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Araujo, José O.; Bratten, Tim; Maiarú, Cesar L.

doi:https://doi.org/10.1155/2015/808520

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Volume 2015 | Article ID 808520 | https://doi.org/10.1155/2015/808520

Irreducible Modular Representations of the Reflection Group

José O. Araujo,¹Tim Bratten,¹and Cesar L. Maiarú¹

Academic Editor: Li Guo

Received30 Jul 2015

Accepted19 Oct 2015

Published09 Nov 2015

Abstract

In an article published in 1980, Farahat and Peel realized the irreducible modular representations of the symmetric group. One year later, Al-Aamily, Morris, and Peel constructed the irreducible modular representations for a Weyl group of type . In both cases, combinatorial methods were used. Almost twenty years later, using a geometric construction based on the ideas of Macdonald, first Aguado and Araujo and then Araujo, Bigeón, and Gamondi also realized the irreducible modular representations for the Weyl groups of types and . In this paper, we extend the geometric construction based on the ideas of Macdonald to realize the irreducible modular representations of the complex reflection group of type .

1. Introduction

The irreducible modular representations for the symmetric group were first realized by Farahat and Peel in the manuscript [1] and one year later Al-Aamily et al. [2], using similar methods, constructed the irreducible modular representations for a Weyl group of type .

In an article published in 1972 [3], Macdonald introduced a geometric construction of some irreducible representations for the Weyl groups built from the action of a Weyl group on the associated root systems. Every irreducible representation of a Weyl group of type or can be realized as Macdonald’s representation (see Carter [4] or Lusztig [5]). Following Macdonald’s ideas, the irreducible modular representations for the symmetric group were constructed in [6] and in [7]; a similar circle of ideas produces the irreducible modular representations for a Weyl group of type . Along these lines, a construction of a Gelfand model for the complex reflection group was given in [8], also based on Macdonald’s ideas (see [9]).

The main result of this study is an extension of the construction given in [6] to obtain the irreducible modular representations of the group (see Theorem 25).

2. Notation and Preliminary Results

In this section, we introduce the main ingredients used in our construction and their respective notations. In particular, will denote the group , which is also known as the generalized symmetric group. will denote a field of characteristic . will denote the polynomial ring , and, for every , denotes the subspace of homogenous polynomials of degree . We let denote the set and indicates the permutation group of .

Initially we assume that contains all th roots of the unity; in particular, we assume that and are coprime. In order to deal with the general case, this hypothesis will be reconsidered at the end of the paper.

For convenience, the group will be presented as the semidirect product:where is the group of the th roots of the unity.

Each element has a unique decompositionwhere and .

In order to simplify notation, in certain cases, will be identified with and with , that is, with and with .

2.1. The Character

The following character will be utilized for defining projectors associated with subgroups of . Let be the linear character given bywhere is the sign map.

Remark 1. In the case where , the field of complex numbers, this character is, precisely, the determinant of the geometric representation of , realized as a group generated by unitary reflections (see [10]). Thus, we have

2.2. Subgroups

As in the case of characteristic zero, the irreducible representations are associated with subgroups of . We now introduce the type of subgroup that will be used for building irreducible modules.

If , let denote the subgroup of given byNote thatwhere is the group of all permutations which fix all elements in and is the subgroup of given by

2.3. The Action of the in and the Bilinear Form

From now on, will denote the set of multi-indexes and we use the notation

There is a natural action of in the polynomial ring as well as in its dual space , which gives them -module structure. This action is described as follows: consider acting on the set multi-indexes byThis action, in turn, naturally induces an action of on the polynomial ring given bywhere . On the other hand, for , we have

If is identified with the subgroup of , it is clear that the defined action extends the natural action of the symmetric group on the polynomial ring .

We will use the -bilinear form, , defined on determined bywhere and is the Kronecker function.

2.4. Multipartitions

The multipartitions are introduced as the natural parameters for enumerating the conjugacy classes of (see Proposition 4).

If is a partition of , it can be denoted as with or as , whereA partition of is called -regular if

A conjugacy class of a finite group is called -regular if does not divide the order of an element of the class.

Remark 2. For the symmetric group , the number of nonequivalent irreducible modular representations coincides with the number of -regular conjugacy classes of [11, Theorem 11.5]. For a general finite group this is true if is a decomposition field for (see [12, Theorem 21.25 and p. 492]).

Definition 3. -partition of is -uple where each is a partition andThe -partition is called -regular if all partitions are -regular.

The following results will be used later on. They are well known and will be presented without proof.

Proposition 4. There is bijective correspondence among the conjugacy classes of and the -partitions of .

Proof. See [13].

Lemma 5. The number of -regular partitions of coincides with the number of -regular conjugacy classes of .

Proof. See [11].

