Irish Math. Soc. Bulletin
Number 68, Winter 2011, 21–21
ISSN 0791-5578
ON THE QUIVER PRESENTATION OF THE
DESCENT ALGEBRA OF TYPE A OR B
MARCUS BISHOP
This is an abstract of the PhD thesis On the Quiver Presentation
of the Descent Algebra of Type A or B written by Marcus Bishop
under the supervision of Götz Pfeiffer at the School of Mathematics,
Statistics, and Applied Mathematics of the National University of
Ireland, Galway and submitted in September 2010.
In this thesis we study the descent algebra of the Coxeter group
of type A or B using a quiver presentation. The descent algebra is
a quotient of the algebra of streets described in the recent paper [1].
To derive the presentation, we introduce an algebra of binary trees.
The trees reflect the structure of the algebra of streets in a natural
way and make it possible to construct a quiver whose path algebra
is essentially the algebra of streets. Then expressing the descent
algebra as a quotient of the algebra of binary trees and transferring
the quotient to the path algebra provides the desired presentation.
We then use the presentation to calculate the Cartan invariants and
the projective indecomposable modules of the descent algebra.
References
[1] Götz Pfeiffer: A quiver presentation for Solomon’s descent algebra. Adv. Math.
220 (5) (2009) 1428–1465. URL http://arxiv.org/pdf/0709.3914v3.
Ruhr-Universität Bochum, Fakultät für Mathematik, Universitätsstraße 150, D-44780, Bochum, Germany.
E-mail address: marcus.bishop@rub.de
2010 Mathematics Subject Classification. 20F55, 16G20.
Key words and phrases. Coxeter group, quiver, presentation, path algebra.
Received on 18-10-2011; revised 20-2-2012.
Support from the Irish Research Council for Science, Engineering, and Technology through the Embark Postgraduate Scholarship Scheme is gratefully
acknowledged.
c
2011
Irish Mathematical Society
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