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Basic abstract algebra by Otto F.G. Schilling, W.Stephen Piper

By Otto F.G. Schilling, W.Stephen Piper

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Bu /. 5 to be the minimal diagonal entry of p d Cu 1 and check (probably by computer) the defined box for smaller values. bu / (in small dimensions). 3). bu /. 4 is quite powerful as we will see in Part III. We also add a related result by Robinson which goes in the opposite direction. cij / up to basic sets. bu / ) which are coprime to p. e. B/-Conjecture. B/ D 21. B/. 4. cij / 2 Zl l be the Cartan matrix of a p-block with defectP d . bu / / D 1Äi Äj Äl qij xi xj we have X qij cij > p d : 1Äi Äj Äl Then prove that xpd C 1 x T l for all 0 ¤ x 2 Zl .

We are going to apply equation . / in [172]. In order to do so, we will bound the trace of A from above and the sum a12 C a23 C : : : C al 1;l from below. B/ D 2. 2 we have a12 ¤ 0 and a12 > 0 after a suitable change of signs (i e. replacing A by an equivalent matrix). 2) 3 Quadratic Forms 29 p Since 2jaij j Ä minfaii ; ajj g, we have 2 Ä ˛, and ˛ Ä ˇ yields ˛ Äp2 p d =3. ˛/ takes its maximal value in the interval Œ2; 2 p d =3pon one of the two borders. 2 p d =3/ for p d 9. In case p d Ä 6 only ˛ D 2 is possible.

For example we can take a permutation matrix PQ with signs for P . In other words we freely arrange the order and signs of the rows of the generalized decomposition matrix. With the matrix S above we can realize elementary column operations on Q2 . We will often apply these reductions without an explicit reference. Finally, after we have a list of possible Cartan matrices C for B, we can check if the elementary divisors are correct by computing lower defect groups (see Sect. 8). We can decrease the list further by reducing C as a quadratic form.

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