Remark 6. Each element in can be uniquely factorized, except for reordering, as a product of disjoint cycles, where there are distinct classes of these cycles. This is the reason for which the conjugacy classes of are indexed by -partitions of and, consequently, the -regular conjugacy classes are indexed by -regular -partitions; see [13].

3. The Modules and

As has been indicated above, it will be assumed that does not divide until the last section. After that, the extension to the general case does not present a major difficulty and will be treated at the end.

In this section two modules will be introduced: , associated with the symmetric group , and , associated with the group . Then, we will define spaces and of linear functionals in and , respectively, on which we will realize the irreducible modular representations of and . It is convenient to have in account what happens in the case of the symmetric group, as it can be extended to the construction for the group , from the key fact given in Theorem 12(i).

When is field of characteristic , it is not difficult to show that as -modules and as -modules.

For each nonempty set of , where , will denote the usual Vandermonde determinantTo each partition of , we associate the subset of given byIt is clear that is -orbit in with the action given in (9).

For each , , we consider the subset given bywhere is the conjugate partition of .

Now, we define and as follows:

Proposition 7. Let be the subgroup of defined asthen

Proof. The result is implied by following fact: given a nonempty subset of , where , is defined bythen

To each -orbit in , we associate the subspace of defined by

Definition 8. If is partition of , one defines to be the subspace of generated by the -orbit of . This means

Remark 9. If is a field of characteristic and , then coincides with the product of linear functionals associated with a set of positive roots of a subgroup of Gelfand of typeIn this case, according to [3], gives a Macdonald representation for , considered as a Weyl group of type , which is associated with a subgroup of the type given in (27) (see [4] or [5]). The -module can also be realized in the following way. Let be the differential operator defined bythenconforming with the statement of Theorem 4.2 of [14].

Let be -partition of and, for each , , and suppose that Define asand define bywhere is as in (20).

Now, if is the -orbit of in , it is clear that , where is as in (25).

Remark 10. Provided that the mapping which sends a partition to a multi-index is injective, the mapping which sends -partition to a multi-index is also injective.

Definition 11. If is -partition of , one defines as the subspace of generated by the -orbit of . This means that

Let be the differential operatorAs in the case of the symmetric group, by Theorem 2.5 in [8] it turns out that

3.1. The Subspaces and

We identify the symmetric group with .

For each partition of , consider (the dual subspace of ) defined byUsing the canonical action of on , we define as the subspace of generated by the -orbit of . This means thatAnalogously, for each -partition of , we consider given byWith respect to the canonical action of on , we define as the subspace of generated by the -orbit of . This means thatWith a different notation, the following result can be found in [6] Lemma 2.2 and Theorem 2.3.

Theorem 12. Keeping the previous notation one has the following:(i)If is -regular, then .(ii)If , is an irreducible -module.(iii)If and are -regular partitions and , then .(iv)Every simple -module is isomorphic to for some -regular partition of .

In the proof of Theorem 12, an idempotent in plays a central role. Analogously, in order to establish a similar result in the case of , an idempotent in is defined as follows.

Given , -partition of , withand as in (18), we define for each , , the subset given byThen, we haveLet be the subgroup defined bywhere is defined in (5). Since each element has a unique decomposition as with and , the subgroup also has a decomposition aswhereEach can be factorized uniquely aswhere is in . We let denote the linear character of given byFinally we define the operators and in by

Remark 13. A reflection in a vector space is a diagonalizable endomorphism such that the space of fixed points of is a hyperplane. Acting on , is realized as a group of reflections (this means it is generated by reflections). If is the field of complex numbers, from (42) and (43), the polynomial can be factorized as a product of linear functionals associated with a system of roots of the subgroup of where some of them, precisely the ones associated with the roots , are taken with certain multiplicities, in a way such that

Proposition 14. With the previous notations, for each -partition of , we have

(i) (ii) (iii)

Proof. (i) This identity is clear from the definition of .
(ii) It is a consequence of the decomposition (44).
(iii) By (22), we haveSince each element can be factorized as , wherewe haveHence,and (iii) is proved.

Using the same notation as in (42) we have the following lemma.

Lemma 15. Let be a reflection of order in and a root of . If is such thatfor some , then is a factor of .

Proof. From the hypothesis, we have with a primitive th root of unity. Fix , a base of , where is a base of the reflective hyperplane of . Thus, for . We can express aswhere .
As , in the identity it can be assumed that . It follows thatwhereIf has the form , with , so that whenever , then is a factor of .

Corollary 16. If , then is a factor of .

Proof. Because of the result established in Proposition 14(i), it follows thatTaking into account the reflections in and the expression of in (49), from Lemma 15 it follows that all distinct factors of in the setare factors of . But as they are irreducible factors not associated with , the fact that is factor of results.

We let denote the degree of the polynomial .

