## 16-dimensional compact projective planes with a large group by Hahl H., Salzmann H.

By Hahl H., Salzmann H.

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3. (Mackey Decomposition). Let H and S be subgroups of G, let T be a full set of double coset representatives for (S, H ) in G and let V be a n FaH-module. Then An Invitation to Projective Characters 44 Proof. Let { g l , . . ,gn} be a left tra,nsversal for H in G. Then VG = \$r=lgg; 8 V (direct sum of F-spaces) Put X = {g; €4 V11 5 i 5 n}. Then G and, in particular S , acts on X . Moreover, gi @ V and g j 8 V lie in the same S-orbit if and only if gi and gj belong to the same double (S, H)-coset.

2(ii), I ( V ) is a subgroup of G containing H . We refer t o I ( V ) as the inertia group of V . If H a G and G = I ( V ) ,then we say that V is G-invariant. If p is an a-representation of H afforded by V , then I ( V ) consists precisely of all those g E N G ( H ) for which p and g p are linearly equivalent If x is the a-character of H afforded by V and g E G , then we write g x for the a-character of gHg-l afforded by g V . We refer to g x as the g-conjugate of x. If gHg-' = H and g x = x , then we say that x is g-invariant.

E. with the case of ordinary characters. Let x be an irreducible a-character of G over F . Assume that F is a splitting field for FOG or that charF = 0. Then the degree of x , written d e g x , is defined to be the F-dimension of a simple FOG-module which affords x. 3, degx is well defined. It is clear that if charF = 0, then d e g x = x(1) 20 An Invitation to Projective Characters Of course, if charF = 0 then we can unambigiously define the degree of an arbitrary a-character x of G over F to be the F-dimension of any F"Gmodule which affords x.