The Integral Tree Representation of the Symmetric Group
Sarah Whitehouse
Universitéd'Artois-Pôle de Lens Laboratoire de Géométrie-Algèbre Rue Jean Souvraz S.P. 18-62307 Lens France
DOI: 10.1023/A:1011264315849
Abstract
Let T n be the space of fully-grown n-trees and let V n and V n \mathbb Z S n + 1 \mathbb{Z}Σ_{n + 1} -modules, giving a description of V n 
in terms of V n and V n+1. This short exact sequence may also be deduced from work of Sundaram.
in terms of V n and V n+1. This short exact sequence may also be deduced from work of Sundaram. Modulo a twist by the sign representation, V n is shown to be dual to the Lie representation of 
n , Lie n . Therefore we have an explicit combinatorial description of the integral representation of n+1 on Lie n and this representation fits into a short exact sequence involving Lie n and Lie n+1.
n , Lie n . Therefore we have an explicit combinatorial description of the integral representation of n+1 on Lie n and this representation fits into a short exact sequence involving Lie n and Lie n+1.Pages: 317–326
Keywords: symmetric group representation; free Lie algebra
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References
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9. S. Sundaram, “Decompositions of Sn-submodules in the free Lie algebra,” J. Algebra 154 (1993), 507-558.
10. S. Sundaram, “Homotopy of non-modular partitions and the Whitehouse module,” J. Alg. Combin. 9 (1999), 251-269.
11. S. Whitehouse, “The Eulerian representations of n as restrictions of representations of n+1,” J. Pure Appl. Algebra 115 (1997), 309-320.
12. S.A. Whitehouse, “ -(co)homology of commutative algebras and some related representations of the symmetric group,” Ph.D. Thesis, University of Warwick, 1994.
2. H. Barcelo, “On the action of the symmetric group on the free Lie algebra and the partition lattice,” J. Combin. Theory (A) 55 (1990), 93-129.
3. A.M. Garsia, “Combinatorics of the free Lie algebra and the symmetric group,” in Analysis, research papers published in honour of J. Moser's 60th birthday, P.H. Rabinowitz and E. Zehnder (Eds.), Academic Press, San Diego, 1990.
4. E. Getzler and M.M. Kapranov, “Cyclic operads and cyclic homology,” in Conference Proceedings and Lecture Notes in Geometry and Topology, Vol. VI, S.-T. Yau (Ed.), International Press, Cambridge, MA, 1995, pp. 167-201.
5. P. Hanlon, “Otter's method and the homology of homeomorphically irreducible k-trees,” J. Combin. Theory (A) 74(2) (1996), 301-320.
6. O. Mathieu, “Hidden n+1-actions,” Commun. Math. Phys. 176 (1996), 467-474.
7. A. Robinson and S. Whitehouse, “The tree representation of n+1,” J. Pure Appl. Algebra 111 (1996), 245-253.
8. A. Robinson and S. Whitehouse, “Operads and -homology of commutative rings,” Math. Proc. Combin. Phil. Soc. Preprint, Université Paris 13, 1998.
9. S. Sundaram, “Decompositions of Sn-submodules in the free Lie algebra,” J. Algebra 154 (1993), 507-558.
10. S. Sundaram, “Homotopy of non-modular partitions and the Whitehouse module,” J. Alg. Combin. 9 (1999), 251-269.
11. S. Whitehouse, “The Eulerian representations of n as restrictions of representations of n+1,” J. Pure Appl. Algebra 115 (1997), 309-320.
12. S.A. Whitehouse, “ -(co)homology of commutative algebras and some related representations of the symmetric group,” Ph.D. Thesis, University of Warwick, 1994.
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