3.3 that for any G-module M the (dual) Orlik-Solomon space (U N M ) * = (C N ⊗ M * ) N as in Definition 2.9 can be considered as an H-module, or (equivalently) as a G-module of N -invariants.
The following result is useful in determining the Orlik-Solomon space U N M in particular examples (cf. Section 7) up to graded G-module isomorphism.
Lemma 3.6. Let η N : S(V * ) → C N denote the G-equivariant projection onto the space of N -harmonic polynomials. Let M be a G-module of rank m, and suppose thatũ 1 , . . . ,ũ m form a homogeneous basis for Let y 1 , . . . , y m be a basis for M * , and writeũ i := m j=1ã ij ⊗ y j and u i = m j=1 a ij ⊗ y j , wherẽ a ij ∈ S(V * ) and a ij ∈ C N , and every non-zeroã ij and a ij is homogeneous of degree e N i (M ). Since theũ i and the u i form bases for Since the kernel of η N : S(V * ) → C N is precisely the ideal generated by homogeneous N -invariants of positive degree, it follows that 3.2. Numerology. The following consequence of Proposition 3.3 establishes the first equation of Theorem 1.3 in more generality.
Corollary 3.7. Let M be a G-module and define U N M as in Definition 2.9. For a suitable choice of indexing we have . Proof. Applying Proposition 3.3, we see that Let m be the rank of M , and let y 1 , . . . , y m be a basis of Remark 3.8. Regardless of whether a G-module M of rank m is amenable, we always have a natural S(V * ) G -linear injective homomorphism The amenability of M as a G-module is therefore precisely the requirement that a ∈ C be a constant polynomial. For brevity and convenience, we will summarize this equivalent characterization of amenability in the following Lemma 3.9 (cf. [BLM06, Theorem 2.10]).
The decomposition C G ≃ C H ⊗ C N from Proposition 3.3 and its Corollary 3.7 have many useful consequences. Although the following result can be proved more directly by appealing to [BLM06,Corollary 8.7], we provide a full proof.
By Lemma 3.9, the amenability of M as a G-module and the amenability of M as an H-module are respectively equivalent to which proves (ii).
Remark 3.11. Since the Orlik-Solomon space U N M of Definition 2.9 is trivial as an N -module, it follows from Lemma 3.10 that U N M is amenable as a G-module if and only if it is amenable as an H-module. From now on we will just say that U N M is amenable whenever these equivalent conditions hold.
Proposition 3.12. Let M be a G-module and define U N M as in Definition 2.9. Then: Proof. Let m denote the rank of M . Define the amenability defects Summing the first equation over i = 1, . . . , m and subtracting the second equation, we obtain , which together with Equation (6) implies that η ≤ γ. Therefore if M is amenable as a G-module then U N M is amenable. If M is amenable as an N -module, so that ν = 0, then we see that , and we obtain from Equation (6) that η = γ in this case. Hence if M is amenable as an N -module and U N M is amenable, then M is amenable as a G-module.
Remark 3.13. It is not true in general that M being amenable as a G-module implies that M is amenable as an N -module. For a counterexample, let G = C a = c and N = C d = c e with a = de, acting on V = C in the standard reflection representation by c → ζ a , a primitive a-th root of unity. Consider the G-module M is amenable as a G-module. However, 2 M is trivial as an N -module, and therefore Suppose M is an amenable G-module of rank m. Then by Proposition 3.12 and Lemma 3.10, the Orlik-Solomon space U N M of Definition 2.9 is amenable (considered either as a G-module or as an H-module). Let us again write from which we obtain more generally a graded G-module decomposition of This results in an obvious bigrading of (U N M ) * and a corresponding tri-grading of the associative algebra S( We will follow the usual strategy of computing this Poincaré series in two different ways to deduce combinatorial formulas. Let us denote as before d G 1 , . . . , d G r the degrees of the fundamental G-invariants generating S(V * ) G as a polynomial algebra.
Proof. We proceed as in the proof of Corollary 3.7: let y 1 , . . . , y m be a basis of , our result follows.
To simplify notation, for e ∈ E N M we denote by (U N M ) e the homogeneous component of U N M corresponding to the dual of (U N M ) * e , rather than the more natural but cumbersome graded dual Proof. For g ∈ G, let us write We know that ℓ≥0 tr g|S For each e ∈ E N M we have that p≥0 tr g| Our result follows after taking the average over g ∈ G on each side.

