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

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

1995-09-11

arxiv.org/abs/q-alg/9509012v1

Mathematics

Quantum Algebra

8 pages, Latex-file and 1 figure. Uuencoded, compressed and tared archive of plain tex file and postscript figure file. Upon u

Scientific paper

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 $H_{n}(q)$.

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