Medical, Pharma, Engineering, Science, Technology and Business

**H. Refaghat ^{a}^{,b} and M. Shahryari^{b}**

^{a}Department of Mathematics, Islamic Azad University (Tabriz Branch), Tabriz, Iran **E-mail: **[email protected]

^{b}Department of Pure Mathematics, Faculty of Mathematical Sciences, University of Tabriz, Tabriz, Iran **E-mail:** [email protected]

**Received date:** February 12, 2010; **Revised date:** April 26, 2010

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Journal of Generalized Lie Theory and Applications

In this article, we give a formula for the number of Gelfand-Zetlin patterns, using dimensions of the symmetry classes of tensors.

Suppose is a sequence of decreasing non-negative integers. A Gelfand- Zetlin pattern based on is an array of integers:

such that for all *i*,

We denote the set of all Gelfand-Zetlin patterns based on by. The set has an important role in the representation theory of general (equivalently, special) and orthogonal linear Lie algebras. For example, let be a dominant weight for the Lie algebra and suppose is the corresponding irreducible representation with the highest weight . In [5], Gelfand and Zetlin have proved that the set can be viewed as a basis for . For the matrix representations of the elements of the Chevalley basis of with respect to , see [1] or [3]. The set is also important from the point of view of *branching rules*. Branching rules are descriptions of the reduction of irreducible representations upon restriction to a subalgebra (subgroup). The rst branching rule discovered is possibly the well-known branching rule of representations of the symmetric group *S _{m}* in the early twenties. Since then, it was an exciting job to discover other kinds of branching rules for finite groups, Lie groups, and Lie algebras. We can employ the set to describe the branching rule for type . Suppose we like to restrict the representation to . For any Gelfand-Zetlin pattern , let

The aim of this article is to compute the number of elements of . Although, one can use the well-known dimension formula of Weyl, but our formula is an alternative one, which uses the irreducible characters of the symmetric group. To give a survey of our main result, suppose

We consider a partition of *m* with the parts:

Let be the irreducible character of the symmetric group *S _{m}* corresponding to , (for standard terms about partitions and characters of

where [ , ] is the inner product of characters in *S _{m}*. In the other words, we will see that

In this section, we are going to review the notion of a symmetry class of tensors. The reader interested in the subject can find a detailed introduction in [9] or [10].

Let *V* be an *n*-dimensional complex inner product space and let *G* be a subgroup of the full symmetric group *S _{m}*. Let denote the tensor product of

by

Suppose that is a complex irreducible character of *G* and define the *symmetrizer*:

The *symmetry class of tensors* associated with *G* and is the image of and it is denoted by . So

For example, if we let *G* = *S _{m}* and , the alternating character, then we get , the

Several monographs and articles have been published on symmetry classes of tensors during the last decades, see for example [9,10].

Let *v _{1}*, ...,

Let {*e** _{1}*, ... ,

for any and . Given two elements , we say that if and only if *α* and *β *lie in the same orbit. Suppose that is a set of representatives of orbits of this action and let denote the stabilizer subgroup of *α*. Define

where [ , ] denotes the inner product of characters (see [7]). It is well known that , if and only if , see for example [10]. Suppose . For any , we have the cyclic subspace:

It is proved that we have the direct sum decomposition:

see [10] for a proof. It is also proved that

and in particular, if is linear then and so the set:

is an orthogonal basis of . Also in the case of linear character , we have . In the general case, let and suppose

is a basis of with . Let

Then, we define . It is clear that

and the set:

is a basis of . Finally, we remember a formula for dimension of symmetry classes. We have

where denotes the number of disjoint cycles (including cycles of length one) in cycle decomposition of .

In this section, we define a Lie module structure on , so let *L* be a complex Lie algebra and suppose that *V* is an *L*-module. For any , define

by

We know that and so is invariant under *D*(*x*). Suppose

where the down arrow denotes restriction.

**Definition 3.1.** Define an action of Lie algebra *L* on by

Then, becomes an *L*-module. In what follows, we will assume that and , the standard module for *L*. Then, becomes an *L*-module. In [8], the irreducible constituents of are determined. In this section, we give a summery of result of [8]. We also assume that *G* = *S _{m}* and , the irreducible character of

A Cartan subalgebra for *L* is

For any , define a linear functional:

by

where , so we have

and hence is a basis for *H*.*

Now let be the fundamental weights corresponding to *H*. It is easy to see that for any *k*:

Let . We define a composition of *m* by , where *m _{i}* is the multiplicity of

So we have

Also we can see that

It is easy to prove that , if and only if . So, for any , we intro- duce a partition , which is just the multiplicity composition with a descending arrangement of entries. In fact, any partition of *m*, with height at most *n*, is of the form , where . For any and , we have

In other words, we have

So the set of weights of is . We also can see by an easy argument that the weight is dominant, if and only if , i.e. is a partition. For two dominant weights and , it is routine to check that appears in as a weight, if and only if majorizes , (for definition of the majorization, see [11]). We are now ready to compute irreducible constituents of . Although, the following theorem is proved in [8] in a more general framework, we prove it here again because what we need is only this special case.

**Theorem 3.2.** *We have*

*where is any element with the property* .

**Proof.** First of all, note that

where denotes the set of all increasing sequences in and denotes majorization. Suppose that is the set of all dominant weights of , ordered in such a way that implies (we say iff is a sum of positive roots). As we saw above, for any *i*, there is such that . Suppose that *r _{i}* equals number of such that . Let

Now we have

so we must show that and for . First, note that if has the property , then and hence we have

Note that *K*_{*,*} denote the well-known Kostka numbers, see [11] or [4] for definition. Now, we compute *c*_{2}; we have . As in the above case, we easily see that , where corresponds to . It is proved that (see [4])

Hence,

By a similar argument, we see that for other "*i*"s and the result follows.

Note that this theorem affords a new method of constructing of all irreducible - modules, namely, let be any integral dominant weight of . As in [10], we have

where is dened as follows: let

be the corresponding representation of with . We introduce the partial symmetrizer by

Now, is precisely the image of . So we have

for all . One of the most important consequences of this construction is the following dimension formula, which is different from Weyl's one.

**Corollary 3.3.** *Let be an irreducible -module with the highest weight and dene the corresponding number m and the partition as in Section 1. Then,*

We are ready now to give the interesting relation between dimension of the symmetry classes of tensors and the number of Gelfand-Zetlin patterns based on . Note that we have the following relations between the weight and the partition :

**Main theorem.** *The number of Gelfand-Zetlin patterns based on is equal to the inner product . Equivalently, we have*

**Remark 4.1.** In [12], the inner product is expressed in term of Kostka numbers. Suppose that is a partition of *m* with distinct parts . Suppose that the multiplicity of *b _{i}* in

and let for . Then the multiplicity of in is equal to

where is the Kostka number.

**Remark 4.2.** Note that we can *normalize* in such a way that we have . To do this, let . Define

Although in general and are non-isomorphic representations, it is clear that . For , we have

and also the corresponding partition has the parts:

Hence, we have also the following normalized formula for the number of Gelfand-Zetlin patterns:

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