arXiv subjects

Hasan R. Karadayi

Publications and source records attributed to Hasan R. Karadayi.

10 recordsLinked to original sources

arXiv2015

Explicit Calculations of Tensor Product Coefficients for E_7

We propose a new method to calculate coupling coefficients of E_7 tensor products. Our method is based on explicit use of E_7 characters in the definition of a tensor product. When applying Weyl character formula for E_7 Lie algebra, one needs to make sums over 2903040 elements of E_7 Weyl group. To implement such enormous sums, we show we have a way which makes their calculations possible. This will be accomplished by decomposing an E_7 character into 72 participating A_7 characters.

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arXiv2013

A note on a theorem of Bourbaki

We have recently show that Poincare series of Hyperbolic Lie algebras have the form of a ratio between Poincare series of a chosen finite Lie algebra and a polynomial of finite degree. By the aid of some properly chosen examples, we now give some remarks on a related theorem of Bourbaki.

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arXiv2012

A Brief Note On The Definition of Signature

It is known that signature of a Weyl group element is defined in terms of the number of its simple Weyl reflections. Actual calculations hence are not always possible especially for Weyl groups with higher order like $E_8$ Weyl group. By extending the concept from signature of a Weyl reflection to signature of a weight, we show that signature of a weight is defined without referring to Weyl reflections, Though both have the same result, the signature of a weight can be calculated for any Lie algebra.

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arXiv2010

On Poincare Polynomials of Hyperbolic Lie Algebras

We have general frameworks to obtain Poincare polynomials for Finite and also Affine types of Kac-Moody Lie algebras. Very little is known however beyond Affine ones, though we have a constructive theorem which can be applied both for finite and infinite cases. One can conclusively said that theorem gives the Poincare polynomial P(G) of a Kac-Moody Lie algebra G in the product form P(G)=P(g) R where g is a precisely chosen sub-algebra of G and R is a rational function. Not in the way which theorem says but, at least for 48 hyperbolic Lie algebras considered in this work, we have shown that there is another way of choosing a sub-algebra in such a way that R appears to be the inverse of a finite polynomial. It is clear that a rational function or its inverse can not be expressed in the form of a finite polynomial. Our method is based on numerical calculations and results are given for each and every one of 48 Hyperbolic Lie algebras. In an illustrative example however, we will give how above-mentioned theorem gives us rational functions in which case we find a finite polynomial for which theorem fails to obtain.

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arXiv2005

On Poincare Polinomials of Hyperbolic Lie Algebras

Poincare polinomials of hyperbolic Lie algebras, which are given by $HA_2$ and $HA_3$ in the Kac's notation, are calculated explicitly. The results show that there is a significant form for hyperbolic Poincare polinomials. Their explicit forms tend to be seen as the ratio of a properly chosen finite Poincare polinomial and a polinomial of finite degree. To this end, by choosing the Poincare polinomials of $D_4$ and $D_5$ Lie algebras, we show that these polinomials come out to be of order 11 and 19 respectively for $HA_2$ and $HA_3$.

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arXiv2003

Special Weights and Roots for Finite Lie Algebras

With the introduction of special roots, we show the existence of some special weights with quite interesting properties for finite Lie algebras. We propose and discuss two statements which lead us to an explicit construction of these special weights and roots. It is seen that special weights provides us an alternative way of expressing the whole action of Weyl groups.

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arXiv1999

Permutation Weights for Affine Lie Algebras

We show that permutation weights, which are previously introduced for finite Lie algebras, can be appropriately defined also for affine Lie algebras. This allows us to classify all the weights of an affine Weyl orbit explicitly. Let $Λ$ be a dominant weight of an affine Lie algebra $ G_N^{(r)} $ for r=1,2,3. At each and every order M of weight depths, the set $\wp_M(Λ)$ of permutation weights is formed out of a finite number of dominant weights of the finite Lie algebra $G_N$. In case of $A_N^{(1)}$ algebras, we give the rules to determine the elements of a $\wp_M(Λ)$ completely. As being a positive test of our proposal, we consider the problem of calculating weight multiplicities for affine Lie algebras and hence our discussions are based on explicit computations of Weyl-Kac character formula. It is known that weight multiplicities are provided by string functions which are defined to be formal power series $\sum_{M=0}^\infty C(M) q^M$ where the order M specifies the depth of weights contributing to C(M). In the conventional calculational schemes which are based on Kac-Peterson form of affine Weyl groups, Weyl-Kac formula includes a sum over a part of the whole root lattice and hence it is seen that the roots of the same length contribute in general to C(M) for several values of M. On the contrary, we will determine, for any fixed value of M, the complete set of weights having depth M and contributing only to C(M). For applications of Weyl-Kac formula, one must also know the signatures which correspond to weights within the Weyl orbits of strictly dominant weights. This is given by the aid of a properly defined index. Another emphasis is that the way of discussion adopted here gives us a possibility for extensions to other infinite dimensional Lie algebras beyond affine Lie algebras.

