A Gelfand model for a finite group

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 [

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

Later on, this type of models was called an

Baddeley’s result implies the existence of involution models for classic Weyl groups, with the exception of the group of type

More recently, in [

A reflection group

Chevalley [

More recently, Araujo and Aguado in [

For the case of Weyl groups of type

In this paper the construction of a Gelfand model for a Weyl group of type

The notation introduced in this section will be used in the remaining of this paper.

Let

For each

Let

It is known that each

Let

Let

Let

Notice that

Below, some properties of

See [

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

Let

See [

In the following sections it will be presented a characterization of the polynomial model for the classical Weyl groups of type

Let

The symmetric group

Let

Two orbits

An orbit

An orbit

See [

For each

Let

Below the main theorem regarding

The Gelfand model for a Weyl group of type

Let

The action of

Let

Two orbits,

An orbit

An orbit

See [

Let

Below the main theorem regarding

See [

Let

It is easy to check that the cardinals

Two orbits

An orbit

Let

given

Let

There are at most two

If

See [

Let

If

Every simple

By considering

See [

Also in [

two, if

one, otherwise.

As before, let

Let

Let

On the other hand, let

From this proposition it is clear that

Let

From the previous considerations it is clear that

Now it will be proved that

Thus, it occurs that

From now on, for a finite set

For each

Let

It follows from Proposition 4 in [

Let

It follows from the action of

It is sufficient to prove that

It will be proved that

Since

Let

Let

Let

An immediate consequence of this proposition is that the

Notice that if

Using the result established in item 4 of Proposition

As a consequence of this decomposition the next lemma follows.

The dimension of

It results from considering the identity

Let

Let

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

If

If

The

As it has been mentioned above, when

When