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ISRN Algebra
VolumeΒ 2012Β (2012), Article IDΒ 658201, 16 pages
http://dx.doi.org/10.5402/2012/658201
Research Article

A Gelfand Model for Weyl Groups of Type 𝐷2𝑛

Departmento de MatemΓ‘tica, Facultad de Ciencias Exactas, UNICEN, B7000 GHG, Tandil, Argentina

Received 27 March 2012; Accepted 17 April 2012

Academic Editors: H.Β Airault, D.Β Sage, A.Β Vourdas, and H.Β You

Copyright Β© 2012 JosΓ© 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.

Abstract

A Gelfand model for a finite group 𝐺 is a complex representation of 𝐺, which is isomorphic to the direct sum of all irreducible representations of 𝐺. When 𝐺 is isomorphic to a subgroup of 𝐺𝐿𝑛(β„‚), where β„‚ is the field of complex numbers, it has been proved that each 𝐺-module over β„‚ is isomorphic to a 𝐺-submodule in the polynomial ring β„‚[π‘₯1,…,π‘₯𝑛], and taking the space of zeros of certain 𝐺-invariant operators in the Weyl algebra, a finite-dimensional 𝐺-space 𝒩𝐺 in β„‚[π‘₯1,…,π‘₯𝑛] can be obtained, which contains all the simple 𝐺-modules over β„‚. This type of representation has been named polynomial model. It has been proved that when 𝐺 is a Coxeter group, the polynomial model is a Gelfand model for 𝐺 if, and only if, 𝐺 has not an irreducible factor of type 𝐷2𝑛, 𝐸7, or 𝐸8. This paper presents a model of Gelfand for a Weyl group of type 𝐷2𝑛 whose construction is based on the same principles as the polynomial model.

1. Introduction

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 [1]. 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 πœ’πΆπ‘˜β†‘π”–π‘›=ξ“πœ’πœ†,(1.1) where πΆπ‘˜ is the centralizer of an involution in 𝔖𝑛 with exactly π‘˜ fixed points, πœ’πΆπ‘˜ is a linear character of πΆπ‘˜, and πœ’πœ† is an irreducible character of 𝔖𝑛 associated with the partition πœ† of 𝑛 with exactly π‘˜ odd terms. From this identity, it follows immediately that ξ“π‘˜πœ’πΆπ‘˜β†‘π”–π‘›=ξ“πœ†πœ’πœ†,(1.2) where the centralizers πΆπ‘˜ are in correspondence with the conjugacy classes of involutions in 𝔖𝑛.

Later on, this type of models was called an involution model by Baddeley [6]. He also proved that if 𝐻 is a finite group that admits an involution model, then so does the semidirect product 𝐻𝑛×𝑠𝔖𝑛.

Baddeley’s result implies the existence of involution models for classic Weyl groups, with the exception of the group of type 𝐷2𝑛. An involution model for a Weyl group of type 𝐴𝑛 is presented in [8] by Inglis et al. and for a Weyl group of type 𝐡𝑛 an involution model is shown in [6, 13]. In [6], Baddeley presents an involution model for a Weyl group of type 𝐷2𝑛+1, and in [14] it is proved that there is no involution model for a Weyl group of type 𝐷2𝑛 with 𝑛β‰₯2. In [6], it is mentioned that is not difficult to prove that there is an involution model for a Weyl group of type 𝐺2 and that it has been checked using computers the non existence of involution models for exceptional Weyl groups of type 𝐹4, 𝐸6, 𝐸7, and 𝐸8. In [15], 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 𝐼𝑛2(𝑛β‰₯3,𝑛≠6) and 𝐻3 and presents a conceptual demonstration of the no existence of an involution model for the group of type 𝐻4.

More recently, in [16] the generalized involution model has been studied in order to include some cases of unitary reflection groups.

A reflection group 𝐺 comes equipped with a canonical representation called the geometric representation of 𝐺. The geometric representation induces a natural action of 𝐺 on the space of polynomial functions.

Chevalley [17], Shephard and Todd [18], Steinberg [19], and others studied the corresponding action on the space ℋ𝐺 of 𝐺-harmonic polynomials proving that ℋ𝐺 is isomorphic to the regular representation of 𝐺, and thus ℋ𝐺 contains a Gelfand model for 𝐺. 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 𝐺-harmonic polynomial space. These representations are known as Madonald representations see [20].

More recently, Araujo and Aguado in [21] have associated with each finite subgroup πΊβŠ‚πΊπΏπ‘›(β„‚) a subspace 𝒩𝐺 of the algebra of polynomials β„‚[π‘₯1,…,π‘₯𝑛], defined as zeros of certain 𝐺-invariant differential operators, and have shown 𝒩𝐺 contains a Gelfand model of 𝐺. This space, called the polynomial model, is a Gelfand model for some Weyl groups. In [3–5], it was proved that 𝒩𝐺 is a Gelfand model for Weyl groups of type 𝐴𝑛, 𝐡𝑛 and 𝐷2𝑛+1. Garge and OesterlΓ© in [22], using the computation of fake degrees of the irreducible characters of a Coxeter group 𝐺, determined that 𝒩𝐺 is a Gelfand model of 𝐺 if, and only if, 𝐺 has not irreducible factors of type 𝐷2𝑛, 𝐸7, or 𝐸8. The fake degrees have been determined due to works of Steinberg [23], when 𝐺 is of type 𝐴𝑛, Lusztig [24], when 𝐺 is of type 𝐡𝑛 or 𝐷𝑛, Beynon and Lusztig [25], when 𝐺 is an exceptional Weyl group, Alvis and Lusztig [26], when 𝐺 is of type 𝐻4, and Macdonald, when 𝐺 is of type 𝐹4 (unpublished). The remaining cases are not difficult.

For the case of Weyl groups of type 𝐷2𝑛, 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 𝐷2𝑛 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.

𝐺 will denote a finite subgroup of 𝐺𝐿𝑛(β„‚) and 𝒫 the polynomial ring β„‚[π‘₯1,…,π‘₯𝑛].

