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Journal of Generalized Lie Theory and Applications
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A formula for the number of Gelfand-Zetlin patterns

H. Refaghata,b and M. Shahryarib

aDepartment of Mathematics, Islamic Azad University (Tabriz Branch), Tabriz, Iran E-mail: [email protected]

bDepartment 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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Abstract

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

1 Introduction

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

image

such that for all i,

image

We denote the set of all Gelfand-Zetlin patterns based on imagebyimage. The set image has an important role in the representation theory of general (equivalently, special) and orthogonal linear Lie algebras. For example, let image be a dominant weight for the Lie algebra image and suppose image is the corresponding irreducible representation with the highest weight image. In [5], Gelfand and Zetlin have proved that the set image can be viewed as a basis for image. For the matrix representations of the elements of the Chevalley basis of image with respect to image, see [1] or [3]. The set image 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 Sm 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 image to describe the branching rule for type image. Suppose we like to restrict the representation image to image. For any Gelfand-Zetlin pattern image, let Mi be the i-th row of M. Then, the restriction of image to imageis equal to the following direct sum decomposition:

image

The aim of this article is to compute the number of elements of image. 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

image

We consider a partition image of m with the parts:

image

Let image be the irreducible character of the symmetric group Sm corresponding to image, (for standard terms about partitions and characters of Sm, see [11]). Also, for any permutation image, let image be the number of disjoint cycles in the cycle decomposition of image. t is clear that the function image is a character of Sm, (for its irreducible constituents, see [12]). Our main result will be

image

where [ , ] is the inner product of characters in Sm. In the other words, we will see that

image

2 Symmetry classes of tensors

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 Sm. Let image denote the tensor product of m copies of V and for anyimage, define the permutation operator:

image

by

image

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

image

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

image

For example, if we let G = Sm and image, the alternating character, then we get image, the m-th Grassman space over V and if G = Sm and image, the principal character, then we obtain V (m), the m-th symmetric power of V , as symmetry classes of tensors.

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

Let v1, ...,vm be arbitrary vectors in V and define the decomposable symmetrized tensor:

image

Let {e1, ... , en} be a basis of V , and suppose that image is the set of all m-tuples of integers image with image. For image, we use the notation image for decomposable symmetrized tensor image. It is clear that image is generated by all image. We define an action of G on image by

image

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

image

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

image

It is proved that we have the direct sum decomposition:

image

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

image

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

image

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

image

is a basis of image with image. Let

image

Then, we define image. It is clear that

image

and the set:

image

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

image

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

3 Symmetry classes as image-modules

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

image

by

image

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

image

where the down arrow denotes restriction.

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

image

Then, image becomes an L-module. In what follows, we will assume that image and image, the standard module for L. Then, imagebecomes an L-module. In [8], the irreducible constituents of image are determined. In this section, we give a summery of result of [8]. We also assume that G = Sm and image, the irreducible character of Sm corresponding a partition image. For simplicity, we denote the symmetry class of tensors byimage. To describe the irreducible constituents of image, it is necessary to introduce some notations.

A Cartan subalgebra for L is

image

For any image, define a linear functional:

image

by

image

where image, so we have

image

and hence image is a basis for H*.

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

image

Let image. We define a composition of m by image, where mi is the multiplicity of i in α. Suppose

image

So we have

image

Also we can see that

image

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

image

In other words, we have

image

So the set of weights of image is image. We also can see by an easy argument that the weight image is dominant, if and only if image, i.e. image is a partition. For two dominant weights image and image, it is routine to check that image appears in image as a weight, if and only if image majorizes image, (for definition of the majorization, see [11]). We are now ready to compute irreducible constituents of image. 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

image

where image is any element with the property image.

Proof. First of all, note that

image

where image denotes the set of all increasing sequences in image and image denotes majorization. Suppose that image is the set of all dominant weights of image, ordered in such a way that image implies image (we say image iff image is a sum of positive roots). As we saw above, for any i, there is image such that image. Suppose that ri equals number of image such that image. Let mij be the multiplicity of image in image. Define a sequence of integers as follows:

image

Now we have

image

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

image

Note that K*,* denote the well-known Kostka numbers, see [11] or [4] for definition. Now, we compute c2; we have image. As in the above case, we easily see that imageimage, where image corresponds to image. It is proved that (see [4])

image

Hence,

image

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

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

image

where image is de ned as follows: let

image

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

image

Now, image is precisely the image of image. So we have

image

for all image. 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 imagebe an irreducible image-module with the highest weight imageand de ne the corresponding number m and the partition image as in Section 1. Then,

image

4 The number of the Gelfand-Zetlin patterns

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

image

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

image

Remark 4.1. In [12], the inner product image is expressed in term of Kostka numbers. Suppose that image is a partition of m with distinct parts image. Suppose that the multiplicity of bi in ρ is ri. If image, define

image

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

image

where image is the Kostka number.

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

image

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

image

and also the corresponding partition image has the parts:

image

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

image

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