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arXiv1998

Summing over the Weyl Groups of E_7 and E_8

It is known that summations over Weyl groups of Lie algebras is a problem which enters in many areas of physics as well as in mathematics. For this, a method which we would like to call {\bf permutation weights} has been previously proposed for pairs $(G_N, A_{N-1})$ of Lie algebras. It is now extended for $(E_7, A_7)$ and also $(E_8, A_8)$. It is clear that these are the most non-trivial ones and hence deserve studying separately. In order to obtain the results of these summations in practice, it is shown that some simplifications occur in the method which is previously proposed for pairs $(A_N, A_{N-1})$ in an unpublished work.

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arXiv1998

Fundamental Weights, Permutation Weights and Weyl Character Formula

For a finite Lie algebra $G_N$ of rank N, the Weyl orbits $W(Λ^{++})$ of strictly dominant weights $Λ^{++}$ contain $dimW(G_N)$ number of weights where $dimW(G_N)$ is the dimension of its Weyl group $W(G_N)$. For any $W(Λ^{++})$, there is a very peculiar subset $\wp(Λ^{++})$ for which we always have $$ dim\wp(Λ^{++})=dimW(G_N)/dimW(A_{N-1}) . $$ For any dominant weight $ Λ^+ $, the elements of $\wp(Λ^+)$ are called {\bf Permutation Weights}. It is shown that there is a one-to-one correspondence between elements of $\wp(Λ^{++})$ and $\wp(ρ)$ where $ρ$ is the Weyl vector of $G_N$. The concept of signature factor which enters in Weyl character formula can be relaxed in such a way that signatures are preserved under this one-to-one correspondence in the sense that corresponding permutation weights have the same signature. Once the permutation weights and their signatures are specified for a dominant $Λ^+$, calculation of the character $ChR(Λ^+)$ for irreducible representation $R(Λ^+)$ will then be provided by $A_N$ multiplicity rules governing generalized Schur functions. The main idea is again to express everything in terms of the so-called {\bf Fundamental Weights} with which we obtain a quite relevant specialization in applications of Weyl character formula.

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arXiv1998

A_N Multiplicity Rules And Schur Functions

We show that a specialization in Weyl character formula can be carried out in such a way that its right-hand side becomes simply a Schur Function. For this, we need the use of fundamental weights. In the generic definition, an Elementary Schur Function $S_Q(x_1,x_2,..,x_Q)$ of degree Q is known to be defined by some polynomial of Q indeterminates $ x_1,x_2,..,x_Q $. It is also known that definition of Elementary Schur Functions can be generalized in such a way that for any partition $(Q_k)$ of weight Q and length k one has a Generalized Schur Function $S_{(Q_k)}(x_1,x_2,..,x_Q)$. When they are considered for $A_{N-1}$ Lie algebras, a kind of degeneration occurs for these generic definitions. This is mainly due to the fact that, for an $A_{N-1}$ Lie algebra, only a finite number of indeterminates, namely (N-1), can be independent. This leads us to define {\bf Degenerated Schur Functions} by taking, for $Q > N-1$, all the indeterminates $x_Q$ to be non-linearly dependent on first (N-1) indeterminates $x_1,x_2,..,x_{N-1}$. With this in mind, we show that for each and every dominant weight of $A_{N-1}$ we always have a (Degenerated) Schur Function which provides the right-hand side of Weyl character formula. Generalized Schur Functions are known to be expressed by determinants of some matrices of Elementary Schur Functions. We would like to call these expressions {\bf multiplicity rules}. This is mainly due to the fact that, to calculate weight multiplicities, these rules give us an efficient method which works equally well no matter how big is the rank of algebras or the dimensions of representations.

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