aa r X i v : . [ m a t h . C O ] J un NORMAL REFLECTION SUBGROUPS
CARLOS E. ARRECHE AND NATHAN F. WILLIAMS
Abstract.
We study normal reflection subgroups of complex reflection groups.Our point of view leads to a refinement of a theorem of Orlik and Solomon tothe effect that the generating function for fixed-space dimension over a reflec-tion group is a product of linear factors involving generalized exponents. Ourrefinement gives a uniform proof and generalization of a recent theorem of thesecond author. Introduction
Hopf proved that the (singular) cohomology of a real connected compact Liegroup G is an exterior algebra on rank( G ) generators of odd degree [Hop64]. ItsPoincaré series is therefore given by Hilb( H ∗ ( G ); q ) = n Y i =1 (1 + q e i +1 ) . Chevalley presented these e i for the exceptional simple Lie algebras in his 1950address at the International Congress of Mathematicians [Che50], and Coxeter rec-ognized them from previous work with real reflection groups [Cox51]. This observa-tion has led to deep relationships between the cohomology of G , and the invarianttheory of the corresponding Weyl group W = N G ( T ) /T [Ree95, RS19]—notably, H ∗ ( G ) ≃ ( H ∗ ( G /T ) × H ∗ ( T )) W ≃ (cid:0) S/S W + ⊗ V V ∗ (cid:1) W . For more information, werefer the reader to the wonderful survey [BG94].It turns out that one method to compute the e i is the generating function forthe dimension of the fixed space fix( w ) = dim(ker(1 − g )) over the Weyl group: X w ∈ W q fix( w ) = r Y i =1 ( q + e i ) . Shephard and Todd [ST54] verified case-by-case that the same sum still factorswhen W is replaced by a complex reflection group G . Let G be a finite complexreflection group, acting by reflections on V . The e i are now determined by thedegrees d i of the fundamental invariants of G on V as e i = d i − . A case-freeproof of this result was given by Solomon [Sol63], mirroring Hopf’s result: writing S = Sym( V ∗ ) and Λ = V ( V ∗ ) , (( S ⊗ Λ) G is a free exterior algebra over the ring of G -invariant polynomials, which gives a factorization of the Poincaré series(1) Hilb(( S ⊗ Λ) G ; q, u ) = r Y i =1 uq e i − q d i . Computing the trace on S ⊗ Λ of the projection to the G -invariants G P g ∈ G g ,specializing to u = q (1 − x ) − , and taking the limit as x → gives the Shephard-Todd result: Date : June 12, 2020.2000
Mathematics Subject Classification.
Primary 20F55; Secondary 05E10.
Key words and phrases. reflection group, normal subgroup, exterior algebra, exponents.
Theorem 1.1 ([ST54, Sol63]) . For any complex reflection group G , (2) X g ∈ G q fix( g ) = r Y i =1 ( q + e i ) . More generally, define the fake degree of an m -dimensional (simple) G -module M to be the polynomial encoding the degrees in which M occurs in the coinvariantring C [ V ] G : f M ( q ) = P i ( C [ V ] iG , M ) q i = P mi =1 q e i ( M ) . Let G ⊂ GL( V ) be a complex reflection group. We say that N ⊳ G is a normalreflection subgroup of G if it is a normal subgroup of G that is generated by reflec-tions. For Weyl groups, nontrivial normal reflection subgroups can be constructedusing root lengths. More generally, normal reflection subgroups are constructed bytaking the union of conjugacy classes of reflections. We give the classification ofnormal reflection subgroups in Section 4, and tie our work with previous work ontheir numerology in Section 5.The following theorem is a special case of results in [BBR02] (where the authorsconsider the more general notion of “bon sous-groupe distingué” in lieu of our normalreflection subgroup N of G ). We emphasize that our proof of this result in Section 2follows the main ideas in [BBR02], specialized to our more restricted setting. Theorem 1.2.
Let N be a normal reflection subgroup of a complex reflection group G acting on V . Then the quotient group H = G/N acts as a reflection group onthe vector space E = V /N . In Section 2 we will build on our proof of Theorem 1.2 to prove the followingnumerological identities.
Theorem 1.3.
