ALGEBRA ISRN Algebra 2090-6293 International Scholarly Research Network 658201 10.5402/2012/658201 658201 Research Article A Gelfand Model for Weyl Groups of Type D2n Araujo José O. Maiarú Luis C. Natale Mauro Airault H. Sage D. Vourdas A. You H. Departmento de Matemática, Facultad de Ciencias Exactas, UNICEN B7000 GHG, Tandil Argentina nicen.edu.ar 2012 7 8 2012 2012 27 03 2012 17 04 2012 2012 Copyright © 2012 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.

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(), where is the field of complex numbers, it has been proved that each G-module over is isomorphic to a G-submodule in the polynomial ring [x1,,xn], and taking the space of zeros of certain G-invariant operators in the Weyl algebra, a finite-dimensional G-space 𝒩G in [x1,,xn] can be obtained, which contains all the simple G-modules over . 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 . Since then, Gelfand models have been developed in several articles; see , 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 (1.1)χCkSn=χλ, where Ck is the centralizer of an involution in 𝔖n with exactly k fixed points, χCk is a linear character of Ck, and χλ is an irreducible character of 𝔖n associated with the partition λ of n with exactly k odd terms. From this identity, it follows immediately that (1.2)kχCkSn=λχλ, where the centralizers Ck are in correspondence with the conjugacy classes of involutions in 𝔖n.

Later on, this type of models was called an involution model by Baddeley . 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  by Inglis et al. and for a Weyl group of type Bn an involution model is shown in [6, 13]. In , Baddeley presents an involution model for a Weyl group of type D2n+1, and in  it is proved that there is no involution model for a Weyl group of type D2n with n2. In , 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 , 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 I2n(n3,n6) and H3 and presents a conceptual demonstration of the no existence of an involution model for the group of type H4.

More recently, in  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 , Shephard and Todd , Steinberg , and others studied the corresponding action on the space G of G-harmonic polynomials proving that G is isomorphic to the regular representation of G, and thus G 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 .

More recently, Araujo and Aguado in  have associated with each finite subgroup GGLn() a subspace 𝒩G of the algebra of polynomials [x1,,xn], 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 , it was proved that 𝒩G is a Gelfand model for Weyl groups of type An, Bn and D2n+1. Garge and Oesterlé in , 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 , when G is of type An, Lusztig , when G is of type Bn or Dn, Beynon and Lusztig , when G is an exceptional Weyl group, Alvis and Lusztig , 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() and 𝒫 the polynomial ring [x1,,xn].

Let In={1,,n} be the set of the first n natural numbers and n the set of multi-index functions: (2.1)Mn={α:InN0}

For each αn the following notation will be used in the rest of this paper: (2.2)αi=α(i),α=(α1,,αn),|α|=i=1nαi.

Let 𝒜=x1,,xn,1,,n be the Weyl algebra of -linear differential operators generated by the multiplication operators xi and partial differential operators i=/xi with 1in.

It is known that each D𝒜 has a unique expression as a finite sum (see ): (2.3)D=λα,βxαβ, where α,βn, λα,β, and (2.4)xα=x1α1x2α2xnαn,β=1β12β2nβn. The degree of D is defined by (2.5)deg(D)=max{i(αi-βi):λα,β0}. The Weyl algebra is a graduated algebra 𝒜=i𝒜i, where (2.6)Ai={α,βMnλα,βxαβ:|α|-|β|=i}. The action of G on 𝒫 induces an action of G on the endomorphism ring End(𝒫), which is defined by (2.7)(gD)(p)=(gDg-1)(p)(gG,DEndC(P)). This action can be restricted to the Weyl algebra 𝒜 noting that each 𝒜i is invariant under the action of G.

Let G be the subalgebra of G-invariant operators in 𝒜, that is, (2.8)IG={DA:gD=D,gG}. Notice that G is contained in the centralizer of G in End(𝒫).

Let G- be the subspace of the Weyl algebra, formed by the G-invariant operators with negative degree (2.9)IG-={DIG:deg(D)<0}.

Definition 2.1.

Let 𝒩G be the subspace of 𝒫 defined by (2.10)NG={pP:D(p)=0,DIG-}.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.

𝒩 G 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 W be its realization as a reflection group. Then, the polynomial model 𝒩W is a Gelfand model for G if, and only if, W 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, Bn and Dn.

2.1. Polynomial Model for a Weyl Group of Type <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M160"><mml:mrow><mml:msub><mml:mrow><mml:mi>A</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:math></inline-formula>

Let G be a Weyl group of type An-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 n by (2.11)σα=ασ-1(σSn,αMn). This action induces a natural homomorphism from 𝔖n in Aut(𝒫) given by (2.12)σ(αMnλαxα)=αMnλαxσα(λαC).

2.1.1. <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M171"><mml:mrow><mml:msub><mml:mrow><mml:mi>𝔖</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:math></inline-formula>-Minimal Orbit

Let 𝒪n be the orbit space of 𝔖n in n. 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 γ~𝔖nδ, if there exists a bijection φ:00 such that (2.13)δ={φα:αγ}.

Definition 2.5.

An orbit γ will be called 𝔖n-minimal if |γ||δ| for all δ𝒪n such that γ~𝔖nδ.

Proposition 2.6.

An orbit γ is 𝔖n-minimal if, and only if, for each αγ, there exists a nonnegative integer h such that

Im (α)={0,1,,h-1},

|α-1(i)||α-1(i+1)| for all 0ih-1(|α-1(i)| being the cardinal of the set α-1(i)).

Proof.

See [4, page 1845].

Definition 2.7.

For each γ𝒪n, let Sγ be the subspace of 𝒫 defined by (2.14)Sγ={αγλαxα:λαC}

2.1.2. The Space <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M207"><mml:mrow><mml:msubsup><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mi>γ</mml:mi></mml:mrow><mml:mrow><mml:mo>∂</mml:mo></mml:mrow></mml:msubsup></mml:mrow></mml:math></inline-formula>

Let be the operator defined by =i=1ni, where i are the partial differential operators as above. For each γ𝒪n, let Sγ be the subspace defined by (2.15)Sγ={PSγ:(P)=0}

2.1.3. The Structure of <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M214"><mml:mrow><mml:msub><mml:mrow><mml:mi>𝒩</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>𝔖</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msub></mml:mrow></mml:math></inline-formula>

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

Theorem 2.8.

S γ is an irreducible G-module, and 𝒩𝔖n can be decomposed as (2.16)NSn=γOSn-minimalSγ. Moreover, 𝒩𝔖n is a Gelfand model of 𝔖n.

2.2. Polynomial Model for a Weyl Group of Type <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M222"><mml:mrow><mml:msub><mml:mrow><mml:mi>B</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:math></inline-formula>

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

Let 𝒞2={1,-1}* be the subgroup of order two. The Weyl group n, of type Bn, can be presented as the semidirect product (2.17)Bn=C2n×sSn, where 𝒞2n=𝒞2××𝒞2 and the semidirect product is induced by the natural action of 𝔖n on 𝒞2n: (2.18)σ(w1,,wn)=(wσ(1),,wσ(n))(σSn,(w1,,wn)C2n).

The action of 𝔖n on n induces a natural homomorphism from n on Aut(𝒫) given by (2.19)(w,σ)(αMnλαxα)=αMnλα(wx)σα(λαC) with (2.20)(wx)σα=i=1n(wixi)(σα)i.

2.2.1. <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M238"><mml:mrow><mml:msub><mml:mrow><mml:mi>ℬ</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:math></inline-formula>-Minimal Orbit

Let 𝒪n be the orbit space of 𝔖n on n, as above.

Definition 2.9.

Two orbits, γ and δ, will be called n-equivalent, denoted by γ~nδ, if there exists a bijection φ:00 such that φ(k) and k have the same parity for all k0,δ={φα:αγ}.

Definition 2.10.

An orbit γ will be called n-minimal if |γ||δ| for all δ𝒪n such that γ~nδ.

Proposition 2.11.

An orbit γ is n-minimal if, and only if, for each αγ and each pair i,j0 with the same parity, one has |α-1(i)||α-1(j)| with 0i<j (|α-1(i)| being the cardinal of the set α-1(i)).

Proof.

See [3, page 365].

2.2.2. The Space <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M265"><mml:mrow><mml:msubsup><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mi>γ</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">Δ</mml:mi></mml:mrow></mml:msubsup></mml:mrow></mml:math></inline-formula>

Let Δ be the Laplacian operator defined by Δ=i=1ni2, where i are the partial differential operators mentioned above. For each γ𝒪n, let SγΔ be the subspace defined by (2.21)SγΔ={PSγ:Δ(P)=0}.

2.2.3. The Structure of <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M272"><mml:mrow><mml:msub><mml:mrow><mml:mi>𝒩</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>ℬ</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msub></mml:mrow></mml:math></inline-formula>

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

Theorem 2.12.

S γ Δ is an irreducible G-module, and 𝒩n can be decomposed as (2.22)NBn=γOBn-minimalSγΔ. Moreover 𝒩n is a Gelfand model of n.

Proof.

See [3, page 371].

2.3. Polynomial Model for a Weyl Group of Type <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M280"><mml:mrow><mml:msub><mml:mrow><mml:mi>D</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:math></inline-formula>

Let 𝒟n be the Weyl group of type Dn naturally included in n. Using the previous notation, for αn the following sets are considered: (2.23)Eα={iIn:αiiseven},Oα={iIn:αiisodd}.

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. <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M292"><mml:mrow><mml:msub><mml:mrow><mml:mi>𝒟</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:math></inline-formula>-Minimal Orbit Definition 2.13.

Two orbits γ and δ will be called 𝒟n-equivalent, denoted by γ~𝒟nδ, if there exists a bijection φ:00 such that

k0, φ(k) and k have the same parity or φ(k) and k have different parities,

δ={φα:αγ}.

Definition 2.14.

An orbit γ will be called 𝒟n-minimal if |γ||δ| for all δ𝒪n such that γ~𝒟nδ.

Proposition 2.15.

Let γ be an orbit, and then the following statements are true

γ is 𝒟n-minimal if, and only if, the following statements are verified:

given αγ and i<j0 with the same parity, then |α-1(i)||α-1(j)|,

|Eγ||Oγ|.

Let π:00 be the involution given by π(2i)=2i+1 and π(2i+1)=2i. The following assertions are equivalent:

γ and πγ are 𝒟n-minimal orbits,

γ is n-minimal,

πγ is n-minimal.

There are at most two 𝒟n-minimal orbits equivalent to γ.

If n is odd, there is only one 𝒟n-minimal orbit equivalent to γ.

γ and πγ are 𝒟n-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.

If γ is 𝒟n-minimal, then (2.24)NBnSγ=NDnSγ and 𝒩𝒟nSγ is a simple 𝒟n-module.

𝒩𝒟n is a Gelfand model for 𝒟n.

Every simple n-module remains simple when it is considered as a 𝒟n-module by restriction.

By considering 𝒩n as a 𝒟n-module by restriction, 𝒩n is isomorphic to 𝒩𝒟n𝒩𝒟n.

Proof.

See [5, page 110].

Also in  it has been proved that if n is even, 𝒩𝒟n is not a Gelfand model for a Weyl group of type Dn. But it does happen that if M is a simple 𝒟n-module, then 𝒩𝒟n contains a copy ofthis, and the multiplicity of M in 𝒩𝒟n is

two, if M is isomorphic to 𝒩𝒟nSγ,γ being a 𝒟n-minimal orbit such that γπγ and |Eγ|=|Oγ|; in this case, as before, π:00 is the involution given by π(2i)=2i+1 and π(2i+1)=2i,

one, otherwise.

3. Gelfand Model for a Weyl Group of Type <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M366"><mml:mrow><mml:msub><mml:mrow><mml:mi>D</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:mi>n</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:math></inline-formula>

As before, let n={α:In0} be the set of multi-index functions. Every αn has an associated vector α^0n, which is obtained by reordering α as follows. (3.1)α^=(αi1,,αin)suchthatαi1αin. Thus, there is defined an order relationship in n given by for all α,βn, αβ if, and only if, α^=β^ or there exists s(1sn) such that (3.2)α^1=β^1,,α^s-1=β^s-1,α^s<β^s,α^i and β^i being the coordinates of the vectors α^ and β^, respectively. Notice that this is the lexicographic order for 0n.

Proposition 3.1.

Let γ𝒪n and α,βn, and then α,βγ if, and only if, α^=β^.

Proof.

Let α,βγ, and therefore there exists σ𝔖n such that β=σα, which implies βi=ασ-1(i) with 1in. Thus, it is easy to see that α^=β^.

On the other hand, let α,βn and α^=β^, say, (3.3)αi1=βj1,,αin=βjn. Let σ𝔖n be given by (3.4)σ-1(ik)=jk(1kn). Then, β=σα, and hence both multi-indexes belong to the same orbit.

From this proposition it is clear that induces a total orderin 𝒪n, which is defined by (3.5)γδαβ(γ,δOn,αγ,βδ). Since the vector α^ is independent of the choice α in γ, it will be denoted by γ^.

Proposition 3.2.

Let αn be defined by αi=i-1, and let γ be the orbit of α. Then, γ is the -maximum of the 𝔖n-minimal orbits and γ^=(n-1,,1,0).

Proof.

From the previous considerations it is clear that γ is an 𝔖n-minimal orbit and γ^=(n-1,,1,0).

Now it will be proved that γ is the -maximum in the set of 𝔖n-minimal orbits. Let δ be an orbit such that δγ and γδ. Then, it should exist an sIn satisfying (3.6)γ^i=δ^i,i<s,γ^s<δ^s, that is, (3.7)δ^1=n-1δ^s-1=n-(s-1),δ^s>n-s.

Thus, it occurs that δ^s=n-(s-1), and from the minimality of δ every number less than n-(s-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, A will denote the set of multi-index functions with domain A, (3.8)MA={α:AN0}, and ı will denote the function in A given by (3.9)ı(i)=1,iA. As in the case of 𝔖n, there is a natural action from the symmetric group 𝔖A in the set of multi-index functions A, defined by (3.10)σα=ασ-1(σSA,αMA). It is possible to extend the concept of 𝔖A-minimal orbit.

For each αn, let us consider the sets (3.11)Eα={iIn:αiiseven}Oα={iIn:αiisodd} 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 αEEα and αOOα.

Proposition 3.3.

Let αn such that α is n-minimal, and then αE/2 is 𝔖Eα-minimal and (αO-1)/2 is 𝔖Oα-minimal.

Proof.

It follows from Proposition 4 in  and the identities (3.12)|(αE2)-1(i)|=|α(i)|(i|Eα|),|(αO-12)-1(j)|=|α(j)|(j|Oα|).

Notation 1.

Let 𝒦 be the subset of n given by (3.13)K={αMn:αisDn-minimal,|Eα|=|Oα|andαE2αO-12}. It will be denoted by the subset of the polynomial ring 𝒫: (3.14)F={αλαxα:λα=0ifαK}. Note that if n is odd, is equal to 𝒫.

Proposition 3.4.

is a 𝒟n-submodule of the polynomial ring 𝒫.

Proof.

It follows from the action of 𝒟n given by (3.15)(w,σ)(αλαxα)=αλα(wx)σα=α±λαxσα and the fact that α𝒦 if, and only if, σα𝒦, and it results that is a 𝒟n-module of 𝒫.

Proposition 3.5.

contains a Gelfand model for the Weyl group of type 𝒟n.

Proof.

It is sufficient to prove that contains a submodule equivalent to the regular module 𝒟n. Effectively, let us consider the polynomial: (3.16)P(x1,,xn)=i=2n[xi(xi+i-1)j=1i-2(xi2-j2)]2n-1[(n-1)!]2i=2nj=1i-2(i-1+j)(i-1-j). Thus P is the interpolating polynomial of the orbit of the regular vector v=(0,1,,n-1), which satisfies (3.17)P(v)=1,P(τv)=0(τDn,τe).

It will be proved that P belongs to . Let λαxα be not a null term of P such that it is 𝒟n-minimal, |Eα|=|Oα|, and αE/2(αO-1)/2. As P was defined in (3.16), it is easy to determine that α1=0 and αj>0 for 1<jn. Then, (αE/2)1=0, and as αE/2 is 𝔖|Eα|-minimal, it is obtained that (3.18)αE2^=(|Eα|-1,|Eα|-2,,0), which by Proposition 3.2 is maximal, which is a contradiction.

Since is a 𝒟n-module, contains the module generated by the orbit of P, which is isomorphic to the regular module. Hence, contains a Gelfand model.

Notation 2.

Let 𝒯 be a subset of the polynomial ring 𝒫 and G a finite subgroup of GLn(); we will denote by 𝒯0 the subset of 𝒯 defined by (3.19)T0={pT:D(p)=0,DIG-},G- 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() 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 S𝒯 be a simple G-module and suppose that S𝒯0. Then, there exists DG- such that D(S)0. Because S is simple and D not null, it follows that D is injective. Thus, D(S)S. If D(S)𝒯0, the proposition is proved; otherwise the procedure will be repeated. As D is an operator of the Weyl algebra 𝒜 with negative degree, the procedure is finite, that is to say, there exists m such that Dm(S)𝒯0 and Dm(S)S.

Remark 3.7.

An immediate consequence of this proposition is that the 𝒟n-module (3.20)F0={fF:D(f)=0,DIDn-} contains a Gelfand model because is a 𝒟n-module containing a Gelfand model.

Remark 3.8.

Notice that if n is odd, then 0=𝒩Dn; instead, if n is even, (3.21)F0FNDn=γOBn-minimal|Eγ|<|Oγ|SγΔ+γOBn-minimal|Eγ|=|Oγ|SγΔ that is (3.22)F0γOBn-minimal|Eγ|<|Oγ|SγΔ+(γO-1^)/2γE/2SγΔ+γE/2=(γO-1^)/2SγΔ.

Using the result established in item 4 of Proposition 2.16 for decomposing 𝒩n, it follows that (3.23)NBn=γOBn-minimalSγΔ=γOBn-minimal|Eγ|<|Oγ|SγΔ+γOBn-minimal|Eγ|=|Oγ|SγΔ+γOBn-minimal|Eγ|>|Oγ|SγΔ. Moreover if |Eγ|=|Oγ|(3.24)γOBn-minimal|Eγ|=|Oγ|SγΔ=γE/2(γO-1)^/2SγΔ+γE/2=(γO-1)^/2SγΔ+(γO-1)^/2γE/2SγΔ.

As a consequence of this decomposition the next lemma follows.

Lemma 3.9.

The dimension of 𝒩n is equal to (3.25)2dim(γOBn-minimal|Eγ|<|Oγ|SγΔ)+2dim((γO-1)^/2γE/2SγΔ)+dim(γE/2=(γO-1)^/2SγΔ).

Proof.

It results from considering the identity (3.26)dim(SγΔ)=dim(SπγΔ) for each γ𝒪n. This identity occurs from the fact that SπΔ and SπγΔ are isomorphic as 𝒟n-modules, see .

Let G be a group; from now on we will be denote by (3.27)Inv(G)={gG:g2=e} the set of involutions of the group G.

Lemma 3.10.

Let G be a Coxeter group and a Gelfand model for G. Then, (3.28)dim(M)=| Inv (G)|.

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 n and the number of involutions of 𝒟n will be given. This will be used in the next theorem.

Lemma 3.11.

If n is even (n=2k), then (3.29)2| Inv (Dn)|-| Inv (Bn)|=(2k)!k!.

Proof.

If σ=(w,π)n, with w𝒞2n, and π𝔖n is an involution, then the cyclic structure of σ looks like (3.30)(±i1,±j1)(±ir,±jr)(±k1)(±ks), where In={i1,ir,j1,,jr,k1,,ks}, π=(i1,j1)(ir,jr)(k1)(ks) is the decomposition of π as product of disjoint cycles, and wil=wjl for 1lr. Thus, the number of involutions of n is (3.31)|Inv(Bn)|=r=0kj=0r-1(n-2j2)r!2r2n-2r If r<k, half of the elements belong to 𝒟n and the other half to n-𝒟n, and therefore (3.32)|Inv(Dn)|=12r=0k-1j=0r-1(n-2j2)r!2r2n-2r+j=0k-1(n-2j2)k!2k2n-2k. Then, (3.33)2|Inv(Dn)|-|Inv(Bn)|=j=0k-1(n-2j2)k!2k2n-2k=(2k)!k!.

Theorem 3.12.

The G-module 0 is a Gelfand model for the group 𝒟n.

Proof.

As it has been mentioned above, when n is odd, 0 is equal to 𝒩𝒟n, and in  it has been proved that 𝒩𝒟n is a Gelfand model for the group 𝒟n.

When n is even, from the fact 0 contains a Gelfand model, only it is necessary to prove that dim(0)|Inv(𝒟n)|. From identity (3.22), it results that (3.34)dim(F0)dim(γOBn-minimal|Eγ|<|Oγ|SγΔ+(γO-1)^/2γE/2SγΔ+γE/2=(γO-1)^/2SγΔ). By Lemma 3.10, it follows that the dimension of the model 𝒩n is equal to the number of involutions of the group n, and thus by the Lemma 3.9 it results that (3.35)|Inv(Bn)|=2dim(γOBn-minimal|Eγ|<|Oγ|SγΔ)+2dim(γE/2(γO-1)^/2SγΔ)+dim(γE/2=(γO-1)^/2SγΔ),|Inv(Bn)|+dim(γE/2=(γO-1)^/2SγΔ)=2dim(γOBn-minimal|Eγ|<|Oγ|SγΔ+(γO-1)^/2γE/2SγΔ),12[|Inv(Bn)|+dim(γE/2=(γO-1)^/2SγΔ)]=dim(vOBn-minimal|Eγ|<|Oγ|SγΔ+(γO-1)^/2γE/2SγΔ),dim(γE/2=(γO-1)^/2SγΔ)=γE/2=(γO-1)^/2dim(SγΔ)=χSn(2nn)χ2(1)=(2n)!n! and then (3.36)12[|Inv(Bn)|+(2n)!n!]=dim(γOBn-minimal|Eγ|<|Oγ|SγΔ+(γO-1)^/2γE/2SγΔ). On the other hand, from identity established in Lemma 3.11, it results that (3.37)|Inv(Dn)|=12[| Inv (Bn)|+(2n)!n!], and using identities (2.2) and (3.36), it is obtained that (3.38)dim(F0)| Inv (Dn)|. Therefore, it has been proved that 0 is a Gelfand model for 𝒟n.

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