Beitr Algebra Geom (2024) 65:319–336 https://doi.org/10.1007/s13366-023-00690-2 ORIG INAL PAPER Centers of Hecke algebras of complex reflection groups Eirini Chavli1 · Götz Pfeiffer2 Received: 3 November 2022 / Accepted: 9 February 2023 / Published online: 14 March 2023 © The Author(s) 2023 Abstract We provide a dual version of the Geck–Rouquier Theorem (Geck and Rouquier in Finite Reductive Groups (Luminy, 1994), Progr. Math., vol. 141, Birkhäuser Boston, Boston, pp. 251–272, 1997) on the center of an Iwahori–Hecke algebra, which also covers the complex case. For the eight complex reflection groups of rank 2, for which the symmetrising trace conjecture is known to be true, we provide a new faithfulmatrix model for their Hecke algebra H. These models enable concrete calculations inside H. For each of the eight groups, we compute an explicit integral basis of the center of H. Keywords Complex reflection group · Hecke algebra · Symmetrizing trace · Coset table Mathematics Subject Classification Primary 20C08; Secondary 20F55 1 Introduction LetW be a finite complex reflection group andH the associated genericHecke algebra, defined over the Laurent polynomial ring R = Z[u± 1 , . . . ,u± k ], where {ui}1�i�k is a set of parameters whose cardinality depends on W. In 1999, Malle (1999, §5) proved that H is split semisimple when defined over the field F = C(v1, . . . , vk), where each parameter vi is a root of ui of rank NW , for some specific NW ∈ N. By Tits’ deformation theorem (Geck and Pfeiffer 2000, Theorem 7.4.6), the specialization vj �→ 1 induces a bijection Irr(H ⊗R F) → Irr(W). B Eirini Chavli eirini.chavli@mathematik.uni-stuttgart.de Götz Pfeiffer goetz.pfeiffer@nuigalway.ie 1 Institute of Algebra and Number Theory, University of Stuttgart, Stuttgart, Germany 2 School of Mathematical and Statistical Sciences, National University of Ireland, Galway, University Road, Galway, Ireland 123 http://crossmark.crossref.org/dialog/?doi=10.1007/s13366-023-00690-2&domain=pdf http://orcid.org/0000-0003-1526-6752 320 Beitr Algebra Geom (2024) 65:319–336 A natural question is how the irreducible representations behave after specializing the parameters ui to arbitrary complex numbers. If the specialized Hecke algebra is semisimple, Tits’ deformation theorem still applies; the simple representations of the specialized Hecke algebra are parametrized again by Irr(W). However, if the specialized algebra is not semisimple one needs to find another way to parametrise the irreducible representations. One main obstacle in this direction is the lack of the description of the center Z(H) of the Hecke algebra H (see Geck and Pfeiffer 2000, Lemma 7.5.10). Apart from the real case (Geck and Rouquier 1997), there is not yet a known precise description of the center of the genericHecke algebraH in the complex case, except for the groups G4 and G(4, 1, 2) provided by Francis (2007). In this paper, we introduce a new general method for computing an R-basis of the center ofH. LetB be an R-basis of H. Expressing an arbitrary element z ∈ Z(H) as a linear combination of B, the conditions sz = zs, one for each generator s of H, give an R-linear system, whose solution describes a basis of Z(H) as linear combinations of the elements of B. This elementary approach has the following three difficulties: • Calculations inside the Hecke algebra are very complicated, even for products of the form sb and bs, for b ∈ B. In the last four years, there is a progress on this direction (see, for example, Boura et al. 2020a, b; Chavli and Chlouveraki 2022). However, it is still unclear if one can completely automate such calculations, since all the attempts so far use a lot of (long) computations made by hand. • Solving the aforementioned R-linear system is not always easy. It is known that Gaussian elimination can suffer from coefficient explosion over a field of rational functions in several variables, such as the quotient field of the ring R of Laurent polynomials. As a result, the choices of dependent and independent variables can be crucial, since wrong choices can lead to dead ends. • Even if the R-linear system can be solved (over the fraction field), the solution cannot be expected to lie in R. In fact, for the case of G12 a first attempt solving this systemprovided uswith a solution not inR (details can be found in the project’s webpage, https://www.eirinichavli.com/center.html). However, we show that it is possible to overcome all these difficulties, as demon- strated by our results for particular examples. A natural first example is the smallest exceptional group, the group G4. For this group we use a new approach on construct- ing an R-basis B (see Example 2.4), which allows us to work with a faithful matrix representation of H, rather than its usual presentation. Hence, we can • automate calculations inside the Hecke algebra of G4 and, in particular, compute the elements sb and bs from above, • solve the R-linear system sz = zs, and • verify that all solutions we obtain for z are in fact R-linear combinations