Algebras associated to the Young-Fibonacci lattice

Okada, Soichi

doi:10.1090/S0002-9947-1994-1273538-7

Algebras associated to the Young-Fibonacci lattice
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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

Schensted-type algorithms for dual graded graphs

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

Reflection-extension of fusion algebras

Applications and extensions of Fomin’s generalization of the Robinson-Schensted correspondence to differential posets

Schensted correspondences for differential posets

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