Let
G, N, H be as in Theorem 1.2. For a suitable choice of indexingof degrees and fake degrees, we have the following equalities: e Ni ( V )+ e Gi ( E ) = e Gi ( V ) d Ni · d Hi = d Gi d Ni · e Hi ( E ) = e Gi ( E ) Example 1.4.
Take G = W ( F ) = G and N to be the normal subgroupgenerated by the reflections corresponding to short roots. Then N ≃ W ( D ) , G/N ≃ W ( A ) = S acting by reflections on C ⊕ C ⊕ C (trivially on C ⊕ C ), sothat Theorem 1.3 gives the equations (1 , , , , , ,
8) = (1 , , , , , , · (1 , , ,
3) = (2 , , , , , , · (0 , , ,
2) = (0 , , , . The following result simultaneously generalizes the results of [Wil20] and theShephard-Todd formula Equation (2) from Theorem 1.1. We state a generalizedversion that incorporates Galois twists in Section 6.
Theorem 1.5.
Let
N ⊳ G be reflection groups acting by reflections on V , and let E = V /N . Then X g ∈ G q fix V g t fix E g = r Y i =1 (cid:0) qt + e Ni ( V ) t + e Gi ( E ) (cid:1) . Example 1.6.
The dihedral group G = G (2 , ,
2) = (cid:10) s, t | s = t = ( st ) = 1 (cid:11) actsas a reflection group on V = C by s = (cid:2) − (cid:3) and t = [ ] . Take N to be thenormal subgroup generated by the reflections conjugate to s . Then N ≃ C × C is ORMAL REFLECTION SUBGROUPS 3 a normal reflection subgroup, isomorphic to the direct product of the cyclic group oforder two with itself, with invariants N = x and N = x , and G acts on E dual to E ∗ = span C { N , N } as the quotient group G/N ≃ C by s = [ ] and t = [ ] . In this case, Theorem 1.5 expresses the equality X g ∈ G q fix V g t fix E g = q t + 2 qt + 2 qt + 2 t + t = ( qt + t )( qt + t + 2) . Acknowledgements.
We thank Theo Douvropoulos for many helpful commentsand suggestions. The first author was partially supported by NSF grant CCF-1815108. The second author was partially supported by Simons Foundation awardnumber 585380.2.
Quotients by Normal Reflection Subgroups
Let V be a complex vector space of dimension r . A reflection is an elementof GL( V ) that fixes some hyperplane pointwise. A complex reflection group G isa finite subgroup of GL( V ) that is generated by reflections. A complex reflectiongroup G is called irreducible if V is a simple G -module; V is then called the reflectionrepresentation of G . A (normal) reflection subgroup of G is a (normal) subgroup of G that is generated by reflections.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 subgroup G of GL( V ) is a complex reflection group if andonly if S ( V ∗ ) G is a polynomial ring, generated by r algebraically independent homo-geneous G -invariant polynomials—the degrees d ≤ · · · ≤ d r of these polynomialsare invariants of G . Theorem 2.1 ([ST54, Che55]) . Let G ≤ GL( V ) be finite. Then G is a complexreflection group if and only if there exist r homogeneous algebraically independentpolynomials G , . . . , G r ∈ S ( V ∗ ) G such that S ( V ∗ ) G = C [ G , . . . , G r ] . In this case, | G | = Q ri =1 d i , where d i = deg( G i ) . Although Theorem 1.2 is a special case of results in [BBR02] (where they considerthe more general notion of “bon sous-groupe distingué”), the proof is more straight-forward in our restricted setting, and also leads directly to a proof of Theorem 1.3.
Theorem 1.2.
Let N be a normal reflection subgroup of a complex reflection group G acting on V . Then the quotient group H = G/N acts as a reflection group onthe vector space E = V /N .Proof.