of B. Our next goal is to explain the coefficients of these R-linear combinations. This allows us to describe the center Z(H) for other complex reflection groups without relying on the solution of the R-linear system. In order to explain our findings, we first need to revisit the real case. LetW be a real reflection group andH its associated Iwahori–Hecke algebra, which admits a standard basis {Tw : w ∈ W}. Denote by Cl(W) the set of conjugacy classes 123 https://www.eirinichavli.com/center.html Beitr Algebra Geom (2024) 65:319–336 321 of W, and choose a set of representatives {wC ∈ C | C ∈ Cl(W)} such that each elementwC has minimal length in its classC. It can be shown that there exist uniquely determined polynomials fw,C ∈ R, independent of the choice of the minimal length representatives wC, the so-called class polynomials (see Geck and Pfeiffer 2000, §8.2), such that χ(Tw) = ∑ C∈Cl(W) fw,C χ(TwC ) for all χ ∈ Irr(W). In other words, the column ( χ(Tw) ) χ of character values of Tw is an R-linear combination of the columns ( χ(TwC ) ) χ of the basis elements of H corresponding to the conjugacy class representatives of minimal length. Clearly, for any choice {vC ∈ C | C ∈ Cl(W)} of conjugacy class representatives, the square matrix ( χ(TvC ) ) χ,C of character values is invertible as it specializes to the character table of W. Hence, for each w ∈ W, there are uniquely determined coefficients ζw,C such that χ(Tw) = ∑ C∈Cl(W) ζw,C χ(TvC ). However, the coefficients ζw,C cannot be expected to belong to R. That is why it is crucial to choose minimal length class representatives. We denote by {T∨ w : w ∈ W} the dual basis of {Tw : w ∈ W} with respect to the standard symmetrising form (for the definition of the dual basis see, for example, Geck and Pfeiffer 2000, Definition 7.1.1). The following theorem has been shown by Geck and Rouquier (1997, §5.1), cf. Geck and Pfeiffer (2000, Theorem 8.2.3 and Corollary 8.2.4). Theorem 1.1 Let W be a finite real reflection group. The elements yC = ∑ w∈W fw,C T∨ w , C ∈ Cl(W), form a basis of the center Z(H). We now examine the complex case. We first assume that the Hecke algebra H admits a symmetrising trace τ. Let {bw : w ∈ W} be a basis of H as R-module and let {b∨ w : w ∈ W} be its dual basis with respect to τ. The lack of length function of a complex reflection group W cannot guarantee us the existence of class polynomials in the sense of the real case. However, it remains true that with respect to any choice of conjugacy class representatives vC, for each element w ∈ W there are coefficients ζw,C ∈ F, which express the column (χ(bw))χ of character values as a linear combination of the columns ( χ(bvC ) ) χ . In this paper, we prove the following: Theorem 1.2 Let W be a finite complex reflection group. For any choice of conjugacy class representatives vC, the elements 123 322 Beitr Algebra Geom (2024) 65:319–336 yC = ∑ w∈W ζw,C b∨ w, C ∈ Cl(W), form a basis of the center Z(H ⊗R F). This theorem generalizes the Theorem 1.1 of Geck–Rouquier in such a way that it includes the complex case, at the expense of working over F in place of R. At the same time, it gains us some flexibility in terms of choosing the conjugacy class repre- sentatives. There is a dual version of the theorem that provides even more flexibility, as follows. For each classC ∈ Cl(W), we choose again a representative vC ∈ C and we define coefficients gw,C ∈ F by the condition χ(b∨ w) = ∑ C∈Cl(W) gw,C χ(b∨ vC ), for all χ ∈ Irr(H ⊗R F). Theorem 1.3 Let W be a finite complex reflection group. For any choice of conjugacy class representatives vC, the elements zC = ∑ w∈W gw,C bw, C ∈ Cl(W), form a basis of the center Z(H ⊗R F). As an illustration, and to state the fact that for the real case Theorems 1.2 and 1.3 give new bases, different from the one of Geck–Rouquier, we apply these theorems to the Coxeter group of type A2 and express the resulting basis of Z(H) as a linear combination of the standard basis {Tw : w ∈ W} of H. Example 1.4 LetW be a finite Coxeter group of typeA2 with generators s, t such that sts = tst. Then {1, s, st} is a set of minimal length class representatives of W, and {1, sts, st} is a set ofmaximal length class representatives. The Iwahori–Hecke algebra H is generated by elements Ts and Tt and defined over the ring R = Z[u± 1 ,u± 2 ]. We set c = u1+u2 andd = −u1u2 (where onlyd is invertible inR). Then T2 s = c Ts+d and T2 t = c Tt + d. We have the following: • Theorem 1.1, with minimal length class representatives yields the basis T1, d−1(Ts + Tt) + d−2 Tsts, d−2(Tst + Tts) + cd−3 Tsts • Theorem 1.2, with maximal length class representatives yields the basis T1, d−2(Ts + Tt) + d−3 Tsts, −cd−2(Ts + Tt) + d−2(Tst + Tts) • Theorem 1.3, with minimal length class representatives yields the basis T1, (Ts + Tt) + d−1 Tsts, (Tst + Tts) + cd−1 Tsts 123 Beitr Algebra Geom (2024) 65:319–336 323 • Theorem 1.3, with maximal length class representatives yields the basis T1, d(Ts + Tt) + Tsts, −c(Ts + Tt) + (Tst + Tts) Note that in all cases the coefficients of the basis elements of Z(H) belong to R = Z[u± 1 ,u± 2 ], and that in the last case they even lie in the polynomial ring Z[u1,u2]. �� It is worth mentioning here that Theorems 1.2 and 1.3 have a general proof, which does not use the case-by-case analysis, based on the classification of complex reflection groups (Shephard and Todd 1954). As mentioned before, Theorems 1.2 and 1.3 provide bases of the center of H over the splitting field F. Choosing an arbitrary basis of the Hecke algebra and arbitrary class representatives, one cannot expect to obtain a basis of the center of H over R. In fact, for G12, there is a choice of conjugacy class representatives vC, where not all the coefficients gw,C of Theorem 1.3 belong to R (for details, see the project’s web page, https://www.eirinichavli.com/center.html). However, for some choice of a basis {bw : w ∈ W} and of class representatives vC, it might turn out that the coefficients gw,C of Theorem 1.3 belong to R, which means that we obtain a basis of the center Z(H). In fact, in this paper we show that for the groups G4, . . . ,G8,G12,G13,G22, i.e., for all exceptional groups of rank 2, whose associated Hecke algebra is known to be symmetric, we can make such choices so that we obtain a basis of the center Z(H). We conjecture that this is true for all complex reflection groups that satisfy the symmetrising trace Conjecture 2.3. The next section of this paper explains in detail how we make these choices. The final section contains our main results. In our calculations we used some programs written in GAP, which one can find in the project’s webpage (https://www.eirinichavli. com/center.html). 2 Choosing a basis 2.1 Hecke algebras A complex reflection group W is a finite subgroup of GLn(C) generated by pseudo- reflections (these are non-trivial elements of W whose fixed points in C n form a hyperplane). Real reflection groups, also known as finite Coxeter groups are particular cases of complex reflection groups. We denote by K the field of definition of W, that is the field generated by the traces on C n of all the elements of W. If K ⊆ R, then W is a finite Coxeter group, and if K = Q, then W is a Weyl group. A complex reflection group W is irreducible if it acts irreducibly on C n and, if that is the case, we call n the rank of W. Each complex reflection group is a direct product of irreducible ones and, hence, the study of reflection groups reduces to the irreducible case. The classification of irreducible complex reflection groups is due to Shephard and Todd (1954) and it is given by the following theorem. 123 https://www.eirinichavli.com/center.html https://www.eirinichavli.com/center.html https://www.eirinichavli.com/center.html 324 Beitr Algebra Geom (2024) 65:319–336 Theorem 2.1 Let W ⊂ GLn(C) be an irreducible complex reflection group. Then, up to conjugacy, W belongs to precisely one of the following classes: • The symmetric group Sn+1. • The infinite family G(de, e,n), where d, e,n ∈ N ∗, such that (de, e,n) = (1, 1,n) and (de, e,n) = (2, 2, 2), of all n × n monomial matrices whose non- zero entries are de-th roots of unity, while the product of all non-zero entries is a d-th root of unity. • The 34 exceptional groups G4, G5, . . . , G37 (ordered with respect to increasing rank). Let W be a complex reflection group. We denote by B the complex braid group associated to W, as defined in Broué et al. (1998, §2 B). A pseudo-reflection s is called distinguished if its only nontrivial eigenvalue onCn equals exp(−2π √ −1/es), where es denotes the order of s inW. Let S denote the set of the distinguished pseudo- reflections of W. For each s ∈ S we choose a set of es indeterminates us,1, . . . ,us,es , such that us,j = ut,j if s and t are conjugate in W. We denote by R the Laurent polynomial ring Z[us,j,u −1 s,j ]. The generic Hecke algebra H associated to W with parameters us,1, . . . ,us,es is the quotient of the group algebra R[B] of B by the ideal generated by the elements of the form (σ − us,1)(σ − us,2) · · · (σ − us,es), (2.1) where s runs over the conjugacy classes of S and σ over the set of braided reflections associated to the pseudo-reflection s (for the standard notion of a braided reflection associated to s one can refer to Broué et al. (1998, §2 B). It is enough to choose one relation of the form described in (2.1) per conjugacy class, since the corresponding braided reflections are conjugate in B. We obtain an equivalent definition of H if we expand the relations (2.1). More precisely, H is the quotient of the group algebra R[B] by the elements of the form σes − as,es−1σes−1 − as,es−2σes−2 − · · · − as,0, (2.2) where as,es−k := (−1)k−1fk(us,1, . . . ,us,es) with fk denoting the k-th elemen- tary symmetric polynomial, for k = 1, . . . , es. Therefore, in the presentation of H, apart from the braid relations coming from the presentation of B, we also have the positive Hecke relations: σes = as,es−1σes−1 + as,es−2σes−2 + · · · + as,0. (2.3) We notice now that as,0 = (−1)es−1us,1us,2 . . . us,es ∈ R×. Hence, σ is invertible in H with σ−1 = a−1 s,0 σes−1 − a−1 s,0 as,es−1σes−2 − a−1 s,0 as,es−2s es−3 − · · · − a−1 s,0 as,1. (2.4) 123 Beitr Algebra Geom (2024) 65:319–336 325 We call relations (2.4) the inverse Hecke relations. IfW is a real reflection group,H is known as the Iwahori–Hecke algebra associated with W (for more details about Iwahori–Hecke algebras one may refer, for example, to Geck and Pfeiffer (2000, §4.4). Iwahori–Hecke algebras admit a standard basis (Tw)w∈W indexed by the elements of W (see Bourbaki 2005, IV, §2). Broué, Malle and Rouquier conjectured a similar result for complex reflection groups (Broué et al. 1998, §4). Conjecture 2.2 (The BMR freeness conjecture) The algebra H is a free R-module of rank |W|. This conjecture is now a theorem, thanks to work of several people who used a case-by-case analysis approach, in order to prove the case of all irreducible complex reflection groups. A detailed state of the art of the proof can be found in Boura (2020b, Theorem 3.5). A symmetrising trace on a free algebra is a tracemapτ that induces a non-degenerate bilinear form, meaning that the determinant of the matrix (τ(bb ′))b,b ′∈B is a unit in the ring over which we define the algebra for some (and hence every) basis B of the algebra. For Iwahori–Hecke algebras there exists a unique symmetrising trace, given by τ(Tw) = δ1w Bourbaki (2005, IV, §2). Broué, Malle and Michel conjectured the existence of a symmetrising trace also for non-real complex reflection groups (Broué et al. 1999, §2.1, Assumption 2(1)). Conjecture 2.3 (The BMM symmetrising trace conjecture) There exists a linear map τ : H → R such that: (1) τ is a symmetrising trace, that is, we haveτ(h1h2) = τ(h2h1) for allh1,h2 ∈ H, and the bilinear map H × H → R, (h1,h2) �→ τ(h1h2) is non-degenerate. (2) τ becomes the canonical symmetrising trace on K[W] when us,j specialises to exp(2π √ −1 j/es) for every s ∈ S and j = 1, . . . , es. (3) τ satisfies τ(Tb−1) ∗ = τ(Tbπ) τ(Tπ) , for all b ∈ B(W), (2.5) where b �→ Tb denotes the restriction of the natural surjection R[B] → H to B, x �→ x∗ is the automorphism of R given by us,j �→ u−1 s,j and π the element z|Z(W)|, with z being the image of a suitable generator of the center of B inside H. Since we have the validity of the BMR freeness conjecture, we know by Broué et al. (1999, §2.1) that if there exists such a linear map τ, then it is unique. If this is the case, we call τ the canonical symmetrising trace on H. Malle andMichel (2010, Proposition 2.7) proved that if the Hecke algebraH admits a basis B ⊂ B consisting of braid group elements that satisfies certain properties (among them that 1 ∈ B), then Condition 2.5 is equivalent to: τ(Tx−1π) = 0, for all x ∈ B \ {1}. (2.6) 123 326 Beitr Algebra Geom (2024) 65:319–336 Apart from the real case, the BMM symmetrising trace conjecture is known to hold for a few exceptional groups and for the infinite family (detailed references can be found in Chavli and Chlouveraki 2022, Conjecture 3.3). We now describe the representation theory of Hecke algebras. In Malle (1999, §5) Malle associates to each complex reflection group W a positive integer NW and he defines for each s ∈ S, a set of es indeterminates vs,1, . . . , vs,es by the property v NW s,j = exp(−2π √ −1 j/es)us,j. (2.7) We denote by F the field C(vs,j) and by extension of scalars we obtain the algebra FH := H ⊗R F, which is split semisimple (Malle 1999, Theorem 5.2). By Tits’ defor- mation theorem (Geck and Pfeiffer 2000, Theorem 7.4.6), the specialization vs,j �→ 1 induces a bijection Irr(FH) → Irr(W). Models of irreducible representations of the Hecke algebra FH associated with certain irreducible complex reflectiongroups havebeen computedbyMalle andMichel (2010), and are readily available in Jean Michel’s development version (Michel 2015) of the CHEVIE package (Geck et al. 1996). In this paper, we use these models to evaluate the irreducible characters on some particular elements, when constructing an explicit basis of the center of the Hecke algebra. 2.2 Coset table Let W be a complex reflection group with associated Hecke algebra H defined over R. The goal of this paper is to provide, at least in some examples, a basis of the center Z(H) of H as R-module. For this purpose, we first find a basis B of the Hecke algebra and then we describe the basis elements of Z(H) as linear combinations of elements of B. In this section we explain the method we use in order to find a basis B for the exceptional groups G4, . . . , G8, G12, G13, G22. These groups are exactly the exceptional complex reflection groups of rank 2 for which we know, apart from the validity of the BMR freeness conjecture 2.2, the validity of the BMM symmetrising trace conjecture 2.3 as well. Our method is not an algorithm and we cannot be sure it works in general since, as we will see in a while, one needs to make some crucial choices, which are a product of experimentation and experience. At the end of this section we give in detail the example of G4, where the reader can see thoroughly our methodology and arguments. We recall that W is generated by distinguished pseudo-reflections s. We choose a particular generator s0 of W and we denote by W ′ the parabolic subgroup of W generated by s0 and by H ′ the subalgebra of H generated by σ0. The action of W on the cosets of W ′ defines a graph with vertex set {W ′x | x ∈ W} and edges W ′x s−→ W ′xs, for s running over the generators of H. In the project’s webpage (https://www.eirinichavli.com/center.html) the reader can find these graphs for the examples we are dealing with in this paper. 