ALGEBRAIC COMBINATORICS

no. 4
On a curious variant of the S n -module Lie n
Sundaram, Sheila1
1 Pierrepont School One Sylvan Road North Westport CT 06880, USA
Algebraic Combinatorics, Volume 3 (2020) no. 4, pp. 985-1009.

We introduce a variant of the much-studied Lie representation of the symmetric group S n , which we denote by Lie n (2) . Our variant gives rise to a decomposition of the regular representation as a sum of exterior powers of the modules Lie n (2) . This is in contrast to the theorems of Poincaré–Birkhoff–Witt and Thrall which decompose the regular representation into a sum of symmetrised Lie modules. We show that nearly every known property of Lie n has a counterpart for the module Lie n (2) , suggesting connections to the cohomology of configuration spaces via the character formulas of Sundaram and Welker, to the Eulerian idempotents of Gerstenhaber and Schack, and to the Hodge decomposition of the complex of injective words arising from Hochschild homology, due to Hanlon and Hersh.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/alco.127
Classification: 05E10, 20C30, 52B30
Keywords: Configuration space, higher Lie module, plethysm, Poincaré–Birkhoff–Witt, Schur positivity, symmetric power, exterior power, Thrall.
Sundaram, Sheila 1

1 Pierrepont School One Sylvan Road North Westport CT 06880, USA
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
@article{ALCO_2020__3_4_985_0,
     author = {Sundaram, Sheila},
     title = {On a curious variant of the $S_n$-module {Lie}$_n$},
     journal = {Algebraic Combinatorics},
     pages = {985--1009},
     publisher = {MathOA foundation},
     volume = {3},
     number = {4},
     year = {2020},
     doi = {10.5802/alco.127},
     language = {en},
     url = {https://alco.centre-mersenne.org/articles/10.5802/alco.127/}
}
TY  - JOUR
AU  - Sundaram, Sheila
TI  - On a curious variant of the $S_n$-module Lie$_n$
JO  - Algebraic Combinatorics
PY  - 2020
SP  - 985
EP  - 1009
VL  - 3
IS  - 4
PB  - MathOA foundation
UR  - https://alco.centre-mersenne.org/articles/10.5802/alco.127/
DO  - 10.5802/alco.127
LA  - en
ID  - ALCO_2020__3_4_985_0
ER  - 
%0 Journal Article
%A Sundaram, Sheila
%T On a curious variant of the $S_n$-module Lie$_n$
%J Algebraic Combinatorics
%D 2020
%P 985-1009
%V 3
%N 4
%I MathOA foundation
%U https://alco.centre-mersenne.org/articles/10.5802/alco.127/
%R 10.5802/alco.127
%G en
%F ALCO_2020__3_4_985_0
Sundaram, Sheila. On a curious variant of the $S_n$-module Lie$_n$. Algebraic Combinatorics, Volume 3 (2020) no. 4, pp. 985-1009. doi : 10.5802/alco.127. https://alco.centre-mersenne.org/articles/10.5802/alco.127/

[1] Cadogan, Charles C. The Möbius function and connected graphs, J. Combinatorial Theory Ser. B, Volume 11 (1971), pp. 193-200 | DOI | Zbl

[2] Foulkes, H. O. Characters of symmetric groups induced by characters of cyclic subgroups, Combinatorics (Proc. Conf. Combinatorial Math., Math. Inst., Oxford, 1972) (1972), pp. 141-154 | MR

[3] Gerstenhaber, Murray; Schack, Samuel D. A Hodge-type decomposition for commutative algebra cohomology, J. Pure Appl. Algebra, Volume 48 (1987) no. 3, pp. 229-247 | DOI | MR | Zbl

[4] Gessel, Ira M.; Restivo, Antonio; Reutenauer, Christophe A bijection between words and multisets of necklaces, European J. Combin., Volume 33 (2012) no. 7, pp. 1537-1546 | DOI | MR | Zbl

[5] Hanlon, Phil The action of S n on the components of the Hodge decomposition of Hochschild homology, Michigan Math. J., Volume 37 (1990) no. 1, pp. 105-124 | DOI | MR | Zbl