Proofs of the Main Theorems
We are now in a position to apply the results of Section 3 to prove the main results announced in the introduction. Fix G ⊂ GL(V ) a complex reflection group acting by reflections on the vector space V of dimension r. Let N G be a normal reflection subgroup with quotient H = G/N , which acts by reflections on E = V /N . For σ ∈ Gal(Q(ζ G )/Q), where ζ G denotes a primitive |G|-th root of unity, write V σ for the Galois twist of V (as defined in Section 2.3). As in Definition 2.9, we write U N for the dual of (C N ⊗ V * ) N , and more generally U N σ for the dual of The first equality is Corollary 3.7 applied to the G-module M = V σ , and the last equality follows from the observations in Remark 3.1.
Let us establish the second equality. Let N σ 1 , . . . , N σ r denote a basis for (E σ ) * as a G-module. We will show that there exist a ij ∈ S(E * ) such that u i := r j=1 a ij ⊗N σ j form an N-homogeneous basis for (C H ⊗ (E σ ) * ) H (so that each non-zero a ij is Nhomogeneous of N-degree e H i (E σ )), and moreover a ij = 0 whenever d N j = d N i and each a ij ∈ S(N d N i ), where as before N d denotes the set of fundamental N -invariants 10, the existence of such a ij will establish our claim, since each non-zero a ij as above will then be It follows from the above observations that When σ = 1, the G-module U N σ in Definition 2.9 admits a more concrete description.   The e G i (E) are the x-degrees of a homogeneous basis for (C G ⊗ E * ) G . By Lemma 3.10, (C G ⊗ E * ) G ≃ (C H ⊗ E * ) H , and therefore the (η H ⊗ 1)(dH i ) serve as a homogeneous basis for (C G ⊗ E * ) G , where η H : S(E * ) → C H denotes the projection onto the space of H-harmonic polynomials (cf. Lemma 4.1). Hence for any k such that ∂Hi ∂N k = 0, the e G i (E) are given by the deg  Note that the exponents and degrees of N ≃ W (D 4 ) must be reordered for the identities to hold (see Remark 3.1).
Remark 4.4. When σ = 1, U N σ ≃ E σ as G-representations by Lemma 4.1-but it is not always the case that U N σ ≃ E σ as G-representations for more general σ. For example, take the cyclic groups G = C a ⊲ C d = N for d|a, with σ being complex conjugation. Then (U N σ ) * = span C {x ⊗ x σ }, on which G acts trivially. We discuss this in more detail in Section 7.
where the λ i (g) are the eigenvalues of g ∈ G acting on V .
We refer to the left-hand side of Theorem 1.4 as the sum side, and to the righthand side as the product side. We will prove Theorem 1.4 by computing the limit as x → 1 of the specialization y → x t and u → qt(1 − x) − 1 of the tri-graded Poincaré series P G σ (x, y, u) := P G V σ (x, y, u) from Definition 3.14 in two different ways to obtain the sum side and the product side separately.
Since M = V σ is amenable as a G-module (and as an N -module) by [OS80, Thm. 2.13], we can apply both Propositions 3.15 and 3.16 in this case to obtain . , e N r (V σ )} denotes the set of V σ -exponents of N as before. The product side of Theorem 1.4 follows immediately.
Proof. We compute: Our argument for the sum side of Theorem 1.4 is more delicate. The reason for this is that for g ∈ G the fixed space of g acting on U N σ often has larger dimension than the fixed space of g acting on V σ , which causes many terms in the term-byterm limit to be zero.
It turns out, as we will now show, that the contributions are correct when taken coset-by-coset. For this, let us define for each coset N g ∈ H = G/N the twisted Poincaré series so that P G σ (x, y, u) = 1 |H| N g∈H P N g σ (x, y, u). The following result is proved along the same lines as Proposition 3.16 and serves as an equivalent definition of P N g σ (x, y, u). Proof. For each n ∈ N , ℓ,e,p≥0 and our result follows from taking the average over n ∈ N .
Definition 4.7. We will adopt the following notation for the rest of this section. Let g ∈ G. We will denote byλ 1 (g), . . . ,λ r (g) the set of eigenvalues of g on V * . We choose once and for all: a g-eigenbasis of fundamental N -invariants N i ∈ E * such that deg x (N i ) = d N i and gN i = ǫ g i (E)N i ; and a g-eigenbasis for the Orlik-Solomon space (see Definition 2.9) u N i ∈ (U N σ ) * such that deg x (u N i ) = e N i (V σ ) and gu N i = ǫ g i (U N σ )u N i . We observe as in [BLM06] that the multisets of pairs ǫ g i (E), d N i i = 1, . . . , r and ǫ g i (U N σ ), e N i (V σ ) i = 1, . . . , r depend only on σ and the coset N g ∈ H, and not on the choice of coset representative g ∈ N g.
Proof. We write P N g σ (x, y, u) as in Lemma 4.6. First observe that, since U N σ is N -invariant, for any n ∈ N we have that Since, for n ∈ N , or equivalently that for each ℓ ≥ 0 we have that 1 |N | n∈N tr ng|Sym ℓ (V * ) = tr g|S(E * ) ℓ .
