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Algebras associated to the Young-Fibonacci lattice

Author: Soichi Okada
Journal: Trans. Amer. Math. Soc. 346 (1994), 549-568
MSC: Primary 05E99; Secondary 06B99
MathSciNet review: 1273538
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Abstract: The algebra $ {\mathcal{F}_n}$ generated by $ {E_1},\; \ldots \;,\;{E_{n - 1}}$ subject to the defining relations $ E_i^2 = {x_i}{E_i}\;(i = 1,\; \ldots \;,\;n - 1),\;{E_{i + 1}}{E_i}{E_{i + 1}}... ...\; \ldots \;,\;n - 2),\;{E_i}{E_j} = {E_j}{E_i}\;(\vert i - j\vert \geqslant 2)$ is shown to be a semisimple algebra of dimension $ n!$ if the parameters $ {x_1},\; \ldots \;,\;{x_{n - 1}},\;{y_1},\; \ldots \;,\;{y_{n - 2}}$ are generic. We also prove that the Bratteli diagram of the tower $ {({\mathcal{F}_n})_{n \geqslant 0}}$ of these algebras is the Hasse diagram of the Young-Fibonacci lattice, which is an interesting example, as well as Young's lattice, of a differential poset introduced by $ \operatorname{R} $. Stanley. A Young-Fibonacci analogue of the ring of symmetric functions is given and studied.

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Keywords: Young-Fibonacci lattice, differential poset, Bratteli diagram
Article copyright: © Copyright 1994 American Mathematical Society

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