Let πˆπ‘›={1,…,𝑛} be the set of the first 𝑛 natural numbers and ℳ𝑛 the set of multi-index functions: ℳ𝑛=ξ€½π›ΌβˆΆπˆπ‘›βŸΆβ„•0ξ€Ύ(2.1)

For each π›Όβˆˆβ„³π‘› the following notation will be used in the rest of this paper: 𝛼𝑖𝛼=𝛼(𝑖),𝛼=1,…,𝛼𝑛,|𝛼|=𝑛𝑖=1𝛼𝑖.(2.2)

Let π’œ=β„‚βŸ¨π‘₯1,…,π‘₯𝑛,πœ•1,…,πœ•π‘›βŸ© be the Weyl algebra of β„‚-linear differential operators generated by the multiplication operators π‘₯𝑖 and partial differential operators πœ•π‘–=πœ•/πœ•π‘₯𝑖 with 1≀𝑖≀𝑛.

It is known that each π·βˆˆπ’œ has a unique expression as a finite sum (see [27]): ξ“πœ†π·=𝛼,𝛽π‘₯π›Όπœ•π›½,(2.3) where 𝛼,π›½βˆˆβ„³π‘›, πœ†π›Ό,π›½βˆˆβ„‚, and π‘₯𝛼=π‘₯𝛼11π‘₯𝛼22β‹―π‘₯𝛼𝑛𝑛,πœ•π›½=πœ•π›½11πœ•π›½22β‹―πœ•π›½π‘›π‘›.(2.4) The degree of 𝐷 is defined by deg(𝐷)=maxπ‘–ξ€·π›Όπ‘–βˆ’π›½π‘–ξ€ΈβˆΆπœ†π›Ό,𝛽.β‰ 0(2.5) The Weyl algebra is a graduated algebra β¨π’œ=π‘–βˆˆβ„€π’œπ‘–, where π’œπ‘–=⎧βŽͺ⎨βŽͺβŽ©ξ“π›Ό,π›½βˆˆβ„³π‘›πœ†π›Ό,𝛽π‘₯π›Όπœ•π›½||𝛽||⎫βŽͺ⎬βŽͺ⎭.∢|𝛼|βˆ’=𝑖(2.6) The action of 𝐺 on 𝒫 induces an action of 𝐺 on the endomorphism ring Endβ„‚(𝒫), which is defined by ξ€·(𝑔⋅𝐷)(𝑝)=π‘”π·π‘”βˆ’1ξ€Έξ€·(𝑝)π‘”βˆˆπΊ,𝐷∈Endβ„‚ξ€Έ.(𝒫)(2.7) This action can be restricted to the Weyl algebra π’œ noting that each π’œπ‘– is invariant under the action of G.

Let ℐ𝐺 be the subalgebra of 𝐺-invariant operators in π’œ, that is, ℐ𝐺={π·βˆˆπ’œβˆΆπ‘”β‹…π·=𝐷,βˆ€π‘”βˆˆπΊ}.(2.8) Notice that ℐ𝐺 is contained in the centralizer of 𝐺 in Endβ„‚(𝒫).

Let β„βˆ’πΊ be the subspace of the Weyl algebra, formed by the 𝐺-invariant operators with negative degree β„βˆ’πΊ=ξ€½π·βˆˆβ„πΊξ€Ύ.∢deg(𝐷)<0(2.9)

Definition 2.1. Let 𝒩𝐺 be the subspace of 𝒫 defined by 𝒩𝐺=ξ€½π‘βˆˆπ’«βˆΆπ·(𝑝)=0,βˆ€π·βˆˆβ„βˆ’πΊξ€Ύ.(2.10)𝒩𝐺 is named the polynomial model of 𝐺.

Notice that 𝒩𝐺 is a 𝐺-module.

Below, some properties of 𝒩𝐺 will be mentioned.

Theorem 2.2. 𝒩𝐺 is a finite-dimensional 𝐺-module, and every simple 𝐺-module has a copy in 𝒩𝐺.

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 𝐺 be a finite irreducible Coxeter group, and let π‘Š be its realization as a reflection group. Then, the polynomial model π’©π‘Š is a Gelfand model for 𝐺 if, and only if, π‘Š is not a Weyl group of type 𝐷2𝑛, 𝐸7, or 𝐸8.

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 𝐴𝑛, 𝐡𝑛 and 𝐷𝑛.

2.1. Polynomial Model for a Weyl Group of Type 𝐴𝑛

Let 𝐺 be a Weyl group of type π΄π‘›βˆ’1. It is known that 𝐺 can be presented as the symmetric group 𝔖𝑛.

The symmetric group 𝔖𝑛 acts on the set of multi-index functions ℳ𝑛 by πœŽβ‹…π›Ό=π›Όβˆ˜πœŽβˆ’1ξ€·πœŽβˆˆπ”–π‘›,π›Όβˆˆβ„³π‘›ξ€Έ.(2.11) This action induces a natural homomorphism from 𝔖𝑛 in Aut(𝒫) given by πœŽβŽ›βŽœβŽœβŽξ“π›Όβˆˆβ„³π‘›πœ†π›Όπ‘₯π›ΌβŽžβŽŸβŽŸβŽ =ξ“π›Όβˆˆβ„³π‘›πœ†π›Όπ‘₯πœŽβ‹…π›Όξ€·πœ†π›Όξ€Έ.βˆˆβ„‚(2.12)

2.1.1. 𝔖𝑛-Minimal Orbit

Let π’ͺ𝑛 be the orbit space of 𝔖𝑛 in ℳ𝑛. It is clear that if two multi-indexes 𝛼 and 𝛽 belong to the same orbit 𝛾, then |𝛼| and |𝛽| take the same value, where |𝛼| and |𝛽| are defined by (2.2), and this value will be denoted by |𝛾|.

Definition 2.4. Two orbits 𝛾 and 𝛿 will be called 𝔖𝑛-equivalent, denoted by π›ΎβˆΌπ”–π‘›π›Ώ, if there exists a bijection πœ‘βˆΆβ„•0β†’β„•0 such that 𝛿={πœ‘βˆ˜π›ΌβˆΆπ›Όβˆˆπ›Ύ}.(2.13)

Definition 2.5. An orbit 𝛾 will be called 𝔖𝑛-minimal if |𝛾|≀|𝛿| for all π›Ώβˆˆπ’ͺ𝑛 such that π›ΎβˆΌπ”–π‘›π›Ώ.

