A Gelfand Model for Weyl Groups of Type
A Gelfand model for a finite group is a complex representation of , which is isomorphic to the direct sum of all irreducible representations of . When is isomorphic to a subgroup of , where is the field of complex numbers, it has been proved that each -module over is isomorphic to a -submodule in the polynomial ring , and taking the space of zeros of certain -invariant operators in the Weyl algebra, a finite-dimensional -space in can be obtained, which contains all the simple -modules over . This type of representation has been named polynomial model. It has been proved that when is a Coxeter group, the polynomial model is a Gelfand model for if, and only if, has not an irreducible factor of type , , or . This paper presents a model of Gelfand for a Weyl group of type whose construction is based on the same principles as the polynomial model.
Gelfand models for a finite group are complex representations whose character is the sum of all irreducible characters of the given group. In this sense, Bernstein et al. have presented Gelfand models for semisimple compact Lie groups, see . Since then, Gelfand models have been developed in several articles; see [2–12], among these there are two types of models that can be associated with reflection groups: the involution model and the polynomial model.
Parallel works, made by Klyachko, on one side, and by Inglis, Richardson, and Saxl, on the other, showed an identity that describes a Gelfand model associated with the symmetric group. The identity is given by where is the centralizer of an involution in with exactly fixed points, is a linear character of , and is an irreducible character of associated with the partition of with exactly odd terms. From this identity, it follows immediately that where the centralizers are in correspondence with the conjugacy classes of involutions in .
Later on, this type of models was called an involution model by Baddeley . He also proved that if is a finite group that admits an involution model, then so does the semidirect product .
Baddeley’s result implies the existence of involution models for classic Weyl groups, with the exception of the group of type . An involution model for a Weyl group of type is presented in  by Inglis et al. and for a Weyl group of type an involution model is shown in [6, 13]. In , Baddeley presents an involution model for a Weyl group of type , and in  it is proved that there is no involution model for a Weyl group of type with . In , it is mentioned that is not difficult to prove that there is an involution model for a Weyl group of type and that it has been checked using computers the non existence of involution models for exceptional Weyl groups of type , , , and . In , Vinroot does some research about involution models for irreducible non crystallographic Coxeter groups. He proves the existence of an involution model for groups of type and and presents a conceptual demonstration of the no existence of an involution model for the group of type .
More recently, in  the generalized involution model has been studied in order to include some cases of unitary reflection groups.
A reflection group comes equipped with a canonical representation called the geometric representation of . The geometric representation induces a natural action of on the space of polynomial functions.
Chevalley , Shephard and Todd , Steinberg , and others studied the corresponding action on the space of -harmonic polynomials proving that is isomorphic to the regular representation of , and thus contains a Gelfand model for . On the other hand, Macdonald found irreducible representations of a Weyl group associated with the root systems of the reflection subgroups that can be naturally realized in the -harmonic polynomial space. These representations are known as Madonald representations see .
More recently, Araujo and Aguado in  have associated with each finite subgroup a subspace of the algebra of polynomials , defined as zeros of certain -invariant differential operators, and have shown contains a Gelfand model of . This space, called the polynomial model, is a Gelfand model for some Weyl groups. In [3–5], it was proved that is a Gelfand model for Weyl groups of type , and . Garge and Oesterlé in , using the computation of fake degrees of the irreducible characters of a Coxeter group , determined that is a Gelfand model of if, and only if, has not irreducible factors of type , , or . The fake degrees have been determined due to works of Steinberg , when is of type , Lusztig , when is of type or , Beynon and Lusztig , when is an exceptional Weyl group, Alvis and Lusztig , when is of type , and Macdonald, when is of type (unpublished). The remaining cases are not difficult.
For the case of Weyl groups of type , neither the polynomial model nor the involution model provides a Gelfand model.
In this paper the construction of a Gelfand model for a Weyl group of type will be presented. It will be built upon a light modification of the polynomial model.
2. Polynomial Model
The notation introduced in this section will be used in the remaining of this paper.
will denote a finite subgroup of and the polynomial ring .
Let be the set of the first natural numbers and the set of multi-index functions:
For each the following notation will be used in the rest of this paper:
Let be the Weyl algebra of -linear differential operators generated by the multiplication operators and partial differential operators with .
It is known that each has a unique expression as a finite sum (see ): where , , and The degree of is defined by The Weyl algebra is a graduated algebra , where The action of on induces an action of on the endomorphism ring , which is defined by This action can be restricted to the Weyl algebra noting that each is invariant under the action of .
Let be the subalgebra of -invariant operators in , that is, Notice that is contained in the centralizer of in .
Let be the subspace of the Weyl algebra, formed by the -invariant operators with negative degree
Definition 2.1. Let be the subspace of defined by is named the polynomial model of .