To see this, note that the operator on Sym ℓ (V * ) given by Proof. The first equality is a special case of [BLM06, Theorem 3.1] (where we note that the change in sign from the −u in their notation to +u in our notation is harmless), and the second equality follows directly from Proposition 4.8.
We obtain the following crucial specialization of Proposition 4.8, which exploits the similarity between Proposition 4.9 and Proposition 4.8, and is inspired by [BLM06, Theorem 3.3].
Proposition 4.10. For g ∈ G, with notation as in Definition 4.7, Proof. Let us agree to index the pairs (ǫ g i (E), d N i ) and (ǫ g i (U N σ ), e N i (V σ )) in the multisets from Definition 4.7 such that ǫ g . By Proposition 4.8, the left-hand side of Equation (7) is By [BLM06, Proposition 3.2], fix U N σ (g) ≥ fix E (g). We will compute the above limit for the partial products ranging over .
If the inequality fix U N σ (g) > fix E (g) is strict, so that ǫ g i (E) = 1 = ǫ g i (U N σ ) for fix E (g) + 1 ≤ i ≤ fix U N σ (g), then we see that for each such i the limit of the corresponding factor is In any case, we have shown that the left-hand side of Equation (7) is t fixE (g) ·P (q) for some P (q) ∈ C[q]. To conclude the proof, it suffices to compare the left-and right-hand sides of Equation (7) at t = 1. For this, we observe as in [BLM06,Theorem 3.3] that, as a consequence of Proposition 4.9 and the arguments of [OS80, Theorem 3.3] that are now standard, Corollary 4.11 (Sum side specialization).
Proof. Let g 1 , . . . , g |H| ∈ G be a full set of coset representatives for G/N = H. Since P G σ (x, y, u) = 1 |H| |H| j=1 P N gj σ (x, y, u) and fix E (g j ) = fix E (ng j ) for any n ∈ N , it follows from Proposition 4.10 that and our result follows after replacing g with g −1 .
Remark 4.12. As mentioned in the introduction, the formula of Theorem 1.4 corresponding to the special case σ = 1 becomes [AW20a, Theorem 1.5]: which recovers Equation (2) for the reflection group G by evaluating at t = 1, since E ≃ U N as G-modules in this case by Lemma 4.1 and e N i (V ) + e G i (E) = e G i (V ) by Theorem 1.3, as discussed in Remark 4.2.
On the other hand, specializing Equation (8) at q = 1 and dividing by |N | on both sides again recovers Equation (2), but this time for the reflection group H: the sum-side follows from observing that fix E (N g) = fix E (g) for every N g ∈ H. The product side follows from the equality In fact, it is also possible to recover Equation (2) for the reflection group N from Equation (8). Since H acts faithfully on E, we have N = {g ∈ G | fix E (g) = r}, and therefore applying 1 r! ∂ r ∂t r to the sum-side of Equation (8) recovers the sum-side of Equation (2) for N . That the analogous result obtains for the product side follows from the well-known higher Leibniz rule for the Hasse-Schmidt derivations Similarly, as we mentioned in the introduction, for arbitrary σ ∈ Gal(Q(ζ G )/Q) the formula of Theorem 1.4 recovers Theorem 1.1 by evaluating at t = 1, since e N i (V σ ) + e G i (U N σ ) = e G i (V σ ) by Theorem 1.3. The above arguments also show that we recover Theorem 1.1 for the reflection group N again by applying 1 r! ∂ r ∂t r to both sides of Equation (9). It would be desirable to recover Theorem 1.1 for the reflection group H from Theorem 1.4 in analogy with the case σ = 1, by evaluating Equation (9) at q = 1 and dividing by |N | on both sides. But in this case we obtain something else: letting g 1 , . . . , g |H| ∈ G be a full set of coset representatives for H = G/N , evaluating Equation (9) at q = 1 yields which does not immediately compare to the statement of Theorem 1.1 for the reflection group H: where compatibly with Definition 4.7 theǭ gj i (E) denote the eigenvalues of g j ∈ G acting on E. We can also explain the two exceptional factorizing representations from [Wil19]:

Reflexponents revisited
• Following the conventions of [Mic], G = G 13 = s, t, u was observed in [Wil19, Section 5.1] to have a two-dimensional representation with the factorizing property. The group N = gsg −1 : g ∈ G (fixing the conjugacy class H s ) is a normal subgroup isomorphic to G(4, 2, 2) and the quotient G/N ≃ W (A 2 ) ≃ S 3 gives the unexplained two-dimensional factorizing representation in this case. • For G = G(ab, b, r) = s, t 2 , t ′ 2 , t 3 , . . . , t r with a, b > 1 and r > 2, we can take N = gsg −1 : g ∈ G . N is a normal subgroup of G isomorphic to (C a ) r (it consists of diagonal matrices whose diagonal entries are a-th roots of unity). The quotient G/N ≃ G(b, b, r) gives the unexplained rdimensional factorizing representation in [Wil19, Section 5.2].