123 https://www.eirinichavli.com/center.html Beitr Algebra Geom (2024) 65:319–336 327 Wenow choose class representatives xi as follows:We choose some representatives as the anchor coset representatives, and we pick representatives for the remaining cosets along the spanning tree. We always choose x1 = 1. The coset representatives xi are in fact explicit words in generators of W. These generators are in one to one correspondence with generators of the Hecke algebra H, by sending s to σ. Hence, we can obtain from the elements xi corresponding elements inside the Hecke algebra, which we also denote by xi. Our goal now is to prove that this chosen set {xi : i = 1, . . . , |W/W ′|} is a basis of H as H ′-module. Since H is a free H ′-module of dimension |W/W ′| (see, for example, Chavli and Chlouveraki 2022), we only have to prove that {xi} is a spanning set for H. By construction, we always have x1 = 1 and, hence, it is enough to prove that for each xi, the elements xi.σ are linear combinations of the form ∑ i hi · xi, where hi ∈ H ′. In order to prove that, we construct a coset table, where we list this linear combina- tion not only for the elements xi.σ, but also for the elements xi.σ −1. The reason we calculate these extra linear combinations is because they are prerequisite to calculating the linear combinations for the elements xi.σ. In order to fill the coset table we use a program created in GAP. This program uses the positive and inverse Hecke relations (2.3) and (2.4) and also some sim- ple hand-calculations.On the project’swebpage (https://www.eirinichavli.com/center. html) one can find these programs for the aforementioned exceptional groups of rank 2. The completed coset table and the fact that H ′ is a free R-module provides a basis for the Hecke algebra H over R as follows: B = {xi, σ0xi, . . . ,σ es−1 0 xi : i = 1, . . . , |W/W ′|}. Our goal now is to express every element of the Hecke algebra as R-linear com- bination of the elements of B. In order to do that, we use again the completed coset table. The expression of xi.σ as linear combination of the form ∑ i hi · xi, hi ∈ H ′ yields a representation ρ for H as a free module over the subalgebra H ′. We use the matrix models of this representation in order to compute the image of each word x in generators σ (and their inverses) inside the Hecke algebraH as the image of the vector 1H = 1H ′x1 + 0x2 + 0x3 + · · · + 0x|W/W ′| = (1H ′, 0, 0, . . . , 0) under the product of the matrices corresponding to the letters of the word x. We can see this as follows: Let x be the word σ m1 i1 · · ·σmr ir , where m1, . . . ,mr ∈ Z ∗. Then the image of x inside H can be computed as the product 1H · ρ(σi1 )m1 · · · ρ(σmr ir ). This is a vector with coefficients in H ′ and, hence, it corresponds to an H ′-linear combination of xi’s and, hence, to an R-linear combination of elements in B. In order to make this clearer to the reader, we give the following example of the exceptional group G4: Example 2.4 Let G4 = 〈s1, s2 | s3 1 = s3 2 = 1, s1s2s1 = s2s1s2〉. The Hecke algebra H of W is defined over R = Z[u± 1 ,u± 2 ,u± 3 ] and it admits the following presentation: H = 〈σ1,σ2 | σ1σ2σ1 = σ2σ1σ2, σ3 1 = aσ2 1 + bσ1 + c, σ3 2 = aσ2 2 + bσ2 + c〉, 123 https://www.eirinichavli.com/center.html https://www.eirinichavli.com/center.html 328 Beitr Algebra Geom (2024) 65:319–336 with suitable a,b, c ∈ R. For the sake of brevity, we set σ ′ 1 = σ−1 1 and σ ′ 2 = σ−1 2 . Let z := (s1s2) 3 ∈ Z(W) and let w := s1s2s1 = s2s1s2. We have z = (s1s2) 3 = (s1s2s1)(s2s1s2) = w2. We also denote by W ′ the parabolic subgroup of W generated by s1 and by H ′ the subalgebra of H generated by σ1. As we have already mentioned, the action ofW on the cosets ofW ′ defines a graph with vertex set {W ′x | x ∈ W} and edges W ′x u−→ W ′xu, labelled by u ∈ {s1, s2}. The following diagram shows this action graph, except for a few edges: 1 3 5 7 2 4 68 Here, instead of labeling the edges with u, we use the colours red and blue. More precisely, blue edges belong to generator s1 and red edges belong to generator s2. Fat edges indicate a spanning tree of the coset graph. The vertices are labeled 1,2 , . . . ,8, corresponding to coset representatives x1 = 1, x2 = s2, x3 = s2s1s2, x4 = s2s1s2s1, x5 = zx1, x6 = zx2, x7 = zx3, x8 = zx4. We make this choice of representatives as follows: We choose x1 = w0, x3 = w1, x5 = w2 and x7 = w3 as anchor coset representatives and we pick representatives for the remaining cosets along the spanning tree: x2� = x2�−1u, � = 1, 2, 3, 4 with generator u = s1, s2 as in the edge connecting coset 2�−1 to coset 2� in the coset graph. By sending s1 and s2 to σ1 and σ2, respectively we can obtain from the elements xi, z and w corresponding elements inside the Hecke algebra, which we also denote by xi, z and w. The elements z and w are also central elements in H (see Bessis 2015, Theorem 12.8 and Broué et al. 1998, Theorem 2.2.4). As we have explained earlier, in order to prove that the set {xi, , i = 1, . . . , 8} is a basis of H as H ′-module we construct the following coset table. xi, i = 1, . . . , 8 xi.