We claim that there exist homogeneous generators N , . . . , N r of S ( V ∗ ) N such that E ∗ = span C { N , . . . , N r } is H -stable. By Theorem 2.1, S ( V ∗ ) N = C [ ˜ N , . . . , ˜ N r ] for some homogeneous algebraically independent ˜ N i . Let I + ⊂ S ( V ∗ ) N be the ideal generated by homogeneous elements of positive degree. Thenboth I + and I are H -stable homogeneous ideals, and therefore the algebraic tan-gent space I + /I to E = V /N at inherits a graded action of H that is compatiblewith the (graded) quotient map π : I + ։ I + /I . Hence there exists a graded H -equivariant section ϕ : I + /I → I + . Letting N i = ϕ ◦ π ( ˜ N i ) we see that N , . . . , N r are still homogeneous algebraically independent generators for S ( V ∗ ) N with deg( N i ) = deg( ˜ N i ) and such that E ∗ = span C { N , . . . , N r } is H -stable, asclaimed.Write G , . . . , G r for the homogeneous generators of S ( V ∗ ) G , again as in The-orem 2.1. Consider the action of H on E ∗ defined by ( gN ) N i := gN i . Since S ( V ∗ ) G = ( S ( V ∗ ) N ) H = S ( E ∗ ) H , there exist polynomials H , . . . , H r ∈ C [ N ] such that H i ( N ) = G i ( x ) , where N = { N , . . . , N r } and x = { x , . . . , x r } denotedual bases for E and V , respectively. C. ARRECHE AND N. WILLIAMS
Since any algebraic relation f ( H , . . . , H r ) = 0 would result in an algebraic rela-tion f ( G , . . . , G r ) = 0 , the H i must be algebraically independent. By Theorem 1.2, H is a complex reflection group. (cid:3) Remark 2.2.
Note that the quotient group H = G/N does not necessarily lift toa reflection subgroup of G nor even a subgroup of G . A counterexample is givenby G (4 , ,
2) =
N ⊳ G = G , so that G/N ≃ S . Remark 2.3.
In the course of the proof of Theorem 1.2 we showed that the vectorspace E = V /N on which H acts by reflections is dual to E ∗ := span C { N , . . . , N r } for a certain choice of fundamental N -invariants N , . . . N r ∈ S ( V ∗ ) N such that E ∗ is G -stable. The resulting action of H on E respects the x -grading on the N -invariants N i ( x ) , and therefore there is a choice of fundamental H -invariants H , . . . , H r ∈ S ( E ∗ ) H = S ( V ∗ ) G such that each H i ( N ) = G i ( x ) is both x -homogeneous of x -degree =: d Gi and N -homogeneous of N -degree =: d Hi . Sincethe action of H on E ∗ respects the homogeneous decomposition of E ∗ according to x -degree, the H i ( N ) may be chosen such that the N -invariants N j ( x ) ∈ N occur-ring non-trivially in H i ( N ) are all of the same x -degree =: d Ni . This relationshipbetween the fundamental invariants for N , G , and H acting on their correspondingreflection representations leads to interesting numerological identities.The following result motivates some of the theoretical ingredients in our proofs. Theorem 2.4 ([Sol63]) . If G ⊂ GL( V ) is a complex reflection group, then the ring ( S ( V ∗ ) ⊗ V V ∗ ) G is a free exterior algebra over the ring of G -invariant polynomials: (cid:16) S ( V ∗ ) ⊗ ^ V ∗ (cid:17) G ≃ S ( V ∗ ) G ⊗ ^ U G , where U G = span C { dG , . . . , dG s } and dG i = P rj =1 ∂G i ∂x j ⊗ x j . Theorem 1.3.
Let
G, N, H be as in Theorem 1.2. For a suitable choice of indexingof degrees and fake degrees, we have the following equalities: e Ni ( V )+ e Gi ( E ) = e Gi ( V ) d Ni · d Hi = d Gi d Ni · e Hi ( E ) = e Gi ( E ) Proof.
Having chosen fundamental N -invariants N , . . . , N r ∈ S ( V ∗ ) N such that E ∗ = span C { N , . . . , N r } is G -stable as in the proof of Theorem 1.2 and Remark 2.3,we have fundamental G -invariants G i ( x ) = H i ( N ) that are x -homogeneous of x -degree d Gi and N -homogeneous of N -degree d Hi , and where the N j ∈ N occurringnon-trivially in H i ( N ) are all of the same x -degree d Ni . The equality d Ni d Hi = d Gi is immediate.Let us show that this same choice of indexing of fundamental invariants for N , G , and H results in the other two equalities. We begin by comparing x -degrees in dG i = r X j =1 ∂G i ∂x j ⊗ x j = r X k =1 ∂H i ∂N k · dN k = r X k =1 r X j =1 ∂H i ∂N k · ∂N k ∂x j ⊗ x j . Recall that e Gi ( V ) = d Gi − x ( dG i ) and e Ni ( V ) = d Ni − x ( dN i ) .Similarly, e Hi ( E ) = d Hi − N ( dH i ) , where this time dH i = P rk =1 ∂H i ∂N k ⊗ N k ∈ ( S ( E ∗ ) ⊗ E ∗ ) H . Since ∂H i ∂N k = 0 whenever deg x ( N k ) = d Ni , it follows that e Gi ( V ) = e Ni ( V ) + d Ni · ( d Hi −
1) = e Ni ( V ) + d Ni · e Hi ( E ) . It remains to show that d Ni e Hi ( E ) = e Gi ( E ) . ORMAL REFLECTION SUBGROUPS 5
The e Gi ( E ) are known to coincide with the x -degrees of any set of homogeneousgenerators for ( S ( V ∗ ) ⊗ E ∗ ) G as a free S ( V ∗ ) G -module. Since E ∗ consists of N -invariants, ( S ( V ∗ ) ⊗ E ∗ ) G = (( S ( V ∗ ) ⊗ E ∗ ) N ) H = ( S ( E ∗ ) ⊗ E ∗ ) H ≃ S ( E ∗ ) H ⊗ U H = S ( V ∗ ) G ⊗ U H , where again U H := span C { dH , . . . , dH r } and the non-trivial isomorphism comesfrom Theorem 2.4 applied to the reflection representation E of H . Hence ( S ( V ∗ ) ⊗ E ∗ ) G is generated by dH i as a free S ( V ∗ ) G -module, whence e Gi ( E ) = deg x ( dH i ) = d Ni e Hi ( E ) . (cid:3) Remark 2.5.
The same argument used in the proof of Theorem 1.3 shows moregenerally: ( S ( V ∗ ) ⊗ ^ E ∗ ) G = (( S ( V ∗ ) ⊗ ^ E ∗ ) N ) H = ( S ( E ∗ ) ⊗ ^ E ∗ ) H ≃ S ( V ∗ ) G ⊗ ^ U H . Poincaré Series and Specializations
Our goal in this section is to prove our main result:
Theorem 1.5.
Let
N ⊳ G be reflection groups acting by reflections on V , and let E = V /N . Then X g ∈ G q fix V g t fix E g = r Y i =1 (cid:0) qt + e Ni ( V ) t + e Gi ( E ) (cid:1) . We refer to the left-hand side of Theorem 1.5 as the sum side , and to the right-hand side as the product side . We prove Theorem 1.5 by computing the Poincaréseries for ( S ( V ∗ ) ⊗ V E ∗ ) G in two different (and standard) ways, keeping trackof the supplemental grading afforded by the x -degrees of N -invariants in E ∗ =span C { N , . . . , N r } : one way corresponds to the product side (Section 3.1), andthe other to the sum side (Section 3.2). A subtlety arises when trying to computethe term-by-term specialization for the sum side, which is dealt with in Sections 3.3and 3.4.A more general version of Theorem 1.5 that incorporates Galois twists is statedin Section 6. For technical reasons that arise in that generalization, we will definethe shifted homogeneous decomposition E ∗ m := span C { N i | deg x ( N i ) = m + 1 } , andsimilarly ( V p E ∗ ) m := span C { N i ∧ · · · ∧ N i p ∈ V p E ∗ | P pj =1 deg x ( N i j ) = m + p } . Writing S ( V ∗ ) ℓ for the homogeneous component of x -degree ℓ , we define the Poincaréseries(3) P ( x, y, u ) := X ℓ,m,p ≥ dim C (( S ( V ∗ ) ℓ ⊗ ( V p E ∗ ) m ) G ) x ℓ y m u p . We write E Nσ = { e N ( V σ ) , . . . , e Nr ( V σ ) } for the set of fake degrees of V σ as an N -representation.3.1. Product Side.
We first obtain the following product formula for the Poincaréseries P ( x, y, u ) defined in Equation (3) from Theorem 1.3 and Remark 2.5, since dH i ∈ S ( V ∗ ) e Gi ( E ) ⊗ E ∗ e Ni ( V ) . Lemma 3.1. P ( x, y, u ) = r Y i =1 x e Gi ( E ) y e Ni ( V ) u − x d Gi i . Corollary 3.2. lim x → P ( x, x t , qt (1 − x ) −
1) = r Y i =1 (cid:0) qt + e Ni ( V ) t + e Gi ( E ) (cid:1) . C. ARRECHE AND N. WILLIAMS
Sum Side.
By taking traces, we now compute a formula for the Poincaréseries P ( x, y, u ) defined in Equation (3) as a sum over elements of G . To simplifynotation, we denote by E m the homogeneous component of E corresponding to thedual of E ∗ m . Lemma 3.3. P ( x, y, u ) = | G | X g ∈ G Q m ∈E Nσ det(1 + uy m g | E m )det(1 − xg | V ) . Proof.
Since ( U Nσ ) ∗ ≃ L m ∈E Nσ E ∗ m , we have that V ( U Nσ ) ∗ ≃ N m ∈E Nσ V E ∗ m as G -modules. hence, for each g ∈ G , X m,p ≥ tr( g | ( V p ( U Nσ ) ∗ ) m ) y m u p = Y m ∈E N X p ≥ tr( g | V p E ∗ m ) y pm u p . For each m ∈ E Nσ we have X p ≥ (tr( g | V p E ∗ m ) y pm u p ) = det(1 + y m ug | E ∗ m ) . There-fore, P ( x, y, u ) = X ℓ ≥ tr( g | ( S ( V ∗ ) ℓ ) x ℓ X m,p ≥ tr( g | ( V p ( U Nσ ) ∗ ) m ) y m u p = Q m ∈E Nσ det(1 + uy m g − | U Nσ m )det(1 − xg − | V ) . The result follows after taking the average over G on each side. (cid:3) The story is not quite so simple as just setting the sum over G from Lemma 3.3equal to the product, and then specializing. The trouble is that in this specializedsum over G from Lemma 3.3, each element of G does not necessarily contribute the“correct amount” specified by the sum side of Theorem 1.5—in particular, ( ng ) | E often has larger fixed space than ( ng ) | V , which causes many terms in the term-by-term limit to be zero. It turns out, as we will now show, that the contributions are correct when taken coset-by-coset.3.3. Sum Side, Coset-by-Coset I.
Fix some g ∈ G . We find a product formulafor Lemma 3.3 restricted to the coset gN . Define P gN ( x, y, u ) := 1 | N | X n ∈ N Q m ∈E Nσ det (cid:0) uy m ( ng ) | E ∗ m (cid:1) det(1 − x ( ng ) | V ∗ ) . Given g ∈ G , we can choose the fundamental N -invariants N , . . . , N r ∈ S ( V ∗ ) G toalso form a g -eigenbasis for E ∗ = span C { N , . . . , N r } , since this space is G -stableand g has finite order. For g ∈ G , let ǫ g , . . . , ǫ gr denote the eigenvalues of g on E ∗ ,so that gN i = ǫ gi N i . Proposition 3.4. P gN ( x, y, u ) = r Y i =1 ǫ gi uy e Ni ( V ) − ǫ gi x d Ni . Proof.
First, Y m ∈E Nσ det(1 + uy m ( ng ) | E ∗ m ) = r Y i =1 (1 + ǫ gi uy e Ni ( V σ ) ) uniformly for any n ∈ N , since E ∗ m is N -invariant. It remains to show that | N | X n ∈ N − x ( ng ) | V ∗ ) = r Y i =1 − ǫ gi x d Ni . ORMAL REFLECTION SUBGROUPS 7
Since S ( E ∗ ) ≃ N m ∈E N Sym( E ∗ m ) , where E ∗ m denotes the span of fundamental N -invariants having x -degree m + 1 and Sym( E ∗ m ) denotes its symmetric algebra,we have X ℓ ≥ tr (cid:0) g | S ( E ∗ ) ℓ (cid:1) x ℓ = Y m ∈E N (cid:16)X ℓ ≥ (cid:0) tr( g | Sym ℓ ( E ∗ m ))( x m +1 ) ℓ (cid:1) , where S ( E ∗ ) ℓ := S ( E ∗ ) ∩ S ( V ∗ ) ℓ and S ( V ∗ ) ℓ as before denotes the homogeneoussubspace of polynomials of x -degree ℓ . On the other hand, Y m ∈E N (cid:16)X ℓ ≥ tr (cid:0) g | Sym ℓ ( E ∗ m ) (cid:1) ( x m +1 ) ℓ (cid:17) = Y m ∈E N − x m +1 ( g | E ∗ m )) = r Y i =1 − ǫ gi x d Ni i , since d Ni = e Ni ( V ) . Therefore, X ℓ ≥ tr (cid:0) g | S ( E ∗ ) ℓ (cid:1) x ℓ = r Y i =1 − ǫ gi x d Ni i . Since for each n ∈ N we have X ℓ ≥ tr (cid:0) ng | S ( V ∗ ) ℓ (cid:1) x ℓ = 1det(1 − x ( ng | V ∗ )) , it remains to show that,for each ℓ ≥ , | N | X n ∈ N (cid:16) tr (cid:0) ng | S ( V ∗ ) ℓ (cid:1)(cid:17) = tr (cid:0) g | S ( E ∗ ) ℓ (cid:1) . To see this, note that the operator | N | X n ∈ N ng = g · | N | X n ∈ N n ! = g ◦ pr Nℓ , where pr Nℓ = | N | P n ∈ N n is the projection from S ( V ∗ ) ℓ onto its g -stable subspace S ( V ∗ ) Nℓ = S ( E ∗ ) ℓ , whence tr (cid:0) ( g ◦ pr Nℓ ) | S ( V ∗ ) ℓ (cid:1) = tr (cid:0) g | S ( E ∗ ) ℓ (cid:1) . (cid:3) Sum Side, Coset-by-Coset II.
We next specialize some results of [BLM06]to the case when N is a normal reflection subgroup of a complex reflection group G . They consider the more general situation when N is translated by an arbitraryelement in the normalizer of N in GL( V ) . Continue to fix some g ∈ G . Proposition 3.5 ([BLM06]) . | N | X n ∈ N det(1 + u ( ng ) | V ∗ )det(1 − x ( ng ) | V ∗ ) = r Y i =1 ǫ gi ux e Ni ( V ) − ǫ gi x d Ni (= P gN ( x, x, u )) . Specializing both sides of Proposition 3.5 to u = q (1 − x ) − and then takingthe limit x → yields the following simple formula for sums over cosets. Corollary 3.6 ([BLM06]) . X n ∈ N q fix V ( ng ) = Y ǫ gi =1 q + e Ni ( V ) Y ǫ gi =1 d Ni . Using Corollary 3.6, we obtain the following crucial specialization of Proposition 3.4,exploiting the fact that Proposition 3.4 gives the series P gN ( x, y, u ) , while Proposition 3.5gives the series P gN ( x, x, u ) . Corollary 3.7. lim x → P gN ( x, x t , qt (1 − x ) −
1) = t fix E g P n ∈ N q fix V ( ng ) . Proof of Theorem 1.5.
We now prove our main theorem.
Proof of Theorem 1.5.
Equating the formulas from Lemmas 3.1 and 3.3 gives P ( x, y, u ) = X g ∈ G Q m ∈E Nσ det(1 + uy m g | E m )det(1 − xg | V ) = | G | r Y i =1 x e Gi ( U Nσ ) y e Ni ( V ) u − x d Gi i . (4) C. ARRECHE AND N. WILLIAMS
Let { g j } | H | j =1 be a set of coset representatives for N in G . By Corollary 3.7, split-ting the sum side of Equation (4) into a sum over the cosets of N and specializinggives lim x → P ( x, x t , qt (1 − x ) −
1) = | H | X j =1 lim x → P gN ( x, x t , qt (1 − x ) − | H | X j =1 t fix E g j X n ∈ N q fix V ng j = X g ∈ G q fix V g t fix E g . The result now follows from Equation (4) by equating this specialization of thesum side with the same specialization of the product side from Corollary 3.2. (cid:3) Classification of Normal Reflection Subgroups
In the interest of space, we restrict our classification of normal reflection sub-groups to rank ≥ . In rank 2, there are two connected posets of imprimitivecomplex reflection groups ordered by normality: one has maximal element G andminimal elements G (4 , , , G , and G , while the other has maximal element G and minimal elements G , G , and G . Theorem 4.1 ([LT09, Corollary 2.18]) . For r ≥ , the normal reflection subgroupsof G ( ab, b, r ) are ( C d ) r and G ( ab, db, r ) for d | a , giving quotients G ( ab, b, r ) / ( C d ) r = G (( a/d ) b, b, r ) and G ( ab, b, r ) /G ( ab, db, r ) = C d . As G and G are the only exceptional reflection groups with more than a singleorbit of reflections, there are three nontrivial exceptional (that is, not imprimitive)examples of normal reflection subgroups in rank greater than two: G (3 , , ⊳ G ,with quotient G ; G ⊳ G , with quotient C ; and G (2 , , ⊳ G ≃ W ( F ) , withquotient S . 5. Reflexponents
