Tuyển tập các báo cáo nghiên cứu khoa học về toán học trên tạp chí toán học quốc tế đề tài: Compression of root systems and the E-sequence. | Compression of root systems and the E-sequence Kevin Purbhoo Submitted Aug 25 2007 Accepted Apr 22 2008 Published Sep 15 2008 Mathematics Subject Classification 17B20 Abstract We examine certain maps from root systems to vector spaces over finite Helds. By choosing appropriate bases the images of these maps can turn out to have nice combinatorial properties which reflect the structure of the underlying root system. The main examples are E6 and E7. 1 Introduction The primary goal of this paper is to provide a convenient way of visualising the root systems E6 and E7. There are two important relations on a root system that one might wish to have a good understanding of the poset structure in which a p if a p is a sum of positive roots and the orthogonality structure in which a p if a and p are orthogonal roots. In our paper on cominuscule Schubert calculus with Frank Sottile 8 we found that our examples required a good simultaneous understanding both these structures. This is easy enough to acquire for the root systems corresponding to the classical Lie groups. In An for example one can visualise the positive roots as the entries of an strictly upper triangular n 1 X n 1 matrix where the ij position represents the root xi Xj. Then a p if and only if a is weakly right and weakly above p. Orthogonality is also straightforward in this picture a and p are non-orthogonal if there is some i such that crossing out the ith row and the ith column succeeds in crossing out both a and p. Figure 1 shows the roots orthogonal to x3 x5 in A5. In type E it is less obvious how to draw such a concrete picture. Separately the two structures have been well studied in the contexts of minuscule posets 7 9 11 and strongly regular graphs see . 1 3 4 . However once one draws the Hasse diagram of the posets the orthogonality structure suddenly becomes mysterious. Of course one can always calculate which pairs of roots are orthogonal but we would prefer a picture which allows us to do it .