Proposition 2.6. An orbit 𝛾 is 𝔖𝑛-minimal if, and only if, for each π›Όβˆˆπ›Ύ, there exists a nonnegative integer β„Ž such that (1)Im(𝛼)={0,1,…,β„Žβˆ’1},(2)|π›Όβˆ’1(𝑖)|β‰₯|π›Όβˆ’1(𝑖+1)| for all 0β‰€π‘–β‰€β„Žβˆ’1(|π›Όβˆ’1(𝑖)| being the cardinal of the set π›Όβˆ’1(𝑖)).

Proof. See [4, page 1845].

Definition 2.7. For each π›Ύβˆˆπ’ͺ𝑛, let 𝑆𝛾 be the subspace of 𝒫 defined by 𝑆𝛾=ξƒ―ξ“π›Όβˆˆπ›Ύπœ†π›Όπ‘₯π›ΌβˆΆπœ†π›Όξƒ°βˆˆβ„‚(2.14)

2.1.2. The Space π‘†πœ•π›Ύ

Let πœ• be the operator defined by βˆ‘πœ•=𝑛𝑖=1πœ•π‘–, where πœ•π‘– are the partial differential operators as above. For each π›Ύβˆˆπ’ͺ𝑛, let π‘†πœ•π›Ύ be the subspace defined by π‘†πœ•π›Ύ=ξ€½π‘ƒβˆˆπ‘†π›Ύξ€ΎβˆΆπœ•(𝑃)=0(2.15)

2.1.3. The Structure of 𝒩𝔖𝑛

Below the main theorem regarding 𝒩𝔖𝑛 is announced without proof. For further details see [4, page 1850].

Theorem 2.8. π‘†πœ•π›Ύ is an irreducible 𝐺-module, and 𝒩𝔖𝑛 can be decomposed as 𝒩𝔖𝑛=ξΆπ›Ύβˆˆπ’ͺ𝔖𝑛-minimalπ‘†πœ•π›Ύ.(2.16) Moreover, 𝒩𝔖𝑛 is a Gelfand model of 𝔖𝑛.

2.2. Polynomial Model for a Weyl Group of Type 𝐡𝑛

The Gelfand model for a Weyl group of type 𝐡𝑛 will be described using the same ideas as the previous section.

Let π’ž2={1,βˆ’1}βŠ‚β„‚βˆ— be the subgroup of order two. The Weyl group ℬ𝑛, of type 𝐡𝑛, can be presented as the semidirect product ℬ𝑛=π’žπ‘›2×𝑠𝔖𝑛,(2.17) where π’žπ‘›2=π’ž2Γ—β‹―Γ—π’ž2 and the semidirect product is induced by the natural action of 𝔖𝑛 on π’žπ‘›2: ξ€·π‘€πœŽβ‹…1,…,𝑀𝑛=ξ€·π‘€πœŽ(1),…,π‘€πœŽ(𝑛)ξ€Έξ€·πœŽβˆˆπ”–π‘›,𝑀1,…,π‘€π‘›ξ€Έβˆˆπ’žπ‘›2ξ€Έ.(2.18)

The action of 𝔖𝑛 on ℳ𝑛 induces a natural homomorphism from ℬ𝑛 on Aut(𝒫) given by βŽ›βŽœβŽœβŽξ“(𝑀,𝜎)π›Όβˆˆβ„³π‘›πœ†π›Όπ‘₯π›ΌβŽžβŽŸβŽŸβŽ =ξ“π›Όβˆˆβ„³π‘›πœ†π›Ό(𝑀π‘₯)πœŽπ›Όξ€·πœ†π›Όξ€Έβˆˆβ„‚(2.19) with (𝑀π‘₯)πœŽπ›Ό=𝑛𝑖=1𝑀𝑖π‘₯𝑖(πœŽπ›Ό)𝑖.(2.20)

2.2.1. ℬ𝑛-Minimal Orbit

Let π’ͺ𝑛 be the orbit space of 𝔖𝑛 on ℳ𝑛, as above.

Definition 2.9. Two orbits, 𝛾 and 𝛿, will be called ℬ𝑛-equivalent, denoted by π›ΎβˆΌβ„¬π‘›π›Ώ, if there exists a bijection πœ‘βˆΆβ„•0β†’β„•0 such that πœ‘(π‘˜) and π‘˜ have the same parity for all π‘˜βˆˆβ„•0,𝛿={πœ‘βˆ˜π›ΌβˆΆπ›Όβˆˆπ›Ύ}.

Definition 2.10. An orbit 𝛾 will be called ℬ𝑛-minimal if |𝛾|≀|𝛿| for all π›Ώβˆˆπ’ͺ𝑛 such that π›ΎβˆΌβ„¬π‘›π›Ώ.

Proposition 2.11. An orbit 𝛾 is ℬ𝑛-minimal if, and only if, for each π›Όβˆˆπ›Ύ and each pair 𝑖,π‘—βˆˆβ„•0 with the same parity, one has |π›Όβˆ’1(𝑖)|β‰₯|π›Όβˆ’1(𝑗)| with 0≀𝑖<𝑗 (|π›Όβˆ’1(𝑖)| being the cardinal of the set π›Όβˆ’1(𝑖)).

Proof. See [3, page 365].

2.2.2. The Space 𝑆Δ𝛾

Let Ξ” be the Laplacian operator defined by βˆ‘Ξ”=𝑛𝑖=1πœ•2𝑖, where πœ•π‘– are the partial differential operators mentioned above. For each π›Ύβˆˆπ’ͺ𝑛, let 𝑆Δ𝛾 be the subspace defined by 𝑆Δ𝛾=ξ€½π‘ƒβˆˆπ‘†π›Ύξ€Ύ.βˆΆΞ”(𝑃)=0(2.21)

2.2.3. The Structure of 𝒩ℬ𝑛

Below the main theorem regarding 𝒩ℬ𝑛 is announced without proof. See references.

