Last update: 22 January 2014
A reflection is an element of that has exactly one eigenvalue not equal to The reflecting hyperplane of a reflection is the subspace which is fixed pointwise. A complex reflection group is a finite subgroup of generated by reflections. The group is irreducible if cannot be written in the form where and are subspaces. The group is a real reflection group if for a real vector space and
The following facts about reflection groups are well known.
Let be an irreducible reflection group.
|(a)||[STo1954, Theorem 5.3] The number of elements such that is where are the exponents of|
|(b)||[Car1972, Lemma 2] If is a real reflection group and with then is the product of two reflections.|
|(c)||[OTe1992, Theorem 6.27] For any the space is the intersection of reflecting hyperplanes.|
Remark. The statement of Lemma 2.1b does not hold for complex reflection groups. Consider the exceptional complex reflection group of rank in the notation of Shephard and Todd [STo1954]. All the reflections have order and Suppose for two reflections and If has eigenvalues and where is a primitive cube root of unity, then has eigenvalues and a contradiction to the assumption that is a reflection. Thus is not a product of two reflections.
Let be a complex reflection group. Let be a graded Hecke algebra for and let Let be the invariants in
|(a)||If and then|
|(b)||If the order of is then|
(a) Let be a nondegenerate Hermitian form on and write where Since and is skew symmetric, restricted to is There is a basis of and constants such that the reflections given by are in Equation (1.6) implies that, for any since (as is skew symmetric). Since for Thus
(b) Since all eigenvalues of are If then by Theorem 1.9b. If then as a linear transformation on By [Ste1964, Theorem 1.5], [Bou1981 V, §5 Ex. 8], the stabilizer, of is a reflection subgroup of and so there is a reflection that is the identity on So and where is restricted to Thus, by Theorem 1.9b,
2A. Real reflection groups
If is a real reflection group then and where is a real vector space. We shall assume that is irreducible.
Let us recall some basic facts about real reflection groups which can be found in [Hum1990] or [Bou1981]. The action of on has fundamental chambers indexed by The roots for are vectors such that the reflections in are the reflections in the hyperplanes For each fundamental chamber the reflections in the hyperplanes that bound form a set of simple reflections for The simple reflections obtained from a different choice of fundamental chamber are
Let be a real reflection group. Let be a set of simple reflections in and let be the order of Then satisfies and for all (the conditions in Theorem 1.9c) if and only if is conjugate to for some
Let and be two roots such that (see Lemma 2.1c). Then has nontrivial intersection with some fundamental chamber for and we may assume that and are walls of the chamber (since is a cone in Since choosing simple reflections with respect to a different chamber corresponds to conjugation by we may assume that the reflections in the hyperplanes and are simple reflections and and
The element is an element of the stabilizer which is a reflection group by [Ste1964, Theorem 1.5]. Since is a rank two real reflection group, and therefore a dihedral group. This dihedral group is generated by the two simple reflections and in the hyperplanes and (restricted to and all reflections have determinant Let be the element restricted to Since and so must be a product of an even number of reflections. Thus or for some where is the order of Since and so is conjugate to for some
Assume that for some Then and so Since is a product of an even number of reflections, The only elements of that are diagonalizable in are and elements with determinant Thus, the eigenvectors of the element (which has distinct eigenvalues since it is not do not lie in only in Let and let denote restricted to Since commutes with and has distinct eigenvalues, and have the same eigenvectors. Hence,
Using Theorem 2.3 and Theorem 1.9b, we can read off the graded Hecke algebras for the irreducible real reflection groups from the Dynkin diagrams. For each irreducible real reflection group, label a set of simple reflections using the Dynkin diagrams below. If nodes and and nodes and are connected by single edges, then is conjugate to via the element
The following table gives representatives of the conjugacy classes of that may have for some graded Hecke algebra We assume that the reflection group is acting on its irreducible reflection representation When is the symmetric group acting on an vector space by permutation matrices, then and, by Lemma 2.2a and Theorem 2.3, for some graded Hecke algebra only if is conjugate to the three cycle (this example is analyzed in Section 3).
2B. Complex reflection groups
The irreducible complex reflection groups were classified by Shephard and Todd [STo1954]. There is one infinite family denoted and a list of exceptional complex reflection groups denoted In this subsection, we classify the graded Hecke algebras for the groups
Let and be positive integers with dividing and let Let be the symmetric group of matrices and let where appears in the entry. Then For let Then the multiplication in is described by the relations where acts on by permuting the factors. Let be the column vector with in the entry and all other entries The group acts on with orthonormal basis as a complex reflection group.
Every real reflection group is a complex reflection group and several of these are special cases of the groups In particular,
|(a)||is the symmetric group|
|(b)||is the Weyl group of type|
|(c)||is the Weyl group of type and|
|(d)||is the dihedral group of order|
The reflections in are where is the transposition in that switches and
Conjugacy in Each element of is conjugate by elements of to a disjoint product of cycles of the form By conjugating this cycle by we have If denote the minimal indices of the cycles and are the numbers for the various cycles, then after conjugating by each cycle becomes where and If is the length of the last cycle, then conjugating the last cycle by gives If we conjugate the last cycle by we have In summary, any element of is conjugate to a product of disjoint cycles where each cycle is of the form except possibly the last cycle, which is of the form where is the length of the last cycle.
