Algebras associated to the Young-Fibonacci lattice
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- by Soichi Okada PDF
- Trans. Amer. Math. Soc. 346 (1994), 549-568 Request permission
Abstract:
The algebra ${\mathcal {F}_n}$ generated by ${E_1},\; \ldots \;,\;{E_{n - 1}}$ subject to the defining relations $E_i^2 = {x_i}{E_i}\;(i = 1,\; \ldots \;,\;n - 1),\;{E_{i + 1}}{E_i}{E_{i + 1}} = {y_i}{E_{i + 1}}\;(i = 1,\; \ldots \;,\;n - 2),\;{E_i}{E_j} = {E_j}{E_i}\;(|i - j| \geqslant 2)$ 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.References
- Eiichi Bannai, Association schemes and fusion algebras (an introduction), J. Algebraic Combin. 2 (1993), no. 4, 327–344. MR 1241504, DOI 10.1023/A:1022489416433
- S. V. Fomin, The generalized Robinson-Schensted-Knuth correspondence, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 155 (1986), no. Differentsial′naya Geometriya, Gruppy Li i Mekh. VIII, 156–175, 195 (Russian); English transl., J. Soviet Math. 41 (1988), no. 2, 979–991. MR 869582, DOI 10.1007/BF01247093 —, Duality of graded graphs, Report No. 15 (1991 /92), Institut Mittag-Leffler. —, Schensted-type algorithms for dual graded graphs, Report No.16 (1991/92), Institut Mittag-Leffler.
- Frederick M. Goodman, Pierre de la Harpe, and Vaughan F. R. Jones, Coxeter graphs and towers of algebras, Mathematical Sciences Research Institute Publications, vol. 14, Springer-Verlag, New York, 1989. MR 999799, DOI 10.1007/978-1-4613-9641-3
- Tom Halverson and Arun Ram, Characters of algebras containing a Jones basic construction: the Temperley-Lieb, Okada, Brauer, and Birman-Wenzl algebras, Adv. Math. 116 (1995), no. 2, 263–321. MR 1363766, DOI 10.1006/aima.1995.1068
- Gordon James and Adalbert Kerber, The representation theory of the symmetric group, Encyclopedia of Mathematics and its Applications, vol. 16, Addison-Wesley Publishing Co., Reading, Mass., 1981. With a foreword by P. M. Cohn; With an introduction by Gilbert de B. Robinson. MR 644144
- Masashi Kosuda and Jun Murakami, Centralizer algebras of the mixed tensor representations of quantum group $U_q(\textrm {gl}(n,\textbf {C}))$, Osaka J. Math. 30 (1993), no. 3, 475–507. MR 1240008
- I. G. Macdonald, Symmetric functions and Hall polynomials, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1979. MR 553598 S. Okada, Reflection-extension of fusion algebras, preprint. T. W. Roby, Applications and extensions of Fomin’s generalization of the Robinson-Schensted correspondence to differential posets, Ph.D. thesis, Massachusetts Institute of Technology, 1991. —, Schensted correspondences for differential posets, preprint.
- Richard P. Stanley, Differential posets, J. Amer. Math. Soc. 1 (1988), no. 4, 919–961. MR 941434, DOI 10.1090/S0894-0347-1988-0941434-9
- Richard P. Stanley, Variations on differential posets, Invariant theory and tableaux (Minneapolis, MN, 1988) IMA Vol. Math. Appl., vol. 19, Springer, New York, 1990, pp. 145–165. MR 1035494
- Richard P. Stanley, Further combinatorial properties of two Fibonacci lattices, European J. Combin. 11 (1990), no. 2, 181–188. MR 1044457, DOI 10.1016/S0195-6698(13)80072-8
- Hans Wenzl, Hecke algebras of type $A_n$ and subfactors, Invent. Math. 92 (1988), no. 2, 349–383. MR 936086, DOI 10.1007/BF01404457
Additional Information
- © Copyright 1994 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 346 (1994), 549-568
- MSC: Primary 05E99; Secondary 06B99
- DOI: https://doi.org/10.1090/S0002-9947-1994-1273538-7
- MathSciNet review: 1273538