[6] Hanlon, Phil; Hersh, Patricia A Hodge decomposition for the complex of injective words, Pacific J. Math., Volume 214 (2004) no. 1, pp. 109-125 | DOI | MR | Zbl

[7] Hersh, Patricia; Reiner, Victor Representation stability for cohomology of configuration spaces in ℝ d , Int. Math. Res. Not. IMRN (2017) no. 5, pp. 1433-1486 (With an appendix written jointly with Steven Sam) | DOI | MR | Zbl

[8] Hyde, Trevor; Lagarias, Jeffrey C. Polynomial splitting measures and cohomology of the pure braid group, Arnold Math. J., Volume 3 (2017) no. 2, pp. 219-249 | DOI | MR | Zbl

[9] Kraśkiewicz, Witold; Weyman, Jerzy Algebra of coinvariants and the action of a Coxeter element, Bayreuth. Math. Schr. (2001) no. 63, pp. 265-284 | MR | Zbl

[10] Lehrer, Gustav I. On the Poincaré series associated with Coxeter group actions on complements of hyperplanes, J. London Math. Soc. (2), Volume 36 (1987) no. 2, pp. 275-294 | DOI | Zbl

[11] Lehrer, Gustav I.; Solomon, Louis On the action of the symmetric group on the cohomology of the complement of its reflecting hyperplanes, J. Algebra, Volume 104 (1986) no. 2, pp. 410-424 | DOI | MR | Zbl

[12] Macdonald, Ian G. Symmetric functions and Hall polynomials, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1995, x+475 pages (With contributions by A. Zelevinsky, Oxford Science Publications) | Zbl

[13] Reiner, Victor; Webb, Peter The combinatorics of the bar resolution in group cohomology, J. Pure Appl. Algebra, Volume 190 (2004) no. 1-3, pp. 291-327 | DOI | MR | Zbl

[14] Reutenauer, Christophe Free Lie algebras, London Mathematical Society Monographs. New Series, 7, The Clarendon Press, Oxford University Press, New York, 1993, xviii+269 pages (Oxford Science Publications) | MR | Zbl

[15] Robinson, Alan; Whitehouse, Sarah The tree representation of Σ n+1 , J. Pure Appl. Algebra, Volume 111 (1996) no. 1-3, pp. 245-253 | DOI | Zbl

[16] Solomon, Louis On the sum of the elements in the character table of a finite group, Proc. Amer. Math. Soc., Volume 12 (1961), pp. 962-963 | DOI | MR | Zbl

[17] Stanley, Richard P. Enumerative combinatorics. Vol. 2, Cambridge Studies in Advanced Mathematics, 62, Cambridge University Press, Cambridge, 1999, xii+581 pages (With a foreword by Gian-Carlo Rota and appendix 1 by Sergey Fomin) | DOI | MR | Zbl

[18] Sundaram, Sheila The homology representations of the symmetric group on Cohen-Macaulay subposets of the partition lattice, Adv. Math., Volume 104 (1994) no. 2, pp. 225-296 | DOI | MR | Zbl

[19] Sundaram, Sheila The conjugacy action of S n and modules induced from centralisers, J. Algebraic Combin., Volume 48 (2018) no. 2, pp. 179-225 | DOI | MR | Zbl

[20] Sundaram, Sheila On a variant of Lie n , Sém. Lothar. Combin., Volume 80B (2018), Paper no. Art. 19, 12 pages | DOI | Zbl

[21] Sundaram, Sheila On the Schur positivity of sums of power sums, Sém. Lothar. Combin., Volume 82B (2019), Paper no. Art. 49, 12 pages | Zbl

[22] Sundaram, Sheila; Welker, Volkmar Group actions on arrangements of linear subspaces and applications to configuration spaces, Trans. Amer. Math. Soc., Volume 349 (1997) no. 4, pp. 1389-1420 | DOI | MR | Zbl

[23] Thrall, Robert M. On symmetrized Kronecker powers and the structure of the free Lie ring, Amer. J. Math., Volume 64 (1942), pp. 371-388 | DOI | MR | Zbl

[24] Whitehouse, Sarah The integral tree representation of the symmetric group, J. Algebraic Combin., Volume 13 (2001) no. 3, pp. 317-326 | DOI | MR | Zbl

Cited by Sources: