A Gelfand Model for Weyl Groups of Type D2n

International Scholarly Research NetworkISRN AlgebraVolume 2012, Article ID 658201, 16 pagesdoi:10.5402/2012/658201

Research ArticleA Gelfand Model for Weyl Groups of Type D2n

Jose O. Araujo, Luis C. Maiaru, and Mauro Natale

Departmento de Matematica, Facultad de Ciencias Exactas, UNICEN, B7000 GHG, Tandil, Argentina

Correspondence should be addressed to Jose O. Araujo, [email protected]

Received 27 March 2012; Accepted 17 April 2012

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

Copyright q 2012 Jose O. Araujo et al. This is an open access article distributed under the CreativeCommons Attribution License, which permits unrestricted use, distribution, and reproduction inany 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 thedirect sum of all irreducible representations of G. When G is isomorphic to a subgroup of GLn(C),where C is the field of complex numbers, it has been proved that each G-module over C is isomor-phic to aG-submodule in the polynomial ringC[x1, . . . , xn], and taking the space of zeros of certainG-invariant operators in the Weyl algebra, a finite-dimensionalG-spaceNG in C[x1, . . . , xn] can beobtained, which contains all the simple G-modules over C. This type of representation has beennamed polynomial model. It has been proved that when G is a Coxeter group, the polynomialmodel is a Gelfand model for G if, and only if, G has not an irreducible factor of type D2n, E7, orE8. This paper presents a model of Gelfand for a Weyl group of type D2n whose construction isbased on the same principles as the polynomial model.

1. Introduction

Gelfand models for a finite group are complex representations whose character is the sumof all irreducible characters of the given group. In this sense, Bernstein et al. have presentedGelfandmodels for semisimple compact Lie groups, see [1]. Since then, Gelfand models havebeen developed in several articles; see [2–12], among these there are two types of models thatcan 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, onthe other, showed an identity that describes a Gelfand model associated with the symmetricgroup. The identity is given by

χCk ↑ Sn =∑

χλ, (1.1)

2 ISRN Algebra

where Ck is the centralizer of an involution in Sn with exactly k fixed points, χCk is a linearcharacter of Ck, and χλ is an irreducible character of Sn associated with the partition λ of nwith exactly k odd terms. From this identity, it follows immediately that

∑

k

χCk ↑ Sn =∑

λ

χλ, (1.2)

where the centralizers Ck are in correspondence with the conjugacy classes of involutions inSn.

Later on, this type of models was called an involution model by Baddeley [6]. He alsoproved that ifH is a finite group that admits an involution model, then so does the semidirectproduct 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 typeAn is presented in [8] by Inglis et al. and for a Weyl group of type Bn an involution modelis shown in [6, 13]. In [6], Baddeley presents an involution model for a Weyl group of typeD2n+1, and in [14] it is proved that there is no involution model for a Weyl group of type D2n

with n ≥ 2. In [6], it is mentioned that is not difficult to prove that there is an involutionmodelfor aWeyl group of typeG2 and that it has been checked using computers the non existence ofinvolutionmodels for exceptionalWeyl groups of type F4, E6, E7, and E8. In [15], Vinroot doessome research about involution models for irreducible non crystallographic Coxeter groups.He proves the existence of an involution model for groups of type In2 (n ≥ 3, n /= 6) and H3

and presents a conceptual demonstration of the no existence of an involution model for thegroup of type H4.

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

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

Chevalley [17], Shephard and Todd [18], Steinberg [19], and others studied the corre-sponding action on the spaceHG of G-harmonic polynomials proving thatHG is isomorphicto the regular representation of G, and thus HG contains a Gelfand model for G. On theother hand,Macdonald found irreducible representations of aWeyl group associatedwith theroot systems of the reflection subgroups that can be naturally realized in the G-harmonicpolynomial space. These representations are known as Madonald representations see [20].

More recently, Araujo and Aguado in [21] have associated with each finite subgroupG ⊂ GLn(C) a subspace NG of the algebra of polynomials C[x1, . . . , xn], defined as zeros ofcertain G-invariant differential operators, and have shown NG contains a Gelfand model ofG. 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, Bn and D2n+1. Gargeand Oesterle in [22], using the computation of fake degrees of the irreducible characters ofa Coxeter group G, determined that NG is a Gelfand model of G if, and only if, G has notirreducible factors of typeD2n,E7, orE8. The fake degrees have been determined due toworksof Steinberg [23], when G is of typeAn, Lusztig [24], when G is of type Bn orDn, Beynon andLusztig [25], when G is an exceptional Weyl group, Alvis and Lusztig [26], when G is oftype H4, and Macdonald, when G is of type F4 (unpublished). The remaining cases are notdifficult.

ISRN Algebra 3

For the case of Weyl groups of typeD2n, neither the polynomial model nor the involu-tion model provides a Gelfand model.

In this paper the construction of a Gelfand model for a Weyl group of typeD2n will bepresented. 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.Gwill denote a finite subgroup of GLn(C) and P the polynomial ring C[x1, . . . , xn].Let In = {1, . . . , n} be the set of the first n natural numbers and Mn the set of multi-

index functions:

Mn = {α : In −→ N0} (2.1)

For each α ∈ Mn the following notation will be used in the rest of this paper:

αi = α(i), α = (α1, . . . , αn), |α| =n∑

i=1

αi. (2.2)

Let A = C〈x1, . . . , xn, ∂1, . . . , ∂n〉 be the Weyl algebra of C-linear differential operatorsgenerated by the multiplication operators xi and partial differential operators ∂i = ∂/∂xi with1 ≤ i ≤ n.

It is known that each D ∈ A has a unique expression as a finite sum (see [27]):

D =∑

λα,βxα∂β, (2.3)

where α, β ∈ Mn, λα,β ∈ C, and

xα = xα11 xα2

2 · · ·xαnn ,

∂β = ∂β11 ∂

β22 · · · ∂βnn .

(2.4)

The degree of D is defined by

deg(D) = max

{∑

i

(αi − βi

): λα,β /= 0

}. (2.5)

The Weyl algebra is a graduated algebra A =⊕

i∈ZAi, where

Ai =

⎧⎨

⎩∑

α,β∈Mn

λα,βxα∂β : |α| −

∣∣β∣∣ = i

⎫⎬

⎭. (2.6)

4 ISRN Algebra

The action of G on P induces an action of G on the endomorphism ring EndC(P), which isdefined by

(g ·D)(p)=(gDg−1

)(p) (

g ∈ G, D ∈ EndC(P)). (2.7)

This action can be restricted to the Weyl algebra A noting that eachAi is invariant under theaction of G.

Let IG be the subalgebra of G-invariant operators inA, that is,

IG ={D ∈ A : g ·D = D, ∀g ∈ G

}. (2.8)

Notice that IG is contained in the centralizer of G in EndC(P).Let I−

G be the subspace of the Weyl algebra, formed by the G-invariant operators withnegative degree

I−G ={D ∈ IG : deg(D) < 0

}. (2.9)

Definition 2.1. Let NG be the subspace of P defined by

NG ={p ∈ P : D

(p)= 0, ∀D ∈ I−

G

}. (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 inNG.

Proof. See [21, page 38].

The analysis of the polynomial model for Coxeter groups has been completely solvedby the following theorem.

Theorem 2.3. Let G be a finite irreducible Coxeter group, and let W be its realization as a reflectiongroup. Then, the polynomial modelNW is a Gelfand model forG if, and only if,W is not a Weyl groupof type D2n, E7, or E8.


In the following sections it will be presented a characterization of the polynomialmodel for the classical Weyl groups of type An, Bn and Dn.

2.1. Polynomial Model for a Weyl Group of Type An

Let G be a Weyl group of type An−1. It is known that G can be presented as the symmetricgroup Sn.

ISRN Algebra 5

The symmetric group Sn acts on the set of multi-index functions Mn by

σ · α = α ◦ σ−1 (σ ∈ Sn, α ∈ Mn). (2.11)

This action induces a natural homomorphism from Sn in Aut(P) given by

σ

(∑

α∈Mn

λαxα

)=∑

α∈Mn

λαxσ·α (λα ∈ C). (2.12)

2.1.1. Sn-Minimal Orbit

Let On be the orbit space of Sn in Mn. It is clear that if two multi-indexes α and β belong tothe 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 thereexists a bijection ϕ : N0 → N0 such that

δ ={ϕ ◦ α : α ∈ γ

}. (2.13)

Definition 2.5. An orbit γ will be called Sn-minimal if |γ | ≤ |δ| for all δ ∈ On such that γ∼Snδ.

Proposition 2.6. An orbit γ is Sn-minimal if, and only if, for each α ∈ γ , there exists a nonnegativeinteger h such that

(1) Im (α) = {0, 1, . . . , h − 1},(2) |α−1(i)| ≥ |α−1(i + 1)| for all 0 ≤ i ≤ h − 1(|α−1(i)| being the cardinal of the set α−1(i)).


Definition 2.7. For each γ ∈ On, let Sγ be the subspace of P defined by

Sγ =

{∑

α∈γλαx

α : λα ∈ C

}. (2.14)

2.1.2. The Space S∂γ

Let ∂ be the operator defined by ∂ =∑n

i=1 ∂i, where ∂i are the partial differential operators asabove. For each γ ∈ On, let S∂

γ be the subspace defined by

S∂γ ={P ∈ Sγ : ∂(P) = 0

}. (2.15)

2.1.3. The Structure ofNSn

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

6 ISRN Algebra

Theorem 2.8. S∂γ is an irreducible G-module, and NSn can be decomposed as

NSn =⊕

γ∈OSn−minimal

S∂γ . (2.16)

Moreover,NSn is a Gelfand model of Sn.

2.2. Polynomial Model for a Weyl Group of Type Bn

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

Let C2 = {1,−1} ⊂ C∗ be the subgroup of order two. The Weyl group Bn, of type Bn,

can be presented as the semidirect product

Bn = Cn2 ×s Sn, (2.17)

where Cn2 = C2 × · · · × C2 and the semidirect product is induced by the natural action of Sn on

Cn2 :

σ · (w1, . . . , wn) =(wσ(1), . . . , wσ(n)

) (σ ∈ Sn, (w1, . . . , wn) ∈ Cn

2

). (2.18)

The action of Sn on Mn induces a natural homomorphism from Bn on Aut(P) givenby

(w,σ)

(∑

α∈Mn

λαxα

)=∑

α∈Mn

λα(wx)σα (λα ∈ C) (2.19)

with

(wx)σα =n∏

i=1

(wixi)(σα)i . (2.20)

2.2.1. Bn-Minimal Orbit

Let On 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 γ∼Bnδ, if thereexists a bijection ϕ : N0 → N0 such that ϕ(k) and k have the same parity for all k ∈ N0,

Definition 2.10. An orbit γ will be called Bn-minimal if |γ | ≤ |δ| for all δ ∈ On such that γ∼Bnδ.

Proposition 2.11. An orbit γ is Bn-minimal if, and only if, for each α ∈ γ and each pair i, j ∈ N0 withthe same parity, one has |α−1(i)| ≥ |α−1(j)|with 0 ≤ i < j (|α−1(i)| being the cardinal of the set α−1(i)).


ISRN Algebra 7

2.2.2. The Space SΔγ

Let Δ be the Laplacian operator defined by Δ =∑n

i=1 ∂2i , where ∂i are the partial differential

operators mentioned above. For each γ ∈ On, let SΔγ be the subspace defined by

SΔγ ={P ∈ Sγ : Δ(P) = 0

}. (2.21)

2.2.3. The Structure ofNBn

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 =⊕

γ∈OBn −minimal

SΔγ . (2.22)

MoreoverNBn is a Gelfand model of Bn.


2.3. Polynomial Model for a Weyl Group of Type Dn

Let Dn be the Weyl group of type Dn naturally included in Bn. Using the previous notation,for α ∈ Mn the following sets are considered:

Eα = {i ∈ In : αi is even}, Oα = {i ∈ In : αi is odd}. (2.23)

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

2.3.1. Dn-Minimal Orbit

Definition 2.13. Two orbits γ and δ will be called Dn-equivalent, denoted by γ∼Dnδ, if thereexists a bijection ϕ : N0 → N0 such that

(1) ∀k ∈ N0, ϕ(k) and k have the same parity or ϕ(k) and k have different parities,

(2) δ = {ϕ ◦ α : α ∈ γ}.

Definition 2.14. An orbit γ will be called Dn-minimal if |γ | ≤ |δ| for all δ ∈ On such that γ∼Dnδ.

Proposition 2.15. Let γ be an orbit, and then the following statements are true

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

(a) given α ∈ γ and i < j ∈ N0 with the same parity, then |α−1(i)| ≥ |α−1(j)|,(b) |Eγ | ≤ |Oγ |.

8 ISRN Algebra

(2) Let π : N0 → N0 be the involution given by π(2i) = 2i + 1 and π(2i + 1) = 2i. The fol-lowing assertions are equivalent:

(a) γ and π ◦ γ are Dn-minimal orbits,(b) γ is Bn-minimal,(c) π ◦ γ is Bn-minimal.

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

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

(5) γ and π ◦ γ are Dn-minimal orbits if, and only if, |Eγ | = |Oγ |.


Proposition 2.16. Let n be odd, and then the following statements are true.

(1) If γ is Dn-minimal, then

NBn ∩ Sγ = NDn ∩ Sγ (2.24)

and NDn ∩ Sγ is a simple Dn-module.

(2) NDn is a Gelfand model for Dn.

(3) Every simple Bn-module remains simple when it is considered as a Dn-module by restric-tion.

(4) By consideringNBn as a Dn-module by restriction,NBn is isomorphic toNDn ⊕NDn .


Also in [5] it has been proved that if n is even, NDn is not a Gelfand model for a Weylgroup of type Dn. But it does happen that if M is a simple Dn-module, then NDn contains acopy of this, and the multiplicity of M inNDn is

(1) two, if M is isomorphic to NDn ∩ Sγ , γ being a Dn-minimal orbit such that γ /=π ◦ γand |Eγ | = |Oγ |; in this case, as before, π : N0 → N0 is the involution given byπ(2i) = 2i + 1 and π(2i + 1) = 2i,

(2) one, otherwise.

3. Gelfand Model for a Weyl Group of Type D2n

As before, let Mn = {α : In → N0} be the set of multi-index functions. Every α ∈ Mn has anassociated vector α ∈ N

n0 , which is obtained by reordering α as follows.

α = (αi1 , . . . , αin) such that αi1 ≥ · · · ≥ αin . (3.1)

Thus, there is defined an order relationship � in Mn given by for all α, β ∈ Mn, α � β if, andonly if, α = β or there exists s (1 ≤ s ≤ n) such that

α1 = β1, . . . , αs−1 = βs−1, αs < βs, (3.2)

ISRN Algebra 9

αi and βi being the coordinates of the vectors α and β, respectively. Notice that this is thelexicographic order for N

n0 .

Proposition 3.1. Let γ ∈ On and α, β ∈ Mn, and then α, β ∈ γ if, and only if, α = β.

Proof. Let α, β ∈ γ , and therefore there exists σ ∈ Sn such that β = σα, which implies βi = ασ−1(i)

with 1 ≤ i ≤ n. Thus, it is easy to see that α = β.On the other hand, let α, β ∈ Mn and α = β, say,

αi1 = βj1 , . . . , αin = βjn . (3.3)

Let σ ∈ Sn be given by

σ−1(ik) = jk (1 ≤ k ≤ n). (3.4)

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

γ � δ ⇐⇒ α � β(γ, δ ∈ On, α ∈ γ, β ∈ δ

). (3.5)

Since the vector α is independent of the choice α in γ , it will be denoted by γ .

Proposition 3.2. Let α ∈ Mn be defined by αi = i−1, and let γ be the orbit of α. Then, γ is the �-maxi-mum 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 δbe an orbit such that δ /= γ and γ � δ. Then, it should exist an s ∈ In satisfying

γi = δi, ∀i < s, γs < δs, (3.6)

that is,

δ1 = n − 1 · · · δs−1 = n − (s − 1), δs > n − s. (3.7)

Thus, it occurs that δs = n−(s−1), and from theminimality of δ every number less thann−(s−1)must appear at least twice, which is a contradiction. And therefore γ is themaximum.

From now on, for a finite set A, SA will denote the symmetric group of A, MA willdenote the set of multi-index functions with domain A,

MA = {α : A −→ N0}, (3.8)

10 ISRN Algebra

and ıwill denote the function in MA given by

ı(i) = 1, ∀i ∈ A. (3.9)

As in the case of Sn, there is a natural action from the symmetric group SA in the set ofmulti-index functions MA, defined by

σ · α = α ◦ σ−1 (σ ∈ SA, α ∈ MA). (3.10)

It is possible to extend the concept of SA-minimal orbit.For each α ∈ Mn, let us consider the sets

Eα = {i ∈ In : αi is even} Oα = {i ∈ In : αi is odd} (3.11)

as defined in the previous section. Then, it is clear that α can be determined from its res-trictions αE and αO to the sets Eα andOα, respectively. Observe that αE ∈ MEα and αO ∈ MOα .

Proposition 3.3. Let α ∈ Mn such that α is Bn-minimal, and then αE/2 is SEα -minimal and (αO −1)/2 is SOα -minimal.

Proof. It follows from Proposition 4 in [3] and the identities

∣∣∣∣(αE

2

)−1(i)∣∣∣∣ = |α(i)| (∀i ∈ |Eα|),

∣∣∣∣∣

(αO − 1

2

)−1(j)∣∣∣∣∣ =∣∣α(j)∣∣ (∀j ∈ |Oα|

).

(3.12)

Notation 1. Let K be the subset of Mn given by

K ={α ∈ Mn : α is Dn-minimal, |Eα| = |Oα| and

αE

2≺ αO − 1

2

}. (3.13)

It will be denoted by F the subset of the polynomial ring P:

F =

{∑

α

λαxα : λα = 0 if α ∈ K

}. (3.14)

Note that if n is odd, F is equal to P.

Proposition 3.4. F is a Dn-submodule of the polynomial ring P.

ISRN Algebra 11

Proof. It follows from the action of Dn given by

(w,σ)

(∑

α

λαxα

)=∑

α

λα(wx)σα =∑

α

± λαxσα (3.15)

and the fact that α ∈ K if, and only if, σα ∈ K, and it results that F is a Dn-module of P.

Proposition 3.5. F contains a Gelfand model for the Weyl group of type Dn.

Proof. It is sufficient to prove that F contains a submodule equivalent to the regular moduleDn. Effectively, let us consider the polynomial:

P(x1, . . . , xn) =

∏ni=2

[xi(xi + i − 1)

∏i−2j=1(x2i − j2)]

2n−1[(n − 1)!]2∏n

i=2∏i−2

j=1(i − 1 + j

)(i − 1 − j

) . (3.16)

Thus P is the interpolating polynomial of the orbit of the regular vector v = (0, 1, . . . , n − 1),which satisfies

P(v) = 1, P(τv) = 0 (∀τ ∈ Dn, τ /= e). (3.17)

It will be proved that P belongs to F. Let λαxα be not a null term of P such that it isDn-minimal, |Eα| = |Oα|, and αE/2 ≺ (αO−1)/2. As P was defined in (3.16), it is easy to determinethat α1 = 0 and αj > 0 for 1 < j ≤ n. Then, (αE/2)1 = 0, and as αE/2 is S|Eα|-minimal, it isobtained that

αE

2= (|Eα| − 1, |Eα| − 2, . . . , 0), (3.18)

which by Proposition 3.2 is maximal, which is a contradiction.Since F is a Dn-module, F contains the module generated by the orbit of P , which is

isomorphic to the regular module. Hence, F contains a Gelfand model.

Notation 2. Let T be a subset of the polynomial ring P and G a finite subgroup of GLn(C); wewill denote by T0 the subset of T defined by

T0 ={p ∈ T : D

(p)= 0, ∀D ∈ I−

G

}, (3.19)

I−G being the set of differential operators invariant in the algebra ofWeyl as it has been defined

in (2.9).

Proposition 3.6. Let G be a finite subgroup of GLn(C) and T a G-module of the polynomial ring Psuch that T contains a model of G, and then T0 also contains a model of G.

Proof. Let S ⊂ T be a simple G-module and suppose that S/⊆T0. Then, there exists D ∈ I−G

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.

12 ISRN Algebra

As D is an operator of the Weyl algebra A with negative degree, the procedure is finite, thatis to say, there existsm ∈ N such that Dm(S) ⊂ T0 and Dm(S) � S.

Remark 3.7. An immediate consequence of this proposition is that the Dn-module

F0 ={f ∈ F : D

(f)= 0, ∀D ∈ I−

Dn

}(3.20)

contains a Gelfand model because F is a Dn-module containing a Gelfand model.

Remark 3.8. Notice that if n is odd, then F0 = NDn ; instead, if n is even,

F0 ⊆ F ∩NDn =⊕

γ∈OBn -minimal

|Eγ |<|Oγ |

SΔγ +

⊕

γ∈OBn -minimal

|Eγ |=|Oγ |

SΔγ

(3.21)

that is

F0 ⊆⊕

γ∈OBn -minimal

|Eγ |<|Oγ |

SΔγ +

⊕(γO−1)/2≺γE/2

SΔγ +

⊕

γE/2=(γO−1)/2

SΔγ .

(3.22)

Using the result established in item 4 of Proposition 2.16 for decomposing NBn , itfollows that

NBn =⊕

γ∈OBn -minimal

SΔγ =

⊕

γ∈OBn -minimal

|Eγ |<|Oγ |

SΔγ +

⊕

γ∈OBn -minimal

|Eγ |=|Oγ |

SΔγ +

⊕

γ∈OBn -minimal

|Eγ |>|Oγ |

SΔγ .

(3.23)

Moreover if |Eγ | = |Oγ |

⊕

γ∈OBn -minimal

|Eγ |=|Oγ |

SΔγ =

⊕

γE/2≺ (γO−1)/2SΔγ +

⊕

γE/2=(γO−1)/2

SΔγ +

⊕

(γO−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

2dim

⎛⎜⎜⎜⎝⊕

γ∈OBn−minimal

|Eγ |<|Oγ |

SΔγ

⎞⎟⎟⎟⎠ + 2dim

⎛⎜⎝

⊕

(γO−1)/2≺γE/2SΔγ

⎞⎟⎠ + dim

⎛⎜⎝

⊕

γE/2=(γO−1)/2

SΔγ

⎞⎟⎠. (3.25)

ISRN Algebra 13

Proof. It results from considering the identity

dim(SΔγ

)= dim(SΔπ◦γ

)(3.26)

for each γ ∈ On. This identity occurs from the fact that SΔπ and SΔ

π◦γ are isomorphic as Dn-modules, see [5].

Let G be a group; from now on we will be denote by

Inv(G) ={g ∈ G : g2 = e

}(3.27)

the set of involutions of the group G.

Lemma 3.10. Let G be a Coxeter group and M a Gelfand model for G. Then,

dim(M) = | Inv (G)|. (3.28)

Proof. It is a consequence from the Frobenius-Schur indicator and the fact that the represen-tations 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 betweenthe number of involutions of Bn and the number of involutions of Dn will be given. This willbe used in the next theorem.

Lemma 3.11. If n is even (n = 2k), then

2| Inv (Dn)| − | Inv (Bn)| =(2k)!k!

. (3.29)

Proof. If σ = (w,π) ∈ Bn, with w ∈ Cn2 , and π ∈ Sn is an involution, then the cyclic structure

of σ looks like

(±i1,±j1

)· · ·(±ir ,±jr

)(±k1) · · · (±ks), (3.30)

where In = {i1, . . . ir , j1, . . . , jr , k1, . . . ,ks}, π = (i1, j1) · · · (ir , jr)(k1) · · · (ks) is the decompositionof π as product of disjoint cycles, and wil = wjl for 1 ≤ l ≤ r. Thus, the number of involutionsof Bn is

|Inv(Bn)| =k∑

r=0

∏r−1j=0

(n−2j2

)

r!2r2n−2r (3.31)

If r < k, half of the elements belong to Dn and the other half to Bn-Dn, and therefore

|Inv(Dn)| =12

k−1∑

r=0

∏r−1j=0

(n−2j2

)

r!2r2n−2r +

∏k−1j=0

(n−2j2

)

k!2k2n−2k. (3.32)

14 ISRN Algebra

Then,

2|Inv(Dn)| − |Inv(Bn)| =

∏k−1j=0

(n−2j2

)

k!2k2n−2k =

(2k)!k!

. (3.33)

Theorem 3.12. The G-module F0 is a Gelfand model for the group Dn.

Proof. As it has been mentioned above, when n is odd, F0 is equal to NDn , and in [5] it hasbeen proved thatNDn is a Gelfand model for the group Dn.

When n is even, from the factF0 contains a Gelfandmodel, only it is necessary to provethat dim(F0) ≤ |Inv(Dn)|. From identity (3.22), it results that

dim(F0)≤ dim

⎛⎜⎜⎜⎝⊕

γ∈OBn−minimal

|Eγ |<|Oγ |

SΔγ +

⊕

(γO−1)/2≺γE/2SΔγ +

⊕

γE/2=(γO−1)/2

SΔγ

⎞⎟⎟⎟⎠. (3.34)

By Lemma 3.10, it follows that the dimension of the model NBn is equal to the number ofinvolutions of the group Bn, and thus by the Lemma 3.9 it results that

|Inv(Bn)| = 2dim

⎛⎜⎜⎝⊕

γ∈OBn−minimal

|Eγ |<|Oγ |

SΔγ

⎞⎟⎟⎠ + 2dim

⎛⎜⎝

⊕

γE/2≺ (γO−1)/2SΔγ

⎞⎟⎠ + dim

⎛⎜⎝

⊕

γE/2=(γO−1)/2

SΔγ

⎞⎟⎠,

|Inv(Bn)| + dim

⎛⎜⎝

⊕

γE/2=(γO−1)/2

SΔγ

⎞⎟⎠ = 2dim

⎛⎜⎜⎜⎜⎜⎜⎝

⊕

γ∈OBn−minimal

|Eγ |<|Oγ |

SΔγ +

⊕

(γO−1)/2�γE/2SΔγ

⎞⎟⎟⎟⎟⎟⎟⎠

,

12

⎡⎢⎣|Inv(Bn)| + dim

⎛⎜⎝

⊕

γE/2=(γO−1)/2

SΔγ

⎞⎟⎠

⎤⎥⎦ = dim

⎛⎜⎜⎜⎝⊕

v∈OBn−minimal

|Eγ |<|Oγ |

SΔγ +

⊕


⎞⎟⎟⎟⎠,

dim

⎛⎜⎝

⊕

γE/2=(γO−1)/2

SΔγ

⎞⎟⎠ =

∑

γE/2=(γO−1)/2

dim(SΔγ

)=∑

χ∈Sn

(2nn

)χ2(1) =

(2n)!n!

(3.35)

ISRN Algebra 15

and then

12

[|Inv(Bn)| +

(2n)!n!

]= dim

⎛⎜⎜⎜⎝⊕

γ∈OBn−minimal

|Eγ |<|Oγ |

SΔγ +

⊕


⎞⎟⎟⎟⎠. (3.36)

On the other hand, from identity established in Lemma 3.11, it results that

|Inv(Dn)| =12

[| Inv (Bn)| +

(2n)!n!

], (3.37)

and using identities (2.2) and (3.36), it is obtained that

dim(F0)≤ | Inv (Dn)|. (3.38)

Therefore, it has been proved that F0 is a Gelfand model for Dn.

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16 ISRN Algebra

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A Gelfand Model for Weyl Groups of Type D2n