Theorem 2.12. 𝑆Δ𝛾 is an irreducible 𝐺-module, and 𝒩ℬ𝑛 can be decomposed as 𝒩ℬ𝑛=ξΆπ›Ύβˆˆπ’ͺℬ𝑛-minimal𝑆Δ𝛾.(2.22) Moreover 𝒩ℬ𝑛 is a Gelfand model of ℬ𝑛.

Proof. See [3, page 371].

2.3. Polynomial Model for a Weyl Group of Type 𝐷𝑛

Let π’Ÿπ‘› be the Weyl group of type 𝐷𝑛 naturally included in ℬ𝑛. Using the previous notation, for π›Όβˆˆβ„³π‘› the following sets are considered: 𝐸𝛼=ξ€½π‘–βˆˆπΌπ‘›βˆΆπ›Όπ‘–isevenξ€Ύ,𝑂𝛼=ξ€½π‘–βˆˆπΌπ‘›βˆΆπ›Όπ‘–isoddξ€Ύ.(2.23)

It is easy to check that the cardinals |𝐸𝛼| and |𝑂𝛼| are equal for all elements in the same orbit 𝛾. Therefore, these values will be denoted by |𝐸𝛾| and |𝑂𝛾|, respectively.

2.3.1. π’Ÿπ‘›-Minimal Orbit

Definition 2.13. Two orbits 𝛾 and 𝛿 will be called π’Ÿπ‘›-equivalent, denoted by π›ΎβˆΌπ’Ÿπ‘›π›Ώ, if there exists a bijection πœ‘βˆΆβ„•0β†’β„•0 such that(1)βˆ€π‘˜βˆˆβ„•0, πœ‘(π‘˜) and π‘˜ have the same parity or πœ‘(π‘˜) and π‘˜ have different parities,(2)𝛿={πœ‘βˆ˜π›ΌβˆΆπ›Όβˆˆπ›Ύ}.

Definition 2.14. An orbit 𝛾 will be called π’Ÿπ‘›-minimal if |𝛾|≀|𝛿| for all π›Ώβˆˆπ’ͺ𝑛 such that π›ΎβˆΌπ’Ÿπ‘›π›Ώ.

Proposition 2.15. Let 𝛾 be an orbit, and then the following statements are true (1)𝛾 is π’Ÿπ‘›-minimal if, and only if, the following statements are verified:(a)given π›Όβˆˆπ›Ύ and 𝑖<π‘—βˆˆβ„•0 with the same parity, then |π›Όβˆ’1(𝑖)|β‰₯|π›Όβˆ’1(𝑗)|,(b)|𝐸𝛾|≀|𝑂𝛾|. (2)Let πœ‹βˆΆβ„•0β†’β„•0 be the involution given by πœ‹(2𝑖)=2𝑖+1 and πœ‹(2𝑖+1)=2𝑖. The following assertions are equivalent:(a)𝛾 and πœ‹βˆ˜π›Ύ are π’Ÿπ‘›-minimal orbits,(b)𝛾 is ℬ𝑛-minimal,(c)πœ‹βˆ˜π›Ύ is ℬ𝑛-minimal. (3)There are at most two π’Ÿπ‘›-minimal orbits equivalent to 𝛾.(4)If 𝑛 is odd, there is only one π’Ÿπ‘›-minimal orbit equivalent to 𝛾.(5)𝛾 and πœ‹βˆ˜π›Ύ are π’Ÿπ‘›-minimal orbits if, and only if, |𝐸𝛾|=|𝑂𝛾|.

Proof. See [5, page 106].

Proposition 2.16. Let 𝑛 be odd, and then the following statements are true. (1)If 𝛾 is π’Ÿπ‘›-minimal, then π’©β„¬π‘›βˆ©π‘†π›Ύ=π’©π’Ÿπ‘›βˆ©π‘†π›Ύ(2.24) and π’©π’Ÿπ‘›βˆ©π‘†π›Ύ is a simple π’Ÿπ‘›-module.(2)π’©π’Ÿπ‘› is a Gelfand model for π’Ÿπ‘›.(3)Every simple ℬ𝑛-module remains simple when it is considered as a π’Ÿπ‘›-module by restriction.(4)By considering 𝒩ℬ𝑛 as a π’Ÿπ‘›-module by restriction, 𝒩ℬ𝑛 is isomorphic to π’©π’Ÿπ‘›βŠ•π’©π’Ÿπ‘›.

Proof. See [5, page 110].

Also in [5] it has been proved that if 𝑛 is even, π’©π’Ÿπ‘› is not a Gelfand model for a Weyl group of type 𝐷𝑛. But it does happen that if 𝑀 is a simple π’Ÿπ‘›-module, then π’©π’Ÿπ‘› contains a copy ofthis, and the multiplicity of 𝑀 in π’©π’Ÿπ‘› is(1)two, if 𝑀 is isomorphic to π’©π’Ÿπ‘›βˆ©π‘†π›Ύ,𝛾 being a π’Ÿπ‘›-minimal orbit such that π›Ύβ‰ πœ‹βˆ˜π›Ύ and |𝐸𝛾|=|𝑂𝛾|; in this case, as before, πœ‹βˆΆβ„•0β†’β„•0 is the involution given by πœ‹(2𝑖)=2𝑖+1 and πœ‹(2𝑖+1)=2𝑖,(2)one, otherwise.

3. Gelfand Model for a Weyl Group of Type 𝐷2𝑛

As before, let ℳ𝑛={π›ΌβˆΆπˆπ‘›β†’β„•0} be the set of multi-index functions. Every π›Όβˆˆβ„³π‘› has an associated vector ξπ›Όβˆˆβ„•π‘›0, which is obtained by reordering 𝛼 as follows. 𝛼𝛼=𝑖1,…,𝛼𝑖𝑛suchthat𝛼𝑖1β‰₯β‹―β‰₯𝛼𝑖𝑛.(3.1) Thus, there is defined an order relationship βͺ― in ℳ𝑛 given by for all 𝛼,π›½βˆˆβ„³π‘›, 𝛼βͺ―𝛽 if, and only if, ̂𝛽𝛼= or there exists 𝑠(1≀𝑠≀𝑛) such that 𝛼1=̂𝛽1,…,ξπ›Όπ‘ βˆ’1=Μ‚π›½π‘ βˆ’1,𝛼𝑠<̂𝛽𝑠,(3.2)𝛼𝑖 and ̂𝛽𝑖 being the coordinates of the vectors 𝛼 and ̂𝛽, respectively. Notice that this is the lexicographic order for ℕ𝑛0.

