×

zbMATH — the first resource for mathematics

A quiver presentation for Solomon’s descent algebra. (English) Zbl 1203.20004
Summary: The descent algebra \(\Sigma(W)\) is a subalgebra of the group algebra \(\mathbb{Q} W\) of a finite Coxeter group \(W\), which supports a homomorphism with nilpotent kernel and commutative image in the character ring of \(W\). Thus \(\Sigma(W)\) is a basic algebra, and as such it has a presentation as a quiver with relations. Here we construct \(\Sigma(W)\) as a quotient of a subalgebra of the path algebra of the Hasse diagram of the Boolean lattice of all subsets of \(S\), the set of simple reflections in \(W\). From this construction we obtain some general information about the quiver of \(\Sigma(W)\) and an algorithm for the construction of a quiver presentation for the descent algebra \(\Sigma(W)\) of any given finite Coxeter group \(W\).

MSC:
20C05 Group rings of finite groups and their modules (group-theoretic aspects)
16G20 Representations of quivers and partially ordered sets
20F55 Reflection and Coxeter groups (group-theoretic aspects)
05E10 Combinatorial aspects of representation theory
05E15 Combinatorial aspects of groups and algebras (MSC2010)
Software:
CHEVIE; GAP; OEIS; ZigZag
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Atkinson, M.D., Solomon’s descent algebra revisited, Bull. London math. soc., 24, 545-551, (1992), MR 93i:20012 · Zbl 0728.20011
[2] 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, 23-44, (1992), MR 93g:20079 · Zbl 0798.20031
[3] T.P. Bidigare, Hyperplane arrangement face algebras and their associated Markov chains, PhD thesis, Univ. Michigan, 1997
[4] Blessenohl, D.; Laue, H., On the descending loewy series of Solomon’s descent algebra, J. algebra, 180, 698-724, (1996), MR 97g:05170 · Zbl 0864.20007
[5] Blessenohl, Dieter; Laue, Hartmut, The module structure of Solomon’s descent algebra, J. aust. math. soc., 72, 317-333, (2002), MR 2003c:20012 · Zbl 1062.20010
[6] Bonnafé, C.; Pfeiffer, G., Around Solomon’s descent algebras, Algebr. represent. theory, 11, 577-602, (2008) · Zbl 1193.20046
[7] Brink, Brigitte; Howlett, Robert B., Normalizers of parabolic subgroups in Coxeter groups, Invent. math., 136, 323-351, (1999), MR 2000b:20048 · Zbl 0926.20024
[8] Brown, Kenneth S., Semigroup and ring theoretical methods in probability, (), 3-26, MR 2005b:60118 · Zbl 1043.60055
[9] Garsia, A.M.; Reutenauer, C., A decomposition of Solomon’s descent algebra, Adv. math., 77, 189-262, (1989), MR 91c:20007 · Zbl 0716.20006
[10] Geck, Meinolf; Hiß, Gerhard; Lübeck, Frank; Malle, Gunter; Pfeiffer, Götz, CHEVIE — A system for computing and processing generic character tables, Appl. algebra engrg. comm. comput., 7, 175-210, (1996), MR 99m:20017 · Zbl 0847.20006
[11] Geck, Meinolf; Pfeiffer, Götz, Characters of finite Coxeter groups and Iwahori-Hecke algebras, London math. soc. monogr. new ser., vol. 21, (2000), Oxford University Press New York, MR 2002k:20017 · Zbl 0996.20004
[12] Howlett, Robert B., Normalizers of parabolic subgroups of reflection groups, J. London math. soc. (2), 21, 62-80, (1980), MR 81g:20094 · Zbl 0427.20040
[13] Frédéric Patras; Schocker, Manfred, Twisted descent algebras and the Solomon-Tits algebra, Adv. math., 199, 151-184, (2006), MR 2006k:16086 · Zbl 1154.16029
[14] Götz Pfeiffer, Zigzag — A GAP3 package for descent algebras of finite Coxeter groups, (2007), electronically available at
[15] Pfeiffer, Götz, Quiver presentations for descent algebras of exceptional type, (2008), preprint
[16] Saliola, Franco V., The loewy length of the descent algebra of type D, (2007), preprint · Zbl 1189.20015
[17] Saliola, Franco V., On the quiver of the descent algebra, J. algebra, 320, 3866-3894, (2008) · Zbl 1200.20003
[18] Schocker, Manfred, The descent algebra of the symmetric group, (), 145-161, MR 2005c:20023 · Zbl 1072.20004
[19] Schocker, Manfred, The module structure of the Solomon-Tits algebra of the symmetric group, J. algebra, 301, 554-586, (2006), MR 2007e:20024 · Zbl 1155.20012
[20] Schönert, Martin, GAP — groups, algorithms, and programming, (1995), Lehrstuhl D für Mathematik RWTH Aachen, home page:
[21] Sloane, N.J.A., The on-line encyclopedia of integer sequences, (2006), published electronically at · Zbl 1274.11001
[22] Solomon, L., A MacKey formula in the group ring of a Coxeter group, J. algebra, 41, 255-268, (1976), MR 56 #3104 · 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.