Notice that is a -module.
Below, some properties of will be mentioned.
Theorem 2.2. is a finite-dimensional -module, and every simple -module has a copy in .
Proof. See [21, page 38].
The analysis of the polynomial model for Coxeter groups has been completely solved by the following theorem.
Theorem 2.3. Let be a finite irreducible Coxeter group, and let be its realization as a reflection group. Then, the polynomial model is a Gelfand model for if, and only if, is not a Weyl group of type , , or .
Proof. See [22, page 7].
In the following sections it will be presented a characterization of the polynomial model for the classical Weyl groups of type , and .
2.1. Polynomial Model for a Weyl Group of Type
Let be a Weyl group of type . It is known that can be presented as the symmetric group .
The symmetric group acts on the set of multi-index functions by This action induces a natural homomorphism from in given by
2.1.1. -Minimal Orbit
Let be the orbit space of in . It is clear that if two multi-indexes and belong to the same orbit , then and take the same value, where and are defined by (2.2), and this value will be denoted by .
Definition 2.4. Two orbits and will be called -equivalent, denoted by , if there exists a bijection such that
Definition 2.5. An orbit will be called -minimal if for all such that .
Proposition 2.6. An orbit is -minimal if, and only if, for each , there exists a nonnegative integer such that (1),(2) for all ( being the cardinal of the set ).
Proof. See [4, page 1845].
Definition 2.7. For each , let be the subspace of defined by
2.1.2. The Space
Let be the operator defined by , where are the partial differential operators as above. For each , let be the subspace defined by
2.1.3. The Structure of
Below the main theorem regarding is announced without proof. For further details see [4, page 1850].
Theorem 2.8. is an irreducible -module, and can be decomposed as Moreover, is a Gelfand model of .
2.2. Polynomial Model for a Weyl Group of Type
The Gelfand model for a Weyl group of type will be described using the same ideas as the previous section.
Let be the subgroup of order two. The Weyl group , of type , can be presented as the semidirect product where and the semidirect product is induced by the natural action of on :
The action of on induces a natural homomorphism from on given by with
2.2.1. -Minimal Orbit
Let be the orbit space of on , as above.
Definition 2.9. Two orbits, and , will be called -equivalent, denoted by , if there exists a bijection such that and have the same parity for all ,.
Definition 2.10. An orbit will be called -minimal if for all such that .
Proposition 2.11. An orbit is -minimal if, and only if, for each and each pair with the same parity, one has with ( being the cardinal of the set ).
Proof. See [3, page 365].
2.2.2. The Space
Let be the Laplacian operator defined by , where are the partial differential operators mentioned above. For each , let be the subspace defined by
2.2.3. The Structure of
Below the main theorem regarding is announced without proof. See references.
Theorem 2.12. is an irreducible -module, and can be decomposed as Moreover is a Gelfand model of .
Proof. See [3, page 371].
2.3. Polynomial Model for a Weyl Group of Type
Let be the Weyl group of type naturally included in . Using the previous notation, for the following sets are considered:
It is easy to check that the cardinals and are equal for all elements in the same orbit . Therefore, these values will be denoted by and , respectively.
2.3.1. -Minimal Orbit
Definition 2.13. Two orbits and will be called -equivalent, denoted by , if there exists a bijection such that(1), and have the same parity or and have different parities,(2).
Definition 2.14. An orbit will be called -minimal if for all such that .
Proposition 2.15. Let be an orbit, and then the following statements are true (1) is -minimal if, and only if, the following statements are verified:(a)given and with the same parity, then ,(b). (2)Let be the involution given by and . The following assertions are equivalent:(a) and are -minimal orbits,(b) is -minimal,(c) is -minimal. (3)There are at most two -minimal orbits equivalent to .(4)If is odd, there is only one -minimal orbit equivalent to .(5) and are -minimal orbits if, and only if, .
Proof. See [5, page 106].
Proposition 2.16. Let be odd, and then the following statements are true. (1)If is -minimal, then and is a simple -module.(2) is a Gelfand model for .(3)Every simple -module remains simple when it is considered as a -module by restriction.(4)By considering as a -module by restriction, is isomorphic to .
Proof. See [5, page 110].
Also in  it has been proved that if is even, is not a Gelfand model for a Weyl group of type . But it does happen that if is a simple -module, then contains a copy ofthis, and the multiplicity of in is(1)two, if is isomorphic to being a -minimal orbit such that and ; in this case, as before, is the involution given by and ,(2)one, otherwise.
3. Gelfand Model for a Weyl Group of Type
As before, let be the set of multi-index functions. Every has an associated vector , which is obtained by reordering as follows. Thus, there is defined an order relationship in given by for all , if, and only if, or there exists such that and being the coordinates of the vectors and , respectively. Notice that this is the lexicographic order for .
