×

zbMATH — the first resource for mathematics

On the quiver presentation of the descent algebra of the symmetric group. (English) Zbl 1291.20003
Summary: We describe a presentation of the descent algebra of the symmetric group \(\mathfrak S_n\) as a quiver with relations. This presentation arises from a new construction of the descent algebra as a homomorphic image of an algebra of forests of binary trees, which can be identified with a subspace of the free Lie algebra. In this setting we provide a short new proof of the known fact that the quiver of the descent algebra of \(\mathfrak S_n\) is given by restricted partition refinement. Moreover, we describe certain families of relations and conjecture that for fixed \(n\in\mathbb N\) the finite set of relations from these families that are relevant for the descent algebra of \(\mathfrak S_n\) generates the ideal of relations of an explicit quiver presentation of that algebra.

MSC:
20C08 Hecke algebras and their representations
16G20 Representations of quivers and partially ordered sets
20C30 Representations of finite symmetric groups
17B01 Identities, free Lie (super)algebras
20F55 Reflection and Coxeter groups (group-theoretic aspects)
05E10 Combinatorial aspects of representation theory
Software:
GAP
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Assem, Ibrahim; Simson, Daniel; Skowroński, Andrzej, Elements of the representation theory of associative algebras, vol. 1, (Techniques of Representation Theory, London Math. Soc. Stud. Texts, vol. 65, (2006), Cambridge University Press Cambridge) · Zbl 1092.16001
[2] Atkinson, M. D., A new proof of a theorem of Solomon, Bull. London Math. Soc., 18, 4, 351-354, (1986) · Zbl 0567.20021
[3] Atkinson, M. D., Solomonʼs descent algebra revisited, Bull. London Math. Soc., 24, 6, 545-551, (1992) · Zbl 0728.20011
[4] Bauer, Thorsten, Über die struktur der Solomon-algebren: ein zugang über einen differential- und integralkalkül, Bayreuth. Math. Schr., 63, 1-102, (2001) · Zbl 1065.20503
[5] Bergeron, F.; Bergeron, N., A decomposition of the descent algebra of the hyperoctahedral group. I, J. Algebra, 148, 1, 86-97, (1992) · Zbl 0798.20008
[6] Bergeron, F.; Bergeron, N.; Howlett, R. B.; Taylor, D. E., A decomposition of the descent algebra of a finite Coxeter group, J. Algebraic Combin., 1, 1, 23-44, (1992) · Zbl 0798.20031
[7] Bergeron, François; Bergeron, Nantel, Orthogonal idempotents in the descent algebra of \(B_n\) and applications, J. Pure Appl. Algebra, 79, 2, 109-129, (1992) · Zbl 0793.20004
[8] Blessenohl, Dieter; Laue, Hartmut, On the descending loewy series of solomonʼs descent algebra, J. Algebra, 180, 3, 698-724, (1996) · Zbl 0864.20007
[9] Blessenohl, Dieter; Laue, Hartmut, The module structure of solomonʼs descent algebra, J. Aust. Math. Soc., 72, 3, 317-333, (2002) · Zbl 1062.20010
[10] Blessenohl, Dieter; Schocker, Manfred, Noncommutative character theory of the symmetric group, (2005), Imperial College Press London · Zbl 1089.20004
[11] Bonnafé, C.; Pfeiffer, G., Around solomonʼs descent algebras, Algebr. Represent. Theory, 11, 6, 577-602, (2008) · Zbl 1193.20046
[12] GAP Group, GAP - Groups, Algorithms, and Programming, Version 4.4.12, 2008.
[13] Garsia, A. M.; Reutenauer, C., A decomposition of solomonʼs descent algebra, Adv. Math., 77, 2, 189-262, (1989) · Zbl 0716.20006
[14] Götz Pfeiffer, Quiver presentations for descent algebras of exceptional type, 2008.
[15] Pfeiffer, Götz, A quiver presentation for solomonʼs descent algebra, Adv. Math., 220, 5, 1428-1465, (2009) · Zbl 1203.20004
[16] Saliola, Franco V., On the quiver of the descent algebra, J. Algebra, 320, 11, 3866-3894, (2008) · Zbl 1200.20003
[17] Saliola, Franco V., The loewy length of the descent algebra of type D, Algebr. Represent. Theory, 13, 2, 243-254, (2010) · Zbl 1189.20015
[18] Schocker, Manfred, The descent algebra of the symmetric group, (Representations of Finite Dimensional Algebras and Related Topics in Lie Theory and Geometry, Fields Inst. Commun., vol. 40, (2004), Amer. Math. Soc. Providence, RI), 145-161 · Zbl 1072.20004
[19] Solomon, Louis, A MacKey formula in the group ring of a Coxeter group, J. Algebra, 41, 2, 255-264, (1976) · Zbl 0355.20007
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.