σ1 xi.σ2 xi.σ ′ 1 xi.σ ′ 2 x1 = z0 σ1 · x1 x2 σ ′ 1 · x1 x2 = z0 σ2 σ ′ 1 · x3 x : σ ′ 1 x1 x3 = z0 σ2σ1σ2 x4 σ1 · x3 σ1 · x2 σ ′ 1 · x3 x4 = z0 σ2σ1σ2σ1 x4 : σ1 σ ′ 1 · x5 x3 x4 : σ ′ 2 x5 = z1 σ1 · x5 x6 σ ′ 1 · x5 σ1 · x4 x6 = z1 σ2 σ ′ 1 · x7 x6 : σ2 x6 : σ ′ 1 x5 x7 = z1 σ2σ1σ2 x8 σ1 · x7 σ1 · x6 σ ′ 1 · x7 x8 = z1 σ2σ1σ2σ1 x8 : σ1 x7 123 Beitr Algebra Geom (2024) 65:319–336 329 In this table, some notation is used as short-hand for more complex expressions. More precisely: • xi : u indicates that xi.u can be computed from other entries in the table by using the relation u = a + bu ′ + cu ′u ′. This relation is obtained from the positive Hecke relation (2.3) if we multiply both sides with u ′u ′. We see how we use such a relation in the example of x4.σ1. We have: x4.σ1 = ax4+bx4.σ ′ 1+cx4.σ ′ 1σ ′ 1. We now notice that x4.σ ′ 1 = x3. Therefore, x4.σ1 = ax4 +bx3 + cx3.σ ′ 1. Since x3.σ ′ 1 = σ1.x2 we have that x4.σ1 = ax4 + bx3 + cσ1.x2. • Similarly, x : u ′ indicates that x.u ′ can be computed by using the inverse Hecke relation (2.4) u ′ = c−1u2 − ac−1u − bc−1. It remains now to compute the entries x1.σ ′ 2, x2.σ2, x8.σ2 and x8.σ ′ 2. We compute these four excluded cases as follows, using braid relations and existing entries from the coset table. x2.σ2 = x2.(σ2σ1σ2)σ ′ 2σ ′ 1 = x2.(σ1σ2σ1)σ ′ 2σ ′ 1 = x4.σ ′ 2σ ′ 1 x8.σ ′ 2 = x8.(σ ′ 2σ ′ 1σ ′ 2)σ2σ1 = x8.(σ ′ 1σ ′ 2σ ′ 1)σ2σ1 = x6.σ2σ1 x1.σ ′ 2 = x1 : σ ′ 2 x8.σ2 = x8 : σ2 The completed coset table and the fact that H ′ is a free R-module with basis {1, σ1, σ2 1} proves the following: Proposition 2.5 With the above notation (i) H is a free H ′-module with basis {xi : i = 1, . . . , 8}. (ii) H is a free R-module with basis {bj : j = 1, . . . , 24} = {xi, σ1xi, σ2 1xi : i = 1, . . . , 8}. We will now express every element of the Hecke algebra as R-linear combination of the elements of {bj : j = 1, . . . , 24}, using the following representation for H as a free module over the subalgebra H ′, which comes from the completed coset table: ρ(σ1) := ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ σ1 . . . . . . . . . σ ′ 1 . . . . . . . . 1 . . . . . cσ1 b a . . . . . . . . σ1 . . . . . . . . . σ ′ 1 . . . . . . . . 1 . . . . . cσ1 b a ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ρ(σ2) := ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ . 1 . . . . . . . . −bc−1 . −ac−1σ ′′ 1 −bc−2σ ′ 1 −ac−2σ ′′ 1 c−2σ ′′ 1 . . σ1 . . . . . . . . . σ ′ 1 . . . . . . . . 1 . . . . . cσ1 b a . . . . . . . . σ . c3σ2 1 bc2σ2 1 bc2 + b2cσ1 bcσ2 1 −a2c + b2σ1 ac −a2 + aσ1 a ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ 123 330 Beitr Algebra Geom (2024) 65:319–336 where σ ′′ 1 = σ−2 1 . We recall that σ ′ = σ−1 = −bc−1−ac−1σ−c−1σ2 and, hence, σ ′′ = −bc−1σ−1 −ac−1 −c−1σ1 = −bc−1(−bc−1 −ac−1σ−c−1σ2)−ac ′ − c ′σ1. Therefore, the entries of the matrices ρ(σ1) and ρ(σ2) involve only positive powers of σ1. In the following example, one can see how we can use ρ in order to express every element of the Hecke algebra as an R-linear combination of elements in the basis {bj : j = 1, . . . , 24} = {xi, σ1xi, σ2 1xi : i = 1, . . . , 8}. σ2 1σ2 2 = (1H′ , 0, 0, 0, 0, 0, 0, 0) · ρ(σ1) · ρ(σ1) · ρ(σ2) · ρ(σ2) = (0, 0, −bc−1σ2 1, 0, −ac−1, −bc−2σ1, −ac−2, c−2) = 0x1 + 0x2 − bc−1σ2 1x3 + 0x4 − ac−1x5 − bc−2σ1x6 − ac−2x7 + c−2x8 = −bc−1b9 − ac−1b13 − bc−2b17 − ac−2b19 + c−2b22. �� 3 Center Let W be a complex reflection group and let H be the associated Hecke algebra, defined over the Laurent polynomial ring R = Z[us,j, u−1 s,j ]. We recall that F denotes the splitting field C(vs,j), with the indeterminates vs,j as defined in (2.7). We also recall that FH denotes the algebra H ⊗R F. As we have mentioned in Sect. 2, we have the validity of the BMR freeness con- jecture 2.2. As a result, we can fix a basis B = {bw : w ∈ W} of H, indexed by the elements of W. We now assume also the validity of the BMM symmetrising trace conjecture 2.3, meaning that H admits a unique symmetrising trace τ. We denote by B∨ = {b∨ w : w ∈ W} the dual basis of B with respect to τ, uniquely determined by the condition τ(bw1 b∨ w2 ) = δw1,w2 for all w1,w2 ∈ W. We denote now by Cl(W) the set of conjugacy classes of elements of W. For each class C ∈ Cl(W), we choose a representative wC ∈ C. The square matrix( χ(bw) ) χ,C of character values is invertible as it specializes to the character table of W. Hence, for each w ∈ W, there are uniquely determined coefficients fw,C ∈ F by the condition χ(bw) = ∑ C∈Cl(W) fw,C χ(bwC ) for all χ ∈ Irr(FH). These coefficients depend on the choice of the elements wC and of the basis B. Remark 3.1 Let W be a real reflection group. Then, we choose B to be the standard basis {Tw : w ∈ W} and wC an element of minimal length in C. We know (Geck and Pfeiffer 2000, §8.2.2 and §8.2.3) that the coefficients fw,C are independent of the actual choice of the elements wC and they belong to R. In this case, fw,C are known as class polynomials. 123 Beitr Algebra Geom (2024) 65:319–336 331 The following theorem has been shown by Geck and Rouquier in the real case (see Geck andRouquier 1997, §5.1 orGeck andPfeiffer 2000, Theorem8.2.3 andCorollary 8.2.4). However, the proof in general for every complex reflection group is slightly different from the original, as the prior existence of class polynomials in the complex case cannot be assumed. Theorem 3.2 Let W be a complex reflection group. The elements yC = ∑ w∈W fw,C b∨ w, C ∈ Cl(W), form a basis of the center Z(FH). Proof For each class C ∈ Cl(W), we define the function fC : H → F, bw �→ fw,C. We first prove that fC is a trace function. For each bw ∈ B and for each χ ∈ Irr(FH) we have: χ(bw) = ∑ C∈Cl(W) fw,C χ(bwC ) = ∑ C∈Cl(W) fC(bw)χ(bwC ) = ∑ C∈Cl(W) χ(bwC ) fC(bw). Therefore, χ = ∑ C∈Cl(W) χ(bwC )fC, for all χ ∈ Irr(FH). (3.1) The matrix (χ(bwC ))χ,C specializes to the character table ofW, which has a nonzero determinant and, hence, it is invertible. We denote its inverse by (i(C,χ))χ,C. There- fore, it follows from (3.1) that fC = ∑ χ∈Irr(FH) i(C,χ)χ, which proves that fC is a trace function. We now prove that the set {fC}C is a basis of the space of trace functions on FH. Since the algebra FH is split semisimple, we know (Geck and Pfeiffer 2000, Exercise 7.4(b)) that the set Irr(FH) is a basis for the space of trace functions on FH. Therefore, from Eq. (3.1) we conclude that {fC}C is a spanning set of this space. We now prove that the set {fC}C is a linearly independent set of trace functions. For this purpose, it suffices to prove that fC(bwC ′ ) = δC,C ′ . By definition, fC(bwC ′ ) = fw,C ′ and fw,C ′ is the coefficient of the columnof the character valueχ(bwC ′ ), when the column χ(bw) is expressed as a linear combination of the character values χ(bwC ). We have χ(bwC ) = 1 · χ(bwC ) + 0, 123 332 Beitr Algebra Geom (2024) 65:319–336 therefore fw,C ′ = 0, for all C ′ = C and fw,C = 1. We conclude that the set {fC}C is a basis of the space of trace functions on FH. We now prove that the set {yC}C is a basis of the center Z(FH). Since we assume that the algebra FH admits a symmetrising trace, we can apply (Geck and Pfeiffer 2000, Lemma 7.1.7), which states that the set {f∗ C}C is a basis of Z(FH), where f∗ C denotes the dual of fC ∈ HomF(FH, F). We have: f∗ C = ∑ w fC(bw)b∨ w = ∑ w fw,Cb∨ w = yC Therefore, the set {yC}C is a basis of the center Z(FH). �� We now prove a dual version of Theorem 3.2. For each class C ∈ Cl(W), we choose again a representative wC ∈ C and we define coefficients gw,C ∈ F by the condition χ(b∨ w) = ∑ C∈Cl(W) gw,C χ(b∨ wC ), for all χ ∈ Irr(FH). Theorem 3.3 The elements zC = ∑ w∈W gw,C bw, C ∈ Cl(W), form a basis of the center Z(FH). Proof For each class C ∈ Cl(W), we define the function gC : H → F, b∨ w �→ gw,C. As in the proof of Theorem 3.2, we prove that {gC}C is a basis of the space of trace functions on FH. For each bw ∈ B and for each χ ∈ Irr(FH) we have: χ(b∨ w) = ∑ C∈Cl(W) gw,C χ(b∨ wC ) = ∑ C gC(b∨ w)χ(b∨ wC ) = ∑ C χ(b∨ wC )gC(b∨ w) Therefore, χ = ∑ C∈Cl(W) χ(b∨ wC )gC, for all χ ∈ Irr(FH). (3.2) The matrix (χ(b∨ wC ))χ,C is invertible, since it specializes to the character table of W, which has a nonzero determinant. We denote its inverse by (j(C,χ))χ,C and Eq. (3.2) becomes: gC = ∑ χ∈Irr(FH) j(C,χ)χ 123 Beitr Algebra Geom (2024) 65:319–336 333 and, hence, gC is a trace function. Using the same arguments again as in the proof of Theorem 3.2, we have gC(b∨ C ′) = δC,C ′ and, hence, together with Eq. (3.2) we conclude that the set {gC}C is a basis of the space of trace functions on FH. We now prove that the set {zC}C is a basis of the center Z(FH). Since we assume that the algebra FH admits a symmetrising trace, we can apply (Geck and Pfeiffer 2000, Lemma 7.1.7), which states that the set {g∗ C}C is a basis of Z(FH), where g∗ C denotes the dual of gC ∈ HomF(FH, F). We have: g∗ C = ∑ w gC(b∨ w)(b∨ w)∨ = ∑ w gw,Cbw = zC Therefore, the set {zC}C is a basis of the center Z(FH). �� Note that the elements zC depend on the choice of the basis B and of the class representatives wC, C ∈ Cl(W). We focus now on the exceptional groups we have described in Sect. 2, namely the groups Gn, where n ∈ {4, . . . , 8, 12, 13, 22}. In Sect. 2 we have made a particular choice of a basis B and we managed with the help of the coset table to express each element of the Hecke algebra as linear combination of elements in B. In particular, we can express any product bb ′, with b,b ′ ∈ B as a linear combination of the elements in B. We use this linear combination in order to give another proof the BMM symmetrising trace conjecture 2.3 (this conjecture is known to hold for these complex reflection groups, as we have mentioned in Sect. 2.1). We define a linear map τ : H → R by setting τ( ∑ b∈B αbb) = α1. We can now calculate the Gram matrix A = (τ(bb ′))b,b ′∈B and prove the following: Proposition 3.4 Let W be one of the groups Gn, where n ∈ {4, . . . , 8, 12, 13, 22}. With the above choice of B, we have: (i) The matrix A is symmetric and its determinant is a unit in R. (ii) Condition (2.6) is satisfied. We note here that in the project’s webpage (https://www.eirinichavli.com/center.html) we give explicitly for each group the determinant of the Gram matrix A. Our goal is to find the coefficientsgw,C. By their definition, we need to find the dual basis B∨. More precisely, we need to express each b∨ i ∈ B∨ as R-linear combination in elements ofB. Letb∨ i = ai 1b1+ai 2b2+· · ·+ai |W|b|W| be this linear combination. Then, we have the following: ⎛ ⎜ ⎜ ⎜ ⎝ τ(b∨ i b1) τ(b∨ i b2) ... τ(b∨ i b|W|) ⎞ ⎟ ⎟ ⎟ ⎠ = A ⎛ ⎜ ⎜ ⎜ ⎝ ai 1 ai 2 ... ai |W| ⎞ ⎟ ⎟ ⎟ ⎠ 123 https://www.eirinichavli.com/center.html 334 Beitr Algebra Geom (2024) 65:319–336 By Proposition 3.4(i) A is invertible in R. Moreover, we know that the dual basis is determined by the condition τ(b∨ k bj) = δkj. Therefore: ⎛ ⎜ ⎜ ⎜ ⎝ ai 1 ai 2 ... ai |W| ⎞ ⎟ ⎟ ⎟ ⎠ = A−1 ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ 0 ... 1 i − place ... 0 = i-th column of A−1 Having now expressed each b∨ i ∈ B∨ as R-linear combination in elements of B, it remains to make a choice of class representativeswC, which will allow us to calculate the coefficients gw,C. We give for each group this particular choice of representa- tives in the webpage of this project (https://www.eirinichavli.com/center.html). The following result is the main theorem of this paper. Theorem 3.5 Let W be the exceptional group Gn, where n ∈ {4, . . . , 8, 12, 13, 22}. There exists a choice of a basis {bw : w ∈ W} of the Hecke algebra H of W and a choice of conjugacy class representatives {wC : C ∈ Cl(W)}, such that the coefficients gw,C belong to R and, hence, the set {zC : C ∈ Cl(W)} is a basis of Z(H). For a better understanding of the above theorem, we revisit the example of G4. Example 3.6 Let W be the complex reflection group G4. In the Example 2.4 we saw that the associated Hecke algebra H admits the basis {b3k+m : k = 0, . . . , 7, m = 1, 2, 3}, defined as b3k+m = σm−1 1 xk+1, where x1 = 1, x2 = σ2, x3 = σ2σ1σ2, x4 = σ2σ1σ2σ1, x5 = zx1, x6 = zx2, x7 = zx3, x8 = zx4. We now make the following choice for the class representatives wC: wC1 = b1, wC2 = b10, wC3 = b13, wC4 = b15, wC5 = b22, wC6 = b23, wC7 = b24. The evaluation of the irreducible characters of H on the dual basis {b∨ i } yields the elements gi,C and hence an explicit basis of Z(FH): z1 = ac3(b3 + b5) + (abc2 + c3)(b6 + b7) + (bc2 + ab2c − a2c2)b8 + ac2(b9 + b11) + (abc + c2)b12 + (2bc + ab2 − a2c)b14 + c(b18 + b19) + bb20 + b24 z2 = c3b3 + bc2(b6 + b7) + b2c b8 + bcb12 + (−ac + b2)b14 + c b17 − ab20 + b21 + b23 z3 = c(b14 + b16) − ab19 + b20 + b22 123 https://www.eirinichavli.com/center.html Beitr Algebra Geom (2024) 65:319–336 335 z4 = c2b5 − acb8 + c(b9 + b11) − ab14 + b15 z5 = b13 z6 = c(b2 + b4) − ab7 + b8 + b10 z7 = b1 As we can see, the elements gi,C are actually elements in R and, hence, the set {z1, . . . , z7} is indeed a basis of Z(H). �� Based on these examples and the fact that complex reflection groups generalize the properties of real reflection groups, we believe that one can find in this way a basis {zC : C ∈ Cl(W)} of Z(H) for each complex reflection group W. Therefore, we state the following conjecture: Conjecture 3.7 Let W be a complex reflection group. There exists a choice of a basis {bw : w ∈ W} of the Hecke algebra H of W, and a choice of conjugacy class representatives {wC : C ∈ Cl(W)} such that the construction of Theorem 3.3 yields polynomial coefficients gw,C ∈ R, and hence a basis {zC : C ∈ Cl(W)} of Z(H). Acknowledgements Work on this project started during the 2-Day Meeting in Stuttgart on Computational Lie Theory in July 2018, supported by DFG (SFB-TRR195). The authors would like to thank Gunter Malle and Ivan Marin for useful comments and discussions. Funding Open Access funding enabled and organized by Projekt DEAL. OpenAccess This article is licensedunder aCreativeCommonsAttribution 4.0 InternationalLicense,which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/. References Bessis, D.: Finite complex reflection arrangements are K(π,1). Ann. 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Note added in proof We mention here that, after this work was completed, Hu and Shi (2022) proved the aforementioned conjecture for the groups of type G(d, 1,n). 123 http://arxiv.org/abs/2211.07069 https://www.eirinichavli.com/center.html Centers of Hecke algebras of complex reflection groups Abstract 1 Introduction 2 Choosing a basis 2.1 Hecke algebras 2.2 Coset table 3 Center Acknowledgements References