arXiv subjects

J. Katriel

Publications and source records attributed to J. Katriel.

4 recordsLinked to original sources

arXiv1995

Recursively minimally-deformed oscillators

A recursive deformation of the boson commutation relation is introduced. Each step consists of a minimal deformation of a commutator $[a,\ad]=f_k(\cdots;\no)$ into $[a,\ad]_{q_{k+1}}=f_k(\cdots;\no)$, where $\cdots$ stands for the set of deformation parameters that $f_k$ depends on, followed by a transformation into the commutator $[a,\ad]=f_{k+1}(\cdots,\, q_{k+1};\no)$ to which the deformed commutator is equivalent within the Fock space. Starting from the harmonic oscillator commutation relation $[a,\ad]=1$ we obtain the Arik-Coon and the Macfarlane-Biedenharn oscillators at the first and second steps, respectively, followed by a sequence of multiparameter generalizations. Several other types of deformed commutation relations related to the treatment of integrable models and to parastatistics are also obtained. The ``generic'' form consists of a linear combination of exponentials of the number operator, and the various recursive families can be classified according to the number of free linear parameters involved, that depends on the form of the initial commutator.

q-alg↗

arXiv1995

On the Fundumental Invariant of the Hecke Algebra $H_{n}(q)$

The fundumental invariant of the Hecke algebra $H_{n}(q)$ is the $q$-deformed class-sum of transpositions of the symmetric group $S_{n}$. Irreducible representations of $H_{n}(q)$, for generic $q$, are shown to be completely characterized by the corresponding eigenvalues of $C_{n}$ alone. For $S_{n}$ more and more invariants are necessary as $n$ inereases. It is pointed out that the $q$-deformed classical quadratic Casimir of $SU(N)$ plays an analogous role. It is indicated why and how this should be a general phenomenon associated with $q$-deformation of classical algebras. Apart from this remarkable conceptual aspect $C_{n}$ can provide powerful and elegant techniques for computations. This is illustrated by using the sequence $C_{2}$, $C_{3}, \cdots,\; C_{n}$ to compute the characters of $H_{n}(q)$.

q-alg↗

arXiv1995

The character table of the Hecke algebra $H_n(q)$ in terms of traces of products of Murphy operators

The traces of the Murphy operators of the Hecke algebra $H_n(q)$, and of products of sets of Murphy operators with non-consecutive indices, can be evaluated by a straightforward recursive procedure. These traces are shown to determine all the reduced traces in this algebra, which, in turn, determine all other traces. To illustrate the procedure we obtain the set of reduced traces for $H_7(q)$ - the lowest order Hecke algebra whose character table has not hitherto been reported. This is preceded by the presentation of an explicit algorithm for the reduction of the trace of an arbitrary element of the Hecke algebra into a linear combination of traces of elements consisting of appropriately defined disjoint cycles; and of a proof, presented in order to make the present article reasonably self-contained, that a reduced trace depends only on the set of lengths of the disjoint cycles that it consists of.

q-alg↗

arXiv1995

The fundamental invariant of the Hecke algebra $H_n(q)$ characterizes the representations of $H_n(q)$, $S_n$, $SU_q(N)$ and $SU(N)$

The irreducible representations (irreps) of the Hecke algebra $H_n(q)$ are shown to be completely characterized by the fundamental invariant of this algebra, $C_n$. This fundamental invariant is related to the quadratic Casimir operator, ${\cal{C}}_2$, of $SU_q(N)$, and reduces to the transposition class-sum, $[(2)]_n$, of $S_n$ when $q\rightarrow 1$. The projection operators constructed in terms of $C_n$ for the various irreps of $H_n(q)$ are well-behaved in the limit $q\rightarrow 1$, even when approaching degenerate eigenvalues of $[(2)]_n$. In the latter case, for which the irreps of $S_n$ are not fully characterized by the corresponding eigenvalue of the transposition class-sum, the limiting form of the projection operator constructed in terms of $C_n$ gives rise to factors that depend on higher class-sums of $S_n$, which effect the desired characterization. Expanding this limiting form of the projection operator into a linear combination of class-sums of $S_n$, the coefficients constitute the corresponding row in the character table of $S_n$. The properties of the fundamental invariant are used to formulate a simple and efficient recursive procedure for the evaluation of the traces of the Hecke algebra. The closely related quadratic Casimir operator of $SU_q(N)$ plays a similar role, providing a complete characterization of the irreps of $SU_q(N)$ and - by constructing appropriate projection operators and then taking the $q\rightarrow 1$ limit - those of $SU(N)$ as well, even when the quadratic Casimir operator of the latter does not suffice to specify its irreps.

q-alg↗