| On the Fundumental Invariant of the Hecke Algebra $H_{n}(q)$ (1995) | |||||||||
Abstract | |||||||||
| 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 $H_{n}(q)$.. Comment: 8 pages, Latex-file and 1 figure. Uuencoded, compressed and tared archive of plain tex file and postscript figure file. Upon uudecoding, uncompressing and taring, tex the file nakai2.tex. Talk presented in the Satellite Meeting, Tianjin, 1995 by A. Chakrabarti | |||||||||
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