Paper Info
Title
THE BRAUER TREE OF THE PRINCIPAL 19-BLOCK OF THE SPORADIC SIMPLE THOMPSON GROUP
Abstract
ULLER Abstract. This paper completes the construction of the Brauer tree of the sporadic simple Thompson group in characteristic 19. Our main computa- tional tool to arrive at this result is a new parallel implementation of the DirectCondense method. 1. Introduction and Results LetTh denote the sporadic simple Thompson group. In (6) the Brauer tree of the principal 19-block of Th has been determined up to two possibilities. In this note, we show which of these is the correct one, and we describe the new computational techniques which enabled us to decide between these two possibilities. We believe that the methods presented here will be powerful enough to solve even more dicult problems in the modular character theory of the sporadic groups. As a general reference for the theory of blocks of cyclic defect, the interpretation of a Brauer tree and its planar embedding see the introduction of (6). The planar embedded Brauer tree of the principal 19-block ofTh is given in Table 1, it coincides with the tree given in (6, p. 277, Case I). Its nodes are labelled by the numbers of the corresponding ordinary irreducible characters, where we use the notation for the ordinary irreducible characters of Th as is given in (3, p. 176), and can also be accessed in GAP (14). In Table 2 we list the ordinary irreducible characters of G lying in the principal 19-block, plus some additional information concerning these. The column headed \CC" contains the entry \r" in rows corresponding to real valued characters. Otherwise it contains the number of the complex conjugate character. The last column of Table 2 contains the values of the characters on elements of class 19A. Characters which are connected on the Brauer tree must have unequal values on this class. The degrees of the irreducible Brauer characters are given in Table 3. A Brauer character corresponding to an edge of the tree connecting i and j, with i < j is denoted by i. To obtain the result of this paper, we had to apply a new condensation technique to a module of dimension 976 841 775, the permutation module on the cosets of the third maximal subgroup ofTh. The condensed module has dimension 1403 overF19 and can be analyzed with the MeatAxe (10), giving the result. The details are given in Section 2. We remark that in order to arrive at only two possibilities for the Brauer tree in (6), we had to rule out several other possibilities using sophisticated techniques involving Green correspondence. We have checked the results of the condensation against these other possible trees. None of them is consistent with the condensation results.
Year
Venue
DocType
1997
Experimental Mathematics
Journal
Volume
Issue
Citations 
6
4
4
PageRank 
References 
Authors
0.94
2
3
Name
Order
Citations
PageRank
Gene Cooperman126735.78
GERHARD HISS2196.22
KLAUS LUX340.94