Centralizers in Let denote the centralizer of Since is a subgroup of for any element Suppose that is an element of which is a product of disjoint cycles of the form and that commutes with Conjugation by effects some combination of the following operations on the cycles of
|(a)||permuting cycles of the same type, and with and|
|(b)||conjugating a single cycle by powers of itself, and|
|(c)||conjugating a single cycle by for any|
Determining the graded Hecke algebras for It follows from Lemma 1.8a that if then has basis Thus, if and then is conjugate to one of the following elements: It is interesting to note that these elements are also representatives of the conjugacy classes of elements in which can be written as a product of two reflections.
We determine conditions on the above elements and on and to give nontrivial graded Hecke algebras:
|(z)||The center of is Since it follows that or whenever|
|(b1)||If the element and has determinant on|
|(b2)||If and mod the element and has determinant on|
|(c1)||If the element and has determinant on|
|(c2)||If and is odd, the element and has determinant on|
|(d1)||If the element and has determinant on|
|(d2)||If the element and has determinant on|
|(ef)||The elements and have order|
We arrive at the following enumeration of the nontrivial graded Hecke algebras for complex reflection groups. (The tensor product algebra always exists and corresponds to the case when all of the skew symmetric forms are zero). The table below gives representatives of the conjugacy classes of that may have for some graded Hecke algebra
2C. Exceptional complex reflection groups
The irreducible exceptional complex reflection groups are denoted in the classification of Shephard and Todd. From Table VII in [STo1954], one sees that the center of is only in the cases and By Schur’s lemma, the center of an irreducible complex reflection group consists of multiples of the identity. Thus, by Corollary 1.11, the only exceptional complex reflection groups that could have a nontrivial graded Hecke algebra (i.e., with some are and (we exclude the real groups). We determine the graded Hecke algebras for these groups using Theorem 1.9b and Lemma 2.2.
The rank group has order and seven conjugacy classes. The following data concerning these conjugacy classes are obtained from the computer software GAP [Sch1995] using the package CHEVIE [GHL1996]. In the following table, is a primitive cube root of unity and denotes the conjugacy class of The elements with determinant and order more than in all have order If is an element of order then and every element of has determinant since is generated by Hence, by Theorem 1.9b and Lemma 2.2, can be nonzero for a graded Hecke algebra for exactly when has order Thus, the dimension of the space of parameters for graded Hecke algebras of is
The rank group has order The computer software GAP provides the following information about the conjugacy classes of If is an element in with order more than and determinant then has order or Let be any element of order Then has determinant and commutes with of order Hence, by Theorem 1.9b, if has order then Let be a representative from the class of elements of order Since is generated by and hence every element of has determinant We can choose as a representative for the conjugacy class of elements of order As and commute, But so is generated by and every element of has determinant Thus, can be nonzero for a graded Hecke algebra for exactly when has order or Thus, the dimension of the space of parameters of graded Hecke algebras for is
The rank group has order Note that since Up to there are two codimension subspaces, and that are equal to for some (see [OTe1992, App. C, Table C.5]). Furthermore, and We need only consider elements of order in and of order in (as the rest have order or In there is only one conjugacy class of elements of order and only one conjugacy class of elements of order and determinant The table below (obtained using GAP) records information about these classes. If has order must contain and and hence is generated by these elements since Thus all elements of have determinant on If has order and determinant then must contain and elements which all have determinant on Since these elements generate and so every element of has determinant on Hence, can be nonzero for a graded Hecke algebra of exactly when has order or has order and determinant Thus, the dimension of the space of parameters for graded Hecke algebras for is
The group is the only exceptional complex reflection group of rank It has order and degrees There are reflecting hyperplanes and the corresponding reflections all have order Up to there are two codimension subspaces, and that are equal to for some (see [OTe1992, App. C, Table C.14]). Furthermore, and We need not consider the case where since then has order and hence for any graded Hecke algebra by Proposition 2.2b.
We use a presentation for in six coordinates from [STo1954]: Let and consider the group generated by order reflections about the hyperplanes where is a primitive cube root of unity. The fixed point space of this (reducible) group is and is just the restriction to Let be the order reflection about Let Then and Let the diagonal matrix with diagonal Then acts as times the identity on as Hence, commutes with But has dimension and has determinant on Thus, by Theorem 1.9b and Lemma 2.2, for any graded Hecke algebra. The same argument applied to shows that has no nontrivial graded Hecke algebras. In summary, the dimension of the space of parameters for graded Hecke algebras for is zero.
This is a typed version of Classification of graded Hecke algebras for complex reflection groups by Arun Ram and Anne V. Shepler.
Research of the first author supported in part by the National Security Agency and by EPSRC Grant GR K99015 at the Newton Institute for Mathematical Sciences. Research of the second author supported in part by National Science Foundation grant DMS-9971099.