Classification of Normal Reflection Subgroups
In this section, we state the classification of normal reflection subgroups of irreducible complex reflection groups.
Recall that G(ab, b, r) is given in its standard reflection representation as the set of r × r monomial matrices whose every non-zero entry is an (ab)-th root of unity and in which the product of the non-zero entries is an a-th root of unity. The following theorem identifies the normal reflection subgroups of the infinite family G(ab, b, r). For rank r = 2, the normal subgroups and quotients are: For rank r ≥ 3, the normal subgroups and quotients are: (r.a) G(ab, b, r)/(C d ) r ≃ G(eb, b, r) and (r.b) G(ab, b, r)/G(ab, db, r) ≃ C d .
In cases (r.a) for r ≥ 1, the polycyclic group N = (C d ) r is included in G as diagonal matrices with each non-zero entry a d-th root of unity. In cases (r.b) for r ≥ 2, the normal reflection subgroup N = G(ab, db, r) is included in G via its standard reflection representation in C r . In case (2.c), the normal reflection subgroup N = G(a, d, 2) occurs twice in G: once via its standard reflection representation, and once as the group generated by the reflections {diag(ζ k e , 1), diag(1, ζ k e ) | k = 1, . . . , e − 1} (when e = 1) along with the reflections 0 Independently of the factorization a = de, these two copies of N are conjugate in GL 2 (C) by the matrix diag(1, ζ 2a ), which normalizes G.
The exceptional (that is, primitive) irreducible complex reflection groups G and their normal reflection subgroups N are listed in Table 1. This classification was computed with Sage [The18] using the code available at [AW20b]. Most examples occur in rank r = 2. In rank r ≥ 3, every reflection has order 2 or 3 [LT09, Theorem 8.4], and the only exceptional groups with more than a single orbit of reflections are G 26 and G 28 [LT09, Table D.2], which leads to the four non-trivial exceptional examples listed in Table 1 in rank r ≥ 3.
An isomorphism type of N is unique up to conjugation in GL r (C) sending isomorphic normal reflection subgroups to each other while stabilizing the reflection representation of G. In fact, there are only two instances in the exceptional groups (G 5 ⊲ G 4 and G 28 ⊲ G (2, 2, 4)) where an isomorphism type appears as a normal subgroup more than once. We gather the situations where the same isomorphism type of N occurs more than once as a normal reflection subgroup of G in the following result, where N (i) denote the different isomorphic copies of the same normal reflection subgroup N ⊳ G.
Theorem 6.2. Suppose G is an irreducible complex reflection group admitting normal reflection subgroups N (1) , . . . , N (k) for k ≥ 2 that are pairwise isomorphic (as abstract groups) but not equal in G. Then k ∈ {2, 3} and the normalizer of G in GL(V ) permutes {N (1) , . . . , N (k) } transitively under conjugation. Moreover, precisely one of the following possibilities occurs.
Proof. By Theorem 6.1 and Table 1, the possibilities above are exhaustive, and it is clear that they are mutually exclusive. It remains to show that any two isomorphic normal reflection subgroups of G are conjugate under an element of the normalizer of G. In every situation listed above, except for G = G 28 , we can find a complex reflection group W that contains G as a normal reflection subgroup (in its standard reflection representation) but which does not contain (the isomorphism type of) N as a normal reflection subgroup. Since there are at most three isomorphic copies of N in each case, they must then form a single conjugacy class in W .
In cases (1) and (2), we can take W to be any of G 6 ; G 7 ; G 8 ; G 9 ; G 10 ; G 11 ; G 13 ; or G 15 ; since all of these contain G = G(4, 2, 2) as their only imprimitive normal reflection subgroup according to Table 1.
In case (4), we can take W to be any of G 7 ; G 10 ; G 11 ; G 14 ; or G 15 ; since all of these contain G 5 as a normal reflection subgroup, but do not normalize G 4 .