Proposition 3.1. Let π›Ύβˆˆπ’ͺ𝑛 and 𝛼,π›½βˆˆβ„³π‘›, and then 𝛼,π›½βˆˆπ›Ύ if, and only if, ̂𝛽𝛼=.

Proof. Let 𝛼,π›½βˆˆπ›Ύ, and therefore there exists πœŽβˆˆπ”–π‘› such that 𝛽=πœŽπ›Ό, which implies 𝛽𝑖=π›ΌπœŽβˆ’1(𝑖) with 1≀𝑖≀𝑛. Thus, it is easy to see that ̂𝛽𝛼=.
On the other hand, let 𝛼,π›½βˆˆβ„³π‘› and ̂𝛽𝛼=, say, 𝛼𝑖1=𝛽𝑗1,…,𝛼𝑖𝑛=𝛽𝑗𝑛.(3.3) Let πœŽβˆˆπ”–π‘› be given by πœŽβˆ’1ξ€·π‘–π‘˜ξ€Έ=π‘—π‘˜(1β‰€π‘˜β‰€π‘›).(3.4) Then, 𝛽=πœŽπ›Ό, and hence both multi-indexes belong to the same orbit.

From this proposition it is clear that βͺ― induces a total orderin π’ͺ𝑛, which is defined by 𝛾βͺ―π›ΏβŸΊπ›Όβͺ―𝛽𝛾,π›Ώβˆˆπ’ͺ𝑛.,π›Όβˆˆπ›Ύ,π›½βˆˆπ›Ώ(3.5) Since the vector 𝛼 is independent of the choice 𝛼 in 𝛾, it will be denoted by ̂𝛾.

Proposition 3.2. Let π›Όβˆˆβ„³π‘› be defined by 𝛼𝑖=π‘–βˆ’1, and let 𝛾 be the orbit of 𝛼. Then, 𝛾 is the βͺ―-maximum of the 𝔖𝑛-minimal orbits and ̂𝛾=(π‘›βˆ’1,…,1,0).

Proof. From the previous considerations it is clear that 𝛾 is an 𝔖𝑛-minimal orbit and ̂𝛾=(π‘›βˆ’1,…,1,0).
Now it will be proved that 𝛾 is the βͺ―-maximum in the set of 𝔖𝑛-minimal orbits. Let 𝛿 be an orbit such that 𝛿≠𝛾 and 𝛾βͺ―𝛿. Then, it should exist an π‘ βˆˆπˆπ‘› satisfying ̂𝛾𝑖=̂𝛿𝑖,βˆ€π‘–<𝑠,̂𝛾𝑠<̂𝛿𝑠,(3.6) that is, ̂𝛿1̂𝛿=π‘›βˆ’1β‹―π‘ βˆ’1̂𝛿=π‘›βˆ’(π‘ βˆ’1),𝑠>π‘›βˆ’π‘ .(3.7)
Thus, it occurs that ̂𝛿𝑠=π‘›βˆ’(π‘ βˆ’1), and from the minimality of 𝛿 every number less than π‘›βˆ’(π‘ βˆ’1) must appear at least twice, which is a contradiction. And therefore 𝛾 is the maximum.

From now on, for a finite set 𝐴, 𝔖𝐴 will denote the symmetric group of 𝐴, ℳ𝐴 will denote the set of multi-index functions with domain 𝐴, ℳ𝐴=ξ€½π›ΌβˆΆπ€βŸΆβ„•0ξ€Ύ,(3.8) and Δ± will denote the function in ℳ𝐴 given by 𝚀(𝑖)=1,βˆ€π‘–βˆˆπ΄.(3.9) As in the case of 𝔖𝑛, there is a natural action from the symmetric group 𝔖𝐴 in the set of multi-index functions ℳ𝐴, defined by πœŽβ‹…π›Ό=π›Όβˆ˜πœŽβˆ’1ξ€·πœŽβˆˆπ”–π΄,π›Όβˆˆβ„³π΄ξ€Έ.(3.10) It is possible to extend the concept of 𝔖𝐴-minimal orbit.

For each π›Όβˆˆβ„³π‘›, let us consider the sets 𝐸𝛼=ξ€½π‘–βˆˆπˆπ‘›βˆΆπ›Όπ‘–iseven𝑂𝛼=ξ€½π‘–βˆˆπˆπ‘›βˆΆπ›Όπ‘–isoddξ€Ύ(3.11) as defined in the previous section. Then, it is clear that 𝛼 can be determined from its restrictions 𝛼𝐸 and 𝛼𝑂 to the sets 𝐸𝛼 and 𝑂𝛼, respectively. Observe that π›ΌπΈβˆˆβ„³πΈπ›Ό and π›Όπ‘‚βˆˆβ„³π‘‚π›Ό.

Proposition 3.3. Let π›Όβˆˆβ„³π‘› such that 𝛼 is ℬ𝑛-minimal, and then 𝛼𝐸/2 is 𝔖𝐸𝛼-minimal and (π›Όπ‘‚βˆ’1)/2 is 𝔖𝑂𝛼-minimal.

Proof. It follows from Proposition 4 in [3] and the identities ||||𝛼𝐸2ξ‚βˆ’1||||=||𝛼||ξ€·||𝐸(𝑖)(𝑖)βˆ€π‘–βˆˆπ›Ό||ξ€Έ,||||ξ‚΅π›Όπ‘‚βˆ’12ξ‚Άβˆ’1||||=||||ξ€·||𝑂(𝑗)𝛼(𝑗)βˆ€π‘—βˆˆπ›Ό||ξ€Έ.(3.12)

Notation 1. Let 𝒦 be the subset of ℳ𝑛 given by 𝒦=π›Όβˆˆβ„³π‘›βˆΆπ›Όisπ’Ÿπ‘›-minimal,||𝐸𝛼||=||𝑂𝛼||and𝛼𝐸2β‰Ίπ›Όπ‘‚βˆ’12ξ‚Ό.(3.13) It will be denoted by β„± the subset of the polynomial ring 𝒫: ℱ=π›Όπœ†π›Όπ‘₯π›ΌβˆΆπœ†π›Ό=0ifξƒ°.π›Όβˆˆπ’¦(3.14) Note that if 𝑛 is odd, β„± is equal to 𝒫.

Proposition 3.4. β„± is a π’Ÿπ‘›-submodule of the polynomial ring 𝒫.

Proof. It follows from the action of π’Ÿπ‘› given by (𝑀,𝜎)π›Όπœ†π›Όπ‘₯𝛼ξƒͺ=ξ“π›Όπœ†π›Ό(𝑀π‘₯)πœŽπ›Ό=ξ“π›ΌΒ±πœ†π›Όπ‘₯πœŽπ›Ό(3.15) and the fact that π›Όβˆˆπ’¦ if, and only if, πœŽπ›Όβˆˆπ’¦, and it results that β„± is a π’Ÿπ‘›-module of 𝒫.

Proposition 3.5. β„± contains a Gelfand model for the Weyl group of type π’Ÿπ‘›.

Proof. It is sufficient to prove that β„± contains a submodule equivalent to the regular module π’Ÿπ‘›. Effectively, let us consider the polynomial: 𝑃π‘₯1,…,π‘₯𝑛=βˆπ‘›π‘–=2π‘₯𝑖π‘₯π‘–ξ€Έβˆ+π‘–βˆ’1π‘–βˆ’2𝑗=1ξ€·π‘₯2π‘–βˆ’π‘—2ξ€Έξ‚„2π‘›βˆ’1[](π‘›βˆ’1)!2βˆπ‘›π‘–=2βˆπ‘–βˆ’2𝑗=1.(π‘–βˆ’1+𝑗)(π‘–βˆ’1βˆ’π‘—)(3.16) Thus 𝑃 is the interpolating polynomial of the orbit of the regular vector 𝑣=(0,1,…,π‘›βˆ’1), which satisfies 𝑃(𝑣)=1,𝑃(πœπ‘£)=0βˆ€πœβˆˆπ’Ÿπ‘›ξ€Έ.,πœβ‰ π‘’(3.17)
It will be proved that 𝑃 belongs to β„±. Let πœ†π›Όπ‘₯𝛼 be not a null term of 𝑃 such that it is π’Ÿπ‘›-minimal, |𝐸𝛼|=|𝑂𝛼|, and 𝛼𝐸/2β‰Ί(π›Όπ‘‚βˆ’1)/2. As 𝑃 was defined in (3.16), it is easy to determine that 𝛼1=0 and 𝛼𝑗>0 for 1<𝑗≀𝑛. Then, (𝛼𝐸/2)1=0, and as 𝛼𝐸/2 is 𝔖|𝐸𝛼|-minimal, it is obtained that ξ‚Šπ›ΌπΈ2=ξ€·||𝐸𝛼||||πΈβˆ’1,𝛼||ξ€Έ,βˆ’2,…,0(3.18) which by Proposition 3.2 is maximal, which is a contradiction.
Since β„± is a π’Ÿπ‘›-module, β„± contains the module generated by the orbit of 𝑃, which is isomorphic to the regular module. Hence, β„± contains a Gelfand model.

Notation 2. Let 𝒯 be a subset of the polynomial ring 𝒫 and 𝐺 a finite subgroup of 𝐺𝐿𝑛(β„‚); we will denote by 𝒯0 the subset of 𝒯 defined by 𝒯0=ξ€½π‘βˆˆπ’―βˆΆπ·(𝑝)=0,βˆ€π·βˆˆβ„βˆ’πΊξ€Ύ,(3.19)β„βˆ’πΊ being the set of differential operators invariant in the algebra of Weyl as it has been defined in (2.9).

Proposition 3.6. Let 𝐺 be a finite subgroup of 𝐺𝐿𝑛(β„‚) and 𝒯 a 𝐺-module of the polynomial ring 𝒫 such that 𝒯 contains a model of 𝐺, and then 𝒯0 also contains a model of 𝐺.

Proof. Let π‘†βŠ‚π’― be a simple 𝐺-module and suppose that π‘†ΜΈβŠ†π’―0. Then, there exists π·βˆˆβ„βˆ’πΊ such that 𝐷(𝑆)β‰ 0. Because 𝑆 is simple and 𝐷 not null, it follows that 𝐷 is injective. Thus, 𝐷(𝑆)≃𝑆. If 𝐷(𝑆)βŠ‚π’―0, the proposition is proved; otherwise the procedure will be repeated. As 𝐷 is an operator of the Weyl algebra π’œ with negative degree, the procedure is finite, that is to say, there exists π‘šβˆˆβ„• such that π·π‘š(𝑆)βŠ‚π’―0 and π·π‘š(𝑆)≃𝑆.

Remark 3.7. An immediate consequence of this proposition is that the π’Ÿπ‘›-module β„±0=ξ‚†π‘“βˆˆβ„±βˆΆπ·(𝑓)=0,βˆ€π·βˆˆβ„βˆ’π’Ÿπ‘›ξ‚‡(3.20) contains a Gelfand model because β„± is a π’Ÿπ‘›-module containing a Gelfand model.

Remark 3.8. Notice that if 𝑛 is odd, then β„±0=𝒩𝐷𝑛; instead, if 𝑛 is even, β„±0βŠ†β„±βˆ©π’©π’Ÿπ‘›=ξΆπ›Ύβˆˆπ’ͺℬ𝑛-minimal||𝐸𝛾||<||𝑂𝛾||𝑆Δ𝛾+ξΆπ›Ύβˆˆπ’ͺℬ𝑛-minimal||𝐸𝛾||=||𝑂𝛾||𝑆Δ𝛾(3.21) that is β„±0βŠ†ξΆπ›Ύβˆˆπ’ͺℬ𝑛-minimal||𝐸𝛾||<||𝑂𝛾||𝑆Δ𝛾+ξΆξ‚΅ξ„žπ›Ύπ‘‚βˆ’1ξ‚Ά/2≺𝛾𝐸/2𝑆Δ𝛾+𝛾𝐸/2=ξ„žπ›Ύπ‘‚βˆ’1ξ‚Ά/2𝑆Δ𝛾.(3.22)

Using the result established in item 4 of Proposition 2.16 for decomposing 𝒩ℬ𝑛, it follows that 𝒩ℬ𝑛=ξΆπ›Ύβˆˆπ’ͺℬ𝑛-minimal𝑆Δ𝛾=ξΆπ›Ύβˆˆπ’ͺℬ𝑛-minimal||𝐸𝛾||<||𝑂𝛾||𝑆Δ𝛾+ξΆπ›Ύβˆˆπ’ͺℬ𝑛-minimal||𝐸𝛾||=||𝑂𝛾||𝑆Δ𝛾+ξΆπ›Ύβˆˆπ’ͺℬ𝑛-minimal||𝐸𝛾||>||𝑂𝛾||𝑆Δ𝛾.(3.23) Moreover if |𝐸𝛾|=|𝑂𝛾|ξΆπ›Ύβˆˆπ’ͺℬ𝑛-minimal||𝐸𝛾||=||𝑂𝛾||𝑆Δ𝛾=𝛾𝐸/2β‰Ίξ„žξ€·π›Ύπ‘‚ξ€Έβˆ’1/2𝑆Δ𝛾+𝛾𝐸/2=ξ„žξ€·π›Ύπ‘‚ξ€Έβˆ’1/2𝑆Δ𝛾+ξΆξ„žξ€·π›Ύπ‘‚ξ€Έβˆ’1/2≺𝛾𝐸/2𝑆Δ𝛾.(3.24)

As a consequence of this decomposition the next lemma follows.

Lemma 3.9. The dimension of 𝒩ℬ𝑛 is equal to βŽ›βŽœβŽœβŽœβŽœβŽξΆ2dimπ›Ύβˆˆπ’ͺℬ𝑛-minimal||𝐸𝛾||<||𝑂𝛾||π‘†Ξ”π›ΎβŽžβŽŸβŽŸβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽœβŽœβŽξΆξ„žξ€·π›Ύ+2dimπ‘‚ξ€Έβˆ’1/2≺𝛾𝐸/2π‘†Ξ”π›ΎβŽžβŽŸβŽŸβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽœβŽœβŽξΆ+dim𝛾𝐸/2=ξ„žξ€·π›Ύπ‘‚ξ€Έβˆ’1/2π‘†Ξ”π›ΎβŽžβŽŸβŽŸβŽŸβŽŸβŽ .(3.25)

Proof. It results from considering the identity 𝑆dimΔ𝛾𝑆=dimΞ”πœ‹βˆ˜π›Ύξ€Έ(3.26) for each π›Ύβˆˆπ’ͺ𝑛. This identity occurs from the fact that π‘†Ξ”πœ‹ and π‘†Ξ”πœ‹βˆ˜π›Ύ are isomorphic as π’Ÿπ‘›-modules, see [5].

Let 𝐺 be a group; from now on we will be denote by Invξ€½(𝐺)=π‘”βˆˆπΊβˆΆπ‘”2ξ€Ύ=𝑒(3.27) the set of involutions of the group 𝐺.

Lemma 3.10. Let 𝐺 be a Coxeter group and β„³ a Gelfand model for 𝐺. Then, ||dim(β„³)=Inv||.(𝐺)(3.28)

Proof. It is a consequence from the FrΓΆbenius-Schur indicator and the fact that the representations of a Coxeter group can be realized over the real numbers, see [28].

With the purpose to establish the central result of this work, a relationship between the number of involutions of ℬ𝑛 and the number of involutions of π’Ÿπ‘› will be given. This will be used in the next theorem.

Lemma 3.11. If 𝑛 is even (𝑛=2π‘˜), then 2||Invξ€·π’Ÿπ‘›ξ€Έ||βˆ’||Invℬ𝑛||=(2π‘˜)!.π‘˜!(3.29)

Proof. If 𝜎=(𝑀,πœ‹)βˆˆβ„¬π‘›, with π‘€βˆˆπ’žπ‘›2, and πœ‹βˆˆπ”–π‘› is an involution, then the cyclic structure of 𝜎 looks like ±𝑖1,±𝑗1ξ€Έβ‹―ξ€·Β±π‘–π‘Ÿ,Β±π‘—π‘Ÿξ€Έξ€·Β±π‘˜1ξ€Έβ‹―ξ€·Β±π‘˜π‘ ξ€Έ,(3.30) where πˆπ‘›={𝑖1,β€¦π‘–π‘Ÿ,𝑗1,…,π‘—π‘Ÿ,π‘˜1,…,k𝑠}, πœ‹=(𝑖1,𝑗1)β‹―(π‘–π‘Ÿ,π‘—π‘Ÿ)(π‘˜1)β‹―(π‘˜π‘ ) is the decomposition of πœ‹ as product of disjoint cycles, and 𝑀𝑖𝑙=𝑀𝑗𝑙 for 1β‰€π‘™β‰€π‘Ÿ. Thus, the number of involutions of ℬ𝑛 is ||Invℬ𝑛||=π‘˜ξ“π‘Ÿ=0βˆπ‘Ÿβˆ’1𝑗=0ξ€·2π‘›βˆ’2𝑗2π‘Ÿ!π‘Ÿ2π‘›βˆ’2π‘Ÿ(3.31) If π‘Ÿ<π‘˜, half of the elements belong to π’Ÿπ‘› and the other half to ℬ𝑛-π’Ÿπ‘›, and therefore ||Invξ€·π’Ÿπ‘›ξ€Έ||=12π‘˜βˆ’1ξ“π‘Ÿ=0βˆπ‘Ÿβˆ’1𝑗=0ξ€·2π‘›βˆ’2𝑗2π‘Ÿ!π‘Ÿ2π‘›βˆ’2π‘Ÿ+βˆπ‘˜βˆ’1𝑗=0ξ€·2π‘›βˆ’2𝑗2π‘˜!π‘˜2π‘›βˆ’2π‘˜.(3.32) Then, 2||Invξ€·π’Ÿπ‘›ξ€Έ||βˆ’||Invℬ𝑛||=βˆπ‘˜βˆ’1𝑗=0ξ€·2π‘›βˆ’2𝑗2π‘˜!π‘˜2π‘›βˆ’2π‘˜=(2π‘˜)!.π‘˜!(3.33)

Theorem 3.12. The 𝐺-module β„±0 is a Gelfand model for the group π’Ÿπ‘›.

