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İstanbul Teknik Üniversitesi / Fen Bilimleri Enstitüsü / Fizik Anabilim Dalı

# Afin lie cebirlerinin karakterlerinde permütasyon ağırlık fonksiyonelleri

## Permutation weights in affine lie algebra characters

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##### Özet:

Sonlu Lie cebirleri için çok katlılıklar ve sonsuz cebirler olan Afin Lie cebirleri içinstring fonksiyonları, permütasyon agırlık fonksiyonelleri kullanılarak hesaplanmıstır.Permütasyon agırlık fonksiyonellerini olusturmak için kullanılan temel agırlıkfonksiyonelleri de ayrıca incelenerek, Weyl yörünge elamanları ortak formlarda ifadeedilmis ve genel olarak Weyl grup elemanları için tanımlanmıs ?isaret? tanımı dabunlar cinsinden yeniden tanımlanmıstır. Klasik olarak Weyl grubu kullanılarakhesaplanan string fonksiyonlarında her bir mertebeye nasıl ve hangi çokluçarpımlardan katkı gelecegini bilemememize karsın bu yeni yöntemde, her bir derinligeait permütasyon agırlık fonksiyonellerinin, string fonksiyonlarının o derinliktekimertebesine yaptıgı, tam katkı açıkça gösterilmistir. Çok katlılık hesabı için 4A ve 4 Bsonlu Lie cebirleri, string fonksiyonları hesabı için (1)4A ve (1)4 B Afin Lie cebirleri örnekolarak seçilmistir. (1)4A 'in bir indirgenemez temsili için sekizinci mertebeye kadar,(1)4 B 'in indirgenemez bir temsili için ise yedinci mertebeye kadar string fonksiyonlarıhesaplanmıstır.Anahtar Kelimeler: Permütasyon agırlık fonksiyonelleri, Afin Lie Cebirleri, Çok katlılık,String fonksiyonları

##### Summary:

Using the classical way, Weyl group summation, it is very hard to calculate character for higher rank finite Lie algebras or string functions for Affine Lie algebras which is a member of infinite dimensional Lie algebras. With this hardness, also in Affine Lie algebras we could?nt know which Weyl group elements contribute to which degree of string functions. Because of this we calculate the characters and string functions with a new approach which is named ?permutation weights?. Before constructing permutation weights, firstly we must define fundamental weights as below.By using this definitions, we can define permutation weights as below.These are elements of a set which we show with . The elements of can be written with permutations of common forms. Because of this common forms we call this set, permutation weight set. To calculate characters we need Weyl orbits and signatures of weights. Each of the Weyl orbits could be written as a direct sum of and shown like this:If we show any permutation weight as below,with the normalization , the signature of this weight is defined as:Now we can define the left hand side of the character equation by using the definition,Character equation is defined with this formula above. The right hand side of the character formula could be defined after defining the character of a Weyl orbit like this:With this definition the right hand side is:By using these definitions firstly we calculate the character of irreducible representation of finite Lie algebra and secondly the character of irreducible representation of finite Lie algebra . After this we find the multiplicities of these representations. In this calculations we show that how the sets of make our work easy. In the other chapters we deal with Affine Lie algebras and we calculate their characters. We show how the Weyl character formula turns to Weyl-Kac character formula and the multiplicities turns to string functions in Affine Lie algebras. Also the Weyl orbits of Affine Lie algebras could be written in some comman forms and we show that in this forms there is finite algebras? weights. In addition we show the conditions to become a member of any orbit in Affine Lie algebras and we define a Lemma for this. The transform of multiplicities of finite Lie algebras to string functions of Affine Lie algebras can be shown as below.Because of this definition the right hand side of the character formula turns to this:For calculating the string functions we also need signatures of Affine and .For the signature are defined asFor the signatures are defined asFinally we show that how the depth of permutation weights contribute to string functions? degree. We need them to calculate a string function at any degree we want. For this we give an example from Affine Lie algebra and find string functions for irreducible representation. When we take the permutation weights up to depth 5 and want to calculate string functions up to 8. degree we see that we find only the first five terms integer, but when we take the permutation weights up to depth 8 and calculate the string functions up to 8. degree we see that all terms are integer. These three string functions are seen as below.And another example is for Affine . We calculate string functions up to 7.degree for irreducible representation as below.