Finally, in case (5) it is not possible to find a complex reflection group W containing G = G 28 as a normal reflection subgroup, since G 28 is the only irreducible complex reflection group admitting a non-trivial normal reflection subgroup in rank r ≥ 4. To see that the two isomorphic copies of N ≃ G (2, 2, 4) in G = G 28 are conjugate under an element of the normalizer of G in GL 4 (C), consider the set of reflecting hyperplanes for G 28 , which are the orthogonal complements (with respect to the standard Hermitian inner product in C 4 ) of the lines in L 1 ∪ L 2 ∪ L 3 defined by (cf. [LT09, Section 7.6.2]): L 2 = C · 1 2 (e 1 ± e 2 ± e 3 ± e 4 ) ; and where e i denotes the standard basis vector with 1 in the i-th entry and 0 elsewhere. There are two orbits of reflecting hyperplanes for G 28 = W (F 4 ) (the Weyl group of type F 4 and a real reflection group), corresponding to L 1 ∪ L 2 (the lines spanned by the short roots) and L 3 (the lines spanned by the long roots). The reflections around the 12 hyperplanes corresponding to L 3 generate the normal reflection subgroup N (1) = G(2, 2, 4) acting in its standard reflection representation. The other normal reflection subgroup N (2) is generated by the reflections around the other 12 hyperplanes corresponding to L 1 ∪ L 2 . To conclude the proof, note that the real orthogonal matrix exchanges the two G 28 -orbits of reflecting hyperplanes L 1 ∪ L 2 ↔ L 3 (identifying L 1 with the set of lines C · (e i ± e j ) ∈ L 3 such that j − i = 2). Hence, P normalizes G 28 and exchanges N (1) and N (2) under conjugation.

Examples
In this section we illustrate our results with examples, beginning with the cyclic groups in Section 7.1, continuing with the infinite family in Section 7.2, and concluding with a non-well-generated exceptional example in Section 7.3. 7.1. Cyclic Groups. Consider G = C a = c , the cyclic group of order a, acting on V = C via c → ζ −1 a , where ζ a is a primitive a-th root of unity. Verifying Theorem 1.1 and determining the Orlik-Solomon space U G σ from Definition 2.9 for σ ∈ Gal(Q(ζ a )/Q) is already an interesting calculation in this case.
Let σ ∈ Gal(Q(ζ a )/Q) act as σ : ζ a → ζ s a , where s ∈ N is coprime to a. Although it is sufficient to only consider s ∈ {1, . . . , a − 1}, it will be essential to allow more general positive exponents s in the description of the action of σ on different roots of unity when we begin considering (normal reflection) subgroups of G shortly. We compute the identity of Theorem 1.1 in this example: On the other hand, we verify that e G i (V σ ) = a s a −s by exhibiting x a⌈ s a ⌉−s ⊗x σ as a basis vector for the dual (U G σ ) * of the Orlik Solomon space U G σ , where x and x σ denote basis vectors for V * and (V σ ) * , respectively. More generally, we have the following.
Lemma 7.1. For G = C a = c the cyclic group of order a acting on its reflection representation V = C by c → ζ −1 a , and for any s ∈ Z (not necessarily coprime to a), the V ⊗s -exponent of G is e G 1 (V ⊗s ) = a s a − s.
Proof. Note that a s a − s ∈ {0, . . . , a − 1} is congruent to −s ( mod a). Letting x s denote a basis vector for V ⊗s , we see that c(x s ) = ζ s a x s and therefore x a⌈ s a ⌉−s ⊗ x s is a basis vector for Suppose now that a = de and consider N = c e ≃ C d , which is a normal reflection subgroup of G with quotient H = G/N ≃ C e . We denote by ζ d := ζ e a and ζ e := ζ d a , so that N acts on V * via c e → ζ d and H = cN acts on Taking again s ∈ N coprime to a and letting σ ∈ Gal(Q(ζ a )/Q) be given by σ : ζ a → ζ s a , we have that E σ is the one-dimensional representation of G defined by c → ζ −ds Using the explicit descriptions of the actions of N and G on V σ , and the actions of G and H on U N σ and E σ described above, the three equalities in Theorem 1.3 become the following numerological statements.
Corollary 7.2. Let a = de and s ∈ N be coprime to a. Then The first (and only non-trivial) equality in this case is equivalent to the identity ⌈ s d ⌉ e = s de , which holds more generally for e ∈ N and s, d ∈ R.
Our Theorem 1.4 in this situation states that the following expressions are equal.
The equality of the coefficients of t was already verified above with a in place of d. To verify the equality of constant terms one could proceed for example by noting that the same arguments show that the left-hand side is equal to and then appealing either to Corollary 7.2 or to more general properties of ceilings to obtain the equality ⌈ s d ⌉ e = s a . We conclude our discussion of the cyclic case with a concrete illustration of the subtlety involved in proving (in Corollary 4.11) that the sum side of Theorem 1.4 provides the correct contribution coset-by-coset, but not term-by-term as in the proof of Theorem 1.1.
To compare the two situations, we compute the trace of ng = c ek · c j ∈ N g (for 0 ≤ k < d and 0 ≤ j < e) acting on S(V * ) ⊗ (V σ ) * and S(V * ) ⊗ (U N σ ) * : Continuing the example with a = 6, d = 2, e = 3, and s = 5, since ⌈ s d ⌉ = e = 3, after specializing y → x t , u → qt(1 − x) − 1, and taking the limit as x → 1, every term except for the identity vanishes-in particular, each element of G does not contribute the "correct amount" specified by the sum side of Theorem 1.4: Here the only coset that provides a non-trivial contribution to the sum side of Theorem 1.4 is the trivial coset N , as predicted by the computation of the sum side of Theorem 1.1, but this non-trivial contribution of qt + t for the whole coset N is concentrated on the identity element c 0 ∈ N alone, which according to the sum side of Theorem 1.4 should have only contributed qt, whereas the non-trivial element c 3 ∈ N did not provide the correct contribution of 1−λ1(c 3 ) σ 1−λ1(c 3 ) q fixV (c 3 ) t fixE (c 3 ) = t specified by the sum side of Theorem 1.4. 7.2. The Infinite Family G(ab, b, r). We begin by defining an ad-hoc operation on G-modules for G = G(ab, b, r) (whose definition was given in Section 6) that will allow us to succintly identify the Orlik-Solomon spaces U N σ of Definition 2.9 for the two kinds of normal reflection subgroups N of G listed in Theorem 6.1 that occur in all ranks r ≥ 2.
Definition 7.3. For n ∈ Z, let µ n : G → G be the group endomorphism obtained by raising each non-zero matrix entry to the n-th power. Given any representation ρ : G → GL(W ), we define the fake tensor power W ⊠n as the representation ρ • µ n : G → GL(W ).
Remark 7.4. Note that in general the "fake n-th power map" µ n used in Definition 7.3 will be neither injective nor surjective when gcd(ab, n) = 1. In the case where gcd(ab, s) = 1, note that V ⊠s ≃ V σ , the Galois twist corresponding to σ : ζ ab → ζ s ab . Although the fake tensor power operation could be expressed in terms of more systematic constructions (interpreting G(ab, b, r) as an index-b subgroup of the wreath product C ab ≀ S r ), we have preferred our ad hoc definition for its simplicity and concreteness.
In the following result, we identify the G-module U N σ when N = C r d as in Theorem 6.1(r.a) for r ≥ 2.
Proof. For this choice of N , the fundamental N -invariants are N i = x d i for 1 ≤ i ≤ r, and we obtain the basis * as in Section 7.1. Fix g ∈ G and ℓ ∈ {1, . . . , r}, and suppose that g(x ℓ ) = ζ k ab x j for some 0 ≤ k < ab and j ∈ {1, . . . , r}. Then from the explicit descriptions of E * and (U N σ ) * above we see that g(N ℓ ) = ζ kd ab N j and g(u N ℓ ) = ζ As a special case of Proposition 7.5, U N σ ≃ E when N = C r d and s < d. x abi j for 1 ≤ i < r and G r = (x 1 · · · x r ) a .
In the following result we determine the Orlik-Solomon space U G σ of Definition 2.9 up to graded cryptomorphism.
Proof. By Lemma 3.6 (applied in the special case where N = G), it suffices to show that theũ G i form a basis for (S(V * ) ⊗ (V σ ) * ) G as a free S(V * ) G -module. Since the group G(ab, b, r) acts on V * by permutation of the x i and multiplication by (ab)-th roots of unity, it is clear that theũ or equivalently such thatũ G i = r j=1 p G ij · u G j . It is clear from the form of theũ G i that they are C-linearly independent, which implies that det(p G ij ) = 0. We claim that for some 0 = c ∈ C. By Gutkin's Theorem [LT09,Theorem 10.13], for some 0 = c ∈ C, where R G denotes the set of reflecting hyperplanes for G, L H ∈ V * denotes a linear form defining H, and C(H, V σ ) is defined as follows.
Denoting by r H = G H < G the cyclic subgroup of G that stabilizes H pointwise, ab x j with i = j and 0 ≤ ℓ < ab − 1, the cyclic generator r H has order 2 and V σ ≃ λ ⊕ C ⊕(r−1) as a G H -module, and therefore C(H, V σ ) = 1 in this case. If a > 1, then we have the additional reflecting hyperplanes defined by L H = x i ; in this case, the cyclic generator r H has order a and (V σ ) * ≃ λ ⊗(⌈ s a ⌉a−s) ⊕ C ⊕(r−1) as a G H -module (cf. Lemma 7.1), and therefore C(H, V σ ) = s a a − s. This concludes the proof of Equation (11).
We see by direct inspection that which implies that deg(det(p G ij )) = 0, and since det(p G ij ) = 0 as we had already seen, it follows that [p G ij ] ∈ GL r (S(V * ) G ), as we wanted to show.
Remark 7.7. Note that when s = ab − 1, so that σ acts by complex conjugation, the basisũ i G for (Ũ G σ ) * in Theorem 7.6 agrees with the one computed in [OS80, Section 6].
In the following result, we identify the G-module U N σ when N = G(ab, db, r) as in Theorem 6.1(r.b) for r ≥ 2.
Proof. Note that for this choice of G and N , with fundamental invariants defined as in Equation (10), we have N i = G i for i = 1, . . . , r − 1, and N r = (x 1 · · · x r ) e while G r = (x 1 · · · x r ) a . The matrix diag(ζ a , 1, . . . , 1) ∈ G maps to the cyclic generator c ∈ C d ≃ H, acting trivially on span C {N 1 , . . . , N r−1 } and mapping c : a is a primitive d-th root of unity. On the other hand, we see that the spaceŨ N σ = span{ũ N 1 , . . . ,ũ N r } constructed in Theorem 7.6 is G-stable and theũ N i form a homogeneous basis for (S(V * ) ⊗ (V σ ) * ) N as a free S(V * ) N -module, and therefore it is isomorphic to (U N σ ) * as a graded G-module by Lemma 3.6. We see that diag(ζ a , 1, . . . , 1) ∈ G acts trivially on span C {ũ N 1 , . . . ,ũ N r−1 } and maps c : r . Note that we do not necessarily have that ⌈ s e ⌉ is relatively prime to d. When d = a so that e = 1 and N = G(ab, ab, r), G/N ≃ C a = c acts by c : u N r → ζ −s a u N r , so that U N σ ≃ E σ . Another special case occurs when s < e, so that U N σ ≃ E.
In the following result, we identify the representation U N σ when G = (2a, 2, 2) and N = G(a, d, 2) as in Theorem 6.1(2.c). In this case it is not possible to write U N σ as a fake tensor power (Definition 7.3) of E in general. Proposition 7.9. Let a = de and N = G(a, d, 2) ⊳ G(2a, 2, 2) = G, and fix σ : where λ 2 is the standard reflection representation of C 2 and λ d is the standard as H-modules.
We see that h i (N j ) = N j if i = j, h 1 (N 1 ) = −N 1 , and h 2 (N 2 ) = ζ d N 2 , and therefore h 1 and h 2 map to the generators of the cyclic factors in the bicyclic quotient group H ≃ C 2 × C d . By Theorem 7.6 and Lemma 3.6,Ũ N σ = span{ũ N 1 ,ũ N 2 } is isomorphic to (U N σ ) * as a graded G-module, wherẽ We see that h i (ũ N j ) =ũ N j if i = j, h 1ũ N 1 = (−1) ⌈ s a ⌉ũN 1 , and h 2 (ũ N 2 ) = ζ ⌈ s e ⌉ dũ N 2 .
As before, note that we do not necessarily have that s e is relatively prime to d. When d = a so that e = 1, we have that U N σ ≃ E σ whenever s < a. When d = 1 so that e = a, we do have U N σ ≃ E ⊠⌈ s a ⌉ as in the previous cases. Another special case occurs when s < e, so that U N σ ≃ E.
7.3. An exceptional example G 15 ⊲ G 12 . To illustrate the choice of indexing of degrees and exponents that is needed in Theorem 1.3, consider where we use the conventions in [LT09, Chapter 3] and the given matrices describe the action of G on V * . We will write the reflection generators of this non-wellgenerated reflection group G as s, t, and u, in the order they appear above. The degrees of G are d G 1 = 12 and d G 2 = 24, with corresponding invariant generators of S(V * ) G given by G 1 (x, y) = (x 5 y − xy 5 ) 2 and G 2 (x, y) = (x 8 + 14x 4 y 4 + y 8 ) 3 .
We then have the following normal reflection subgroup N G generated by the reflections u, sus −1 , and tsus −1 t −1 : where again the above matrices describe the acion of N on its reflection representation V * . The degrees of N are d N 1 = 6 and d N 2 = 8, with corresponding invariant generators of S(V * ) N given by N 1 (x, y) = x 5 y − xy 5 and N 2 (x, y) = x 8 + 14x 4 y 4 + y 8 .
Compatibly with the choices of invariant generators G 1 and G 2 for G and N 1 and N 2 for N , we find invariant generators of S(E * ) H that are N-homogeneous of degrees d H 1 = 2 and d H 2 = 3: H 1 (N 1 , N 2 ) = N 2 1 = G 1 (x, y) and H 2 (N 1 , N 2 ) = N 3 2 = G 2 (x, y), and also x-homogeneous of degrees d G 1 = 12 = d N 1 · d H 1 and d G 2 = 24 = d N 2 · d H 2 . As we can see from the explicit matrices above, the reflection representation of G is actually defined over ζ 24 . For s coprime to 24 we write σ s for the Galois automorphism σ s ∈ Gal(Q(ζ 24 )/Q) defined by σ s : ζ 24 → ζ s 24 . The complete data for every Galois twist V σ for G = G 15 and N = G 12 appears in Table 2. Code for computing similar examples using Sage can be found at [AW20b].
1−λi(g) σ 1−λi(g) q fixV (g) t fixE (g)  Table 2. Data for G = G 15 , N = G 12 , and H = G/N = C 2 × C 3 computed using [AW20b]. The rows are indexed by the Galois twists σ s : ζ 24 → ζ s 24 (for s coprime to 24). The columns contain the degrees of N multiplied by the E σ -exponents of H to obtain the E σ -exponents of G, and the V σ -exponents of N added to the U N σ -exponents of G to obtain the V σ -exponents of G, indexed according to Theorem 1.3. The final column lists the corresponding factorization of the weighted sum over G.
Writing u G i = a G i1 ⊗ X σ + a G i2 ⊗ y σ and u N i = a N i1 ⊗ x σ + a N i2 ⊗ y σ as above, let us verify that a G ij ∈ C G and a N ij ∈ C N . It is clear that a G 11 , a G 12 ∈ C G , because deg(a G ij ) = 1 < d G 1 , d G 2 , and every polynomial of degree smaller than every degree of G belongs to C G . Similarly, we see that a N 21 , a N 22 ∈ C N , since deg(a N 2j ) = 1 < d N 1 , d N 2 . We observe that u G 2 = N 1 N 2 2 · u N 2 (we discuss the meaning of this observation in more detail below), and therefore u G 2 ∈ C G ⊗ (V σ ) * ≃ C H ⊗ C N ⊗ (V σ ) * by Proposition 3.3. To see that u N 1 ∈ C N ⊗ (V σ ) * also, suppose that w N 1 , u N 2 is a homogeneous basis for (U N σ ) * , and let p 1 , p 2 ∈ S(V * ) N be homogeneous such that u N 1 = p 1 w N 1 +p 2 u N 2 . But then p 2 = 0, since no homogeneous N -invariant polynomial has degree 10, which implies that a N 1j is divisible by a homogeneous N -invariant polynomial p 1 with deg(p 1 ) ≤ 11. But the only non-constant choices are p 1 = N 1 or p 1 = N 2 , none of which divide the coefficients a N 1j above, and therefore p 1 ∈ C × . This concludes the proof that the u G i and u N i specified above are indeed bases for (U G σ ) * and (U N σ ) * , respectively, as claimed. Hence, the V σ -exponents of G are e G 1 (V σ ) = 11 and e G 2 (V σ ) = 23, and the V σ -exponents of N are e N 1 (V σ ) = 11 and e N 2 (V σ ) = 1. We compute the action of G on (U N σ ) * and obtain g(u N 1 ) = u N 1 for all g ∈ G, s(u N 2 ) = −u N 2 , t(u N 2 ) = ζ 2 3 u N 2 , and u(u N 2 ) = u N 2 . The resulting G-module structure on U N σ yields the direct sum of the trivial representation with the unique 1-dimensional representation of G with fake degree q 22 , which is G-isomorphic to the C-span of the G-harmonic semi-invariant homogeneous polynomial a H 22 := (x 21 y − xy 21 ) + 27(x 17 y 5 − x 5 y 17 ) + 170(x 13 y 9 − x 9 y 13 ) ∈ C G ⊂ S(V * ), so that a basis for (C G ⊗ (U N σ ) * ) G is given by {1 ⊗ u N 1 , a H 22 ⊗ u N 2 }. On the other hand, the H-module structure on U N σ yields the direct sum of the trivial representation with the 1-dimensional representation of H with fake degree q 3 given by the C-span of the H-harmonic semi-invariant N-homogeneous polynomial a H 22 = N 1 N 2 2 ∈ C H ⊂ S(E * ), so that a basis for (C H ⊗ (U N σ ) * ) H is given by {1 ⊗ u N 1 , N 1 N 2 2 ⊗ u N 2 }. Thus we witness the general isomorphism (C G ⊗ (V σ ) * ) G ≃ (C H ⊗ (U N σ ) * ) H from the proof of Corollary 3.7 in this example, since we obtain a H 22 ⊗ u N 2 → u G 2 by collapsing the first tensor in a H 22 ⊗ u N 2 = (x 5 y − xy 5 )(x 8 + 14x 4 y 4 + y 8 ) 2 ⊗ (y ⊗ x σ − x ⊗ y σ ) ∈ (C H ⊗ (U N σ ) * ) H → (x 5 y − xy 5 )(x 8 + 14x 4 y 4 + y 8 ) 2 (y ⊗ x σ − x ⊗ y σ ) = u G 2 ∈ (C G ⊗ (V σ ) * ) G .

Acknowledgements
We thank Theo Douvropoulos for many helpful remarks and suggestions.