Dual graded graphs and Bratteli diagrams of towers of groups.

*(English)*Zbl 1409.05212Summary: An \(r\)-dual tower of groups is a nested sequence of finite groups, like the symmetric groups, whose Bratteli diagram forms an \(r\)-dual graded graph. Miller and Reiner introduced a special case of these towers in order to study the Smith forms of the up and down maps in a differential poset. Agarwal and the author have also used these towers to compute critical groups of representations of groups appearing in the tower. In this paper I prove that when \(r=1\) or \(r\) is prime, wreath products of a fixed group with the symmetric groups are the only \(r\)-dual tower of groups, and conjecture that this is the case for general values of \(r\). This implies that these wreath products are the only groups for which one can define an analog of the Robinson-Schensted bijection in terms of a growth rule in a dual graded graph.

##### MSC:

05E10 | Combinatorial aspects of representation theory |

05C25 | Graphs and abstract algebra (groups, rings, fields, etc.) |

06A07 | Combinatorics of partially ordered sets |

06A11 | Algebraic aspects of posets |

20C30 | Representations of finite symmetric groups |

##### Keywords:

dual graded graph; differential poset; tower of groups; Schensted correspondence; Bratteli diagram##### References:

[1] | Ayush Agarwal and Christian Gaetz. Differential posets and restriction in critical groups.arXiv:1710.08253, October 2017. |

[2] | Nantel Bergeron, Thomas Lam, and Huilan Li. Combinatorial Hopf algebras and towers of algebras—dimension, quantization and functorality.Algebr. Represent. Theory, 15(4):675–696, 2012. · Zbl 1281.16036 |

[3] | David S. Dummit and Richard M. Foote. Abstract algebra. John Wiley & Sons, Inc., Hoboken, NJ, third edition, 2004. · Zbl 1037.00003 |

[4] | Sergey Fomin. Duality of graded graphs. J. Algebraic Combin., 3(4):357–404, 1994. · Zbl 0810.05005 |

[5] | Sergey Fomin. Schensted algorithms for dual graded graphs. J. Algebraic Combin., 4(1):5–45, 1995. the electronic journal of combinatorics 26(1) (2019), #P1.2511 · Zbl 0817.05077 |

[6] | Christian Gaetz. Critical groups of group representations. Linear Algebra Appl., 508:91–99, 2016. · Zbl 1346.05298 |

[7] | Frederick M. Goodman, Pierre de la Harpe, and Vaughan F. R. Jones. Coxeter graphs and towers of algebras, volume 14 of Mathematical Sciences Research Institute Publications. Springer-Verlag, New York, 1989. · Zbl 0698.46050 |

[8] | Gordon James and Adalbert Kerber. The representation theory of the symmetric group, volume 16 of Encyclopedia of Mathematics and its Applications. AddisonWesley Publishing Co., Reading, Mass., 1981. · Zbl 0491.20010 |

[9] | Alexander Miller and Victor Reiner. Differential posets and Smith normal forms. Order, 26(3):197–228, 2009. · Zbl 1228.05096 |

[10] | Soichi Okada. Wreath products by the symmetric groups and product posets of Young’s lattices. J. Combin. Theory Ser. A, 55(1):14–32, 1990. · Zbl 0707.05062 |

[11] | Jean-Pierre Serre. Linear representations of finite groups. Springer-Verlag, New York-Heidelberg, 1977. Translated from the second French edition by Leonard L. Scott, Graduate Texts in Mathematics, Vol. 42. · Zbl 0355.20006 |

[12] | G. C. Shephard and J. A. Todd. Finite unitary reflection groups. Canadian J. Math., 6:274–304, 1954. · Zbl 0055.14305 |

[13] | Richard P. Stanley. Differential posets. J. Amer. Math. Soc., 1(4):919–961, 1988. · Zbl 0658.05006 |

[14] | Richard P. Stanley. Enumerative combinatorics. Volume 1, volume 49 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, second edition, 2012. · Zbl 1247.05003 |

[15] | Andrey V. Zelevinsky. Representations of finite classical groups, volume 869 of Lecture Notes in Mathematics. Springer-Verlag, Berlin-New York, 1981. · Zbl 0465.20009 |

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