Proposition 17. Let be -partition of and . Considering the action of on given by (10) and on given by the restriction of the usual action on , we have the following:(i), where is a polynomial -invariant.(ii)If , then(iii)For each

Proof. (i) By Proposition 14, for each we haveTherefore,In particular, this identity is valid for a reflection in , so that by Lemma 15 it follows thatOn the other hand, if we apply to both sides of the former identity, we have so that is -invariant.
(ii) By the linearity of , it is sufficient to prove that for each such that we haveBy the identity given in Proposition 14,Thus, if does not belong to the -orbit of , this means thatMoreover, given that , by (i) there exists such thatAs and are in different -orbits, it must be the case that and so (70) is proved when does not belong to -orbit of .
If is in -orbit of , there is such thatThen,Since the coefficient of in the monomial decomposition of is precisely , this completes the proof of the identity (70).
(iii) If and , we have

4. The Bilinear Form

In this section, some properties of the linear form introduced in will be proven, in order to extend the results in [6] to the group .

Definition 18. If is a subset of , and one says that is supported in if for each . One also says that the monomial is supported in if is supported in . If , one says that is supported in if each monomial in is.

If we associate supported in given byIt is clear that is supported in if, and only if, .

If is a partition of and , can be decomposed in unique way as such that if , then

Proposition 19. Let and is a partition of . If and are factorized aswhere and are both supported in for each , then

Proof. The proof can be obtained using induction in , so it is sufficient to treat the case of a partition of two terms. For simplicity, we express . If we writethenwhere is Kronecker’s function. On the other hand, it follows from the factorization thatand this decomposition is unique as was observed in (78); therefore,where are the coefficients of and in and , respectively, and, analogously, are the coefficient of and in and . Then from which the following proposition holds.

Proposition 20. If and is a natural number, then

Proof. This is a consequence of the identityfor each pair .

Given a decomposition of , we havewhere the subsets are pairwise disjoint. We indicate by the set of all such that, for each , , is congruent to module .

Definition 21. If , one will say that has -type if

Proposition 22. Let and be with -type and with -type . If , then .

Proof. If and are different decompositions of , it is clear that and consequently .

Lemma 23. If is a -regular -partition, the bilinear form when restricted to is nonzero.

Proof. Let be the -partition of . For each , , let and suppose thatSince is -regular, each is a -regular partition. By Lemma 2.2 (iv) in [6], there are permutations in such that for each If we put from (31) and Propositions 19 and 20 we have

5. The Modular Representations

Theorem 24. Using the previous notation, we have the following:(i)If is a -regular -partition of , then .(ii)If , is an irreducible -module.(iii)If and are -regular -partitions of and , then .(iv)Every simple -module is isomorphic to for some -regular -partition of .

Proof. (i) This is a consequence of Lemma 23.
(ii) Let be an -partition of , a -submodule of , and , nonzero. Given that , there exists such thatIf for every , the linear functional should be zero, but we have assumed that . Hence, there exists such that . We can suppose, without lack of generality, that , changing by if that was the case. Now, if , and by Proposition 17Since , it follows that , which means that .
(iii) Let and be -regular -partitions of such that . We prove in Lemma 23 that there exists such thatIf in Proposition 17(iii) we change by , we haveSince the operator is nonzero in and it is a linear combination of elements in , , it must be nonzero in . Hence, there exists such thatThus,Hence,but this only occurs if and belong to the same -orbit, and, from Remark 10, it must be .
(iv) It is sufficient to prove that the number of -regular-partitions agrees with the number of -regular conjugacy classes in . But this is a consequence of the fact that there is a bijection between the -partitions and the conjugacy classes of which put in correspondence -regular -partitions with the -regular conjugacy classes of ; see [13].

Finally, considering the generic case where and may not be coprime, write , where and does not divide . Suppose that contains the th roots of the unity. Let us consider , the canonical projection, given by the decompositionThis morphism induces the projection given byBy the previous theorem, for each -regular-partition of we have an irreducible representation of and thus is an irreducible representation of .

Keeping the previous notation, we have following theorem which establishes the irreducible modular representations of in the general case.

Theorem 25. If contains the th roots of the unity, the following statements hold:(i)If is a -regular -partition of , then is an irreducible representation of .(ii)If and are -regular -partitions of and , then and are nonequivalent.(iii)Every irreducible representation of is equivalent to for some -regular -partition of .

Proof. (i) and (ii) follow from Theorem 24.
(iii) By (i) and (ii) we have as many nonequivalent irreducible representations of as -regular -partitions of . On the other hand, it follows from [13] that every -regular conjugacy class of has a representative ; in consequence and have the same number of -regular classes. Since the number of -regular classes of agrees with the number of -regular -partitions of it follows that the representations are all irreducible presentations of .

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

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Copyright © 2015 José O. Araujo et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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