Fix G a complex reflection group of rank r with reflection representation V . Callan r -dimensional representation M of G factorizing if M has dimension r and X g ∈ G q fix V ( g ) t fix M ( g ) = r Y i =1 (cid:16) qt + ( e Gi ( V ) − m i ) t + m i (cid:17) , for some nonnegative integers m , . . . , m r . More generally, call a representation M of G of dimension dim M ≤ r factorizing if it is factorizing in the above sense afteradding in r − dim M copies of the trivial representation.We can now give a uniform explanation for certain ad-hoc identities from [Wil20].Let H be an orbit of reflecting hyperplanes, write R H for the set of reflectionsfixing some L ∈ H , and let N H = hR H i be the subgroup generated by reflectionsaround hyperplanes in H . Since these reflections form a conjugacy class in G , N H is a normal reflection subgroup of G . Furthermore: the quotient G/N H acts as areflection group on the N H -invariants of V ; and this action gives a G -representation M H that is factorizing by Theorem 1.5.6. Galois Twists
Let V be an r -dimensional complex vector space and G ⊂ GL( V ) be a complexreflection group. It is known that G can be realized over Q ( ζ G ) , where ζ G is aprimitive | G | -th root of unity, in the sense that there is a basis for V with respectto which G ⊂ GL r ( Q ( ζ G )) . For σ ∈ Gal( Q ( ζ G ) / Q ) , the Galois twist V σ is therepresentation of G on the same underlying vector space V obtained by applying ORMAL REFLECTION SUBGROUPS 9 σ to the matrix entries of g ∈ GL( Q ( ζ G )) . Orlik and Solomon found a beautifulgeneralization of Equation (2) that takes into account these Galois twists. Belowwe write λ ( g ) , . . . , λ r ( g ) for the eigenvalues of g ∈ G on V . Theorem 6.1 ([OS80]) . Fix G a reflection group and σ ∈ Gal( Q ( ζ G ) / Q ) . Then X g ∈ G Y λ i ( g ) =1 − λ i ( g ) σ − λ i ( g ) q fix V g = r Y i =1 ( q + e i ( V σ )) . Definition 6.2.
Let I G + ⊂ S ( V ∗ ) be the ideal generated by homogeneous G -invariant polynomials of positive degree, and let C G be a G -stable homogeneoussubspace of S ( V ∗ ) such that S ( V ∗ ) ≃ I G + ⊕ C G as G -modules. For σ ∈ Gal( ζ G ) , de-fine the Orlik-Solomon space U σG := ( C G ⊗ ( V σ ) ∗ ) G = span C { u G , . . . , u Gr } , where the u Gi are homogeneous with deg( u Gi ) = e Gi ( V σ ) and such that (cid:0) S ( V ∗ ) ⊗ ( V σ ) ∗ (cid:1) G ≃ S ( V ∗ ) G ⊗ U σG .We can now state the general form of our main theorem. Theorem 1.5.
Let
N ⊳ G be reflection groups acting by reflections on V , and let E = V /N . Let σ ∈ Gal( Q ( ζ G ) / Q ) and define the Orlik-Solomon space U σN as inDefinition 6.2. Then X g ∈ G Y λ i ( g ) =1 − λ i ( g ) σ − λ i ( g ) q fix V g t fix E g = r Y i =1 (cid:0) qt + e Ni ( V σ ) t + e Gi (( U σN ) ∗ ) (cid:1) . Remark 6.3.
The Orlik-Solomon space U σN playes the role of E ∗ in this general-ization of Theorem 1.5. When σ = 1 , a straightforward argument yields a graded G -module homomorphism (of degree − ) E ∗ ≃ U N . This is why we shifted theobvious x -grading of E ∗ by − in Section 3. However, it is not true in general that ( E σ ) ∗ ≃ U σN when σ = 1 . References [BBR02] David Bessis, Cédric Bonnafé, and Raphaïl Rouquier,
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University of Texas at Dallas
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