Proposition 3.1. Let and , and then if, and only if, .
Proof. Let , and therefore there exists such that , which implies with . Thus, it is easy to see that .
On the other hand, let and , say, Let be given by Then, , and hence both multi-indexes belong to the same orbit.
From this proposition it is clear that induces a total orderin , which is defined by Since the vector is independent of the choice in , it will be denoted by .
Proposition 3.2. Let be defined by , and let be the orbit of . Then, is the -maximum of the -minimal orbits and .
Proof. From the previous considerations it is clear that is an -minimal orbit and .
Now it will be proved that is the ⪯-maximum in the set of -minimal orbits. Let be an orbit such that and . Then, it should exist an satisfying that is,
Thus, it occurs that , and from the minimality of every number less than must appear at least twice, which is a contradiction. And therefore is the maximum.
From now on, for a finite set , will denote the symmetric group of , will denote the set of multi-index functions with domain , and ı will denote the function in given by As in the case of , there is a natural action from the symmetric group in the set of multi-index functions , defined by It is possible to extend the concept of -minimal orbit.
For each , let us consider the sets as defined in the previous section. Then, it is clear that can be determined from its restrictions and to the sets and , respectively. Observe that and .
Proposition 3.3. Let such that is -minimal, and then is -minimal and is -minimal.
Proof. It follows from Proposition 4 in  and the identities
Notation 1. Let be the subset of given by It will be denoted by the subset of the polynomial ring : Note that if is odd, is equal to .
Proposition 3.4. is a -submodule of the polynomial ring .
Proof. It follows from the action of given by and the fact that if, and only if, , and it results that is a -module of .
Proposition 3.5. contains a Gelfand model for the Weyl group of type .
Proof. It is sufficient to prove that contains a submodule equivalent to the regular module . Effectively, let us consider the polynomial:
Thus is the interpolating polynomial of the orbit of the regular vector , which satisfies
It will be proved that belongs to . Let be not a null term of such that it is -minimal, , and . As was defined in (3.16), it is easy to determine that and for . Then, , and as is -minimal, it is obtained that which by Proposition 3.2 is maximal, which is a contradiction.
Since is a -module, contains the module generated by the orbit of , which is isomorphic to the regular module. Hence, contains a Gelfand model.
Notation 2. Let be a subset of the polynomial ring and a finite subgroup of ; we will denote by the subset of defined by being the set of differential operators invariant in the algebra of Weyl as it has been defined in (2.9).
Proposition 3.6. Let be a finite subgroup of and a -module of the polynomial ring such that contains a model of , and then also contains a model of .
Proof. Let be a simple -module and suppose that . Then, there exists such that . Because is simple and not null, it follows that is injective. Thus, . If , the proposition is proved; otherwise the procedure will be repeated. As is an operator of the Weyl algebra with negative degree, the procedure is finite, that is to say, there exists such that and .
Remark 3.7. An immediate consequence of this proposition is that the -module contains a Gelfand model because is a -module containing a Gelfand model.
Remark 3.8. Notice that if is odd, then ; instead, if is even, that is
Using the result established in item 4 of Proposition 2.16 for decomposing , it follows that Moreover if
As a consequence of this decomposition the next lemma follows.
Lemma 3.9. The dimension of is equal to
Proof. It results from considering the identity for each . This identity occurs from the fact that and are isomorphic as -modules, see .
Let be a group; from now on we will be denote by the set of involutions of the group .
Lemma 3.10. Let be a Coxeter group and a Gelfand model for . Then,
Proof. It is a consequence from the Fröbenius-Schur indicator and the fact that the representations of a Coxeter group can be realized over the real numbers, see .
With the purpose to establish the central result of this work, a relationship between the number of involutions of and the number of involutions of will be given. This will be used in the next theorem.
Lemma 3.11. If is even , then
Proof. If , with , and is an involution, then the cyclic structure of looks like where , is the decomposition of as product of disjoint cycles, and for . Thus, the number of involutions of is If , half of the elements belong to and the other half to , and therefore Then,
Theorem 3.12. The -module is a Gelfand model for the group .
Proof. As it has been mentioned above, when is odd, is equal to , and in  it has been proved that is a Gelfand model for the group .
When is even, from the fact contains a Gelfand model, only it is necessary to prove that . From identity (3.22), it results that By Lemma 3.10, it follows that the dimension of the model is equal to the number of involutions of the group , and thus by the Lemma 3.9 it results that and then On the other hand, from identity established in Lemma 3.11, it results that and using identities (2.2) and (3.36), it is obtained that Therefore, it has been proved that is a Gelfand model for .
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