Proof. As it has been mentioned above, when 𝑛 is odd, β„±0 is equal to π’©π’Ÿπ‘›, and in [5] it has been proved that π’©π’Ÿπ‘› is a Gelfand model for the group π’Ÿπ‘›.
When 𝑛 is even, from the fact β„±0 contains a Gelfand model, only it is necessary to prove that dim(β„±0)≀|Inv(π’Ÿπ‘›)|. From identity (3.22), it results that ξ€·β„±dim0ξ€ΈβŽ›βŽœβŽœβŽœβŽœβŽξΆβ‰€dimπ›Ύβˆˆπ’ͺβ„¬π‘›βˆ’minimal||𝐸𝛾||<||𝑂𝛾||𝑆Δ𝛾+ξΆξ„žξ€·π›Ύπ‘‚ξ€Έβˆ’1/2≺𝛾𝐸/2𝑆Δ𝛾+𝛾𝐸/2=ξ„žξ€·π›Ύπ‘‚ξ€Έβˆ’1/2π‘†Ξ”π›ΎβŽžβŽŸβŽŸβŽŸβŽŸβŽ .(3.34) By Lemma 3.10, it follows that the dimension of the model 𝒩ℬ𝑛 is equal to the number of involutions of the group ℬ𝑛, and thus by the Lemma 3.9 it results that ||Invℬ𝑛||βŽ›βŽœβŽœβŽœβŽœβŽξΆ=2dimπ›Ύβˆˆπ’ͺβ„¬π‘›βˆ’minimal||𝐸𝛾||<||𝑂𝛾||π‘†Ξ”π›ΎβŽžβŽŸβŽŸβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽœβŽœβŽξΆ+2dim𝛾𝐸/2β‰Ίξ„žξ€·π›Ύπ‘‚ξ€Έβˆ’1/2π‘†Ξ”π›ΎβŽžβŽŸβŽŸβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽœβŽœβŽξΆ+dim𝛾𝐸/2=ξ„žξ€·π›Ύπ‘‚ξ€Έβˆ’1/2π‘†Ξ”π›ΎβŽžβŽŸβŽŸβŽŸβŽŸβŽ ,||Invℬ𝑛||βŽ›βŽœβŽœβŽœβŽœβŽξΆ+dim𝛾𝐸/2=ξ„žξ€·π›Ύπ‘‚ξ€Έβˆ’1/2π‘†Ξ”π›ΎβŽžβŽŸβŽŸβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽœβŽœβŽξΆ=2dimπ›Ύβˆˆπ’ͺβ„¬π‘›βˆ’minimal||𝐸𝛾||<||𝑂𝛾||𝑆Δ𝛾+ξΆξ„žξ€·π›Ύπ‘‚ξ€Έβˆ’1/2βͺ―𝛾𝐸/2π‘†Ξ”π›ΎβŽžβŽŸβŽŸβŽŸβŽŸβŽ ,12⎑⎒⎒⎒⎒⎣||Invℬ𝑛||βŽ›βŽœβŽœβŽœβŽœβŽξΆ+dim𝛾𝐸/2=ξ„žξ€·π›Ύπ‘‚ξ€Έβˆ’1/2π‘†Ξ”π›ΎβŽžβŽŸβŽŸβŽŸβŽŸβŽ βŽ€βŽ₯βŽ₯βŽ₯βŽ₯βŽ¦βŽ›βŽœβŽœβŽœβŽœβŽξΆ=dimπ‘£βˆˆπ’ͺβ„¬π‘›βˆ’minimal||𝐸𝛾||<||𝑂𝛾||𝑆Δ𝛾+ξΆξ„žξ€·π›Ύπ‘‚ξ€Έβˆ’1/2βͺ―𝛾𝐸/2π‘†Ξ”π›ΎβŽžβŽŸβŽŸβŽŸβŽŸβŽ ,βŽ›βŽœβŽœβŽœβŽœβŽξΆdim𝛾𝐸/2=ξ„žξ€·π›Ύπ‘‚ξ€Έβˆ’1/2π‘†Ξ”π›ΎβŽžβŽŸβŽŸβŽŸβŽŸβŽ =𝛾𝐸/2=ξ„žξ€·π›Ύπ‘‚ξ€Έβˆ’1/2𝑆dimΔ𝛾=ξ“πœ’βˆˆπ”–π‘›βŽ›βŽœβŽœβŽπ‘›βŽžβŽŸβŽŸβŽ πœ’2𝑛2(1)=(2𝑛)!𝑛!(3.35) and then 12ξ‚Έ||Invℬ𝑛||+(2𝑛)!ξ‚ΉβŽ›βŽœβŽœβŽœβŽœβŽξΆπ‘›!=dimπ›Ύβˆˆπ’ͺβ„¬π‘›βˆ’minimal||𝐸𝛾||<||𝑂𝛾||𝑆Δ𝛾+ξΆξ„žξ€·π›Ύπ‘‚ξ€Έβˆ’1/2βͺ―𝛾𝐸/2π‘†Ξ”π›ΎβŽžβŽŸβŽŸβŽŸβŽŸβŽ .(3.36) On the other hand, from identity established in Lemma 3.11, it results that ||Invξ€·π’Ÿπ‘›ξ€Έ||=12ξ‚Έ||Invℬ𝑛||+(2𝑛)!ξ‚Ή,𝑛!(3.37) and using identities (2.2) and (3.36), it is obtained that ξ€·β„±dim0≀||Invξ€·π’Ÿπ‘›ξ€Έ||.(3.38) Therefore, it has been proved that β„±0 is a Gelfand model for π’Ÿπ‘›.

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