New PDF release: 16-dimensional compact projective planes with a collineation

New PDF release: 16-dimensional compact projective planes with a collineation

By Salzmann H.

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Additional resources for 16-dimensional compact projective planes with a collineation group of dimension >= 35

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Mρ and s, r = ρ Tj Wρ ρ i,r . 15. 15 we take f = Trσ w, with w ∈ Wσ , then we have [ ρ σ i,j Tr w](gx0 ) = dρ σ ρ ρ T w, Tj ρ(g)vi |G| r L(X) and this is equal to zero if σ = ρ or if σ = ρ but r = j . Note that (ii) may also be deduced from (i) and (iii). 14. 17 ρ ρ (i) i,i is the orthogonal projection onto Ti Wρ . mρ ρ (ii) i=1 i,i is the orthogonal projection onto the isotypic component mρ Wρ . 3). This is a particular case of the theory developed in the previous section. For f1 , f2 ∈ L(G) we define the convolution f1 ∗ f2 ∈ L(G) by setting f1 (gh−1 )f2 (h).

22). 5 The group algebra and the Fourier transform 39 We now show that the map T → T is a ∗-anti-isomorphism. Indeed, for , 1 2 and f ∈ L(G) we have T 1 (T 2 f ) = (f ∗ 2) ∗ 1 =f ∗( 2 ∗ 1) =T 2∗ 1 f, that is, T 1 T 2 = T 2 ∗ 1 (anti-multiplicative property). Moreover, for f1 , f2 and ∈ L(G), we have T f1 , f2 L(G) f1 (gs) (s −1 )f2 (g) = g∈G s∈G f1 (t) (s −1 )f2 (ts −1 ) (setting t = gs) = t∈G s∈G = f1 , T ˇ f2 L(G) , that is (T )∗ = T ˇ . 3 Give a direct proof of the above proposition by showing that if T ∈ HomG (L(X), L(X)), then Tf = f ∗ , where = T (δ1G ).

N} : |A| = k} (we also say that A ∈ n−k,k is a k-subset of {1, 2, . . , n}). The group Sn acts on n−k,k : for π ∈ Sn and A ∈ n−k,k , then π A = {π (j ) : j ∈ A}. Fix A ∈ n−k,k and denote by K its stabilizer. Clearly, K is isomorC phic to Sn−k × Sk , where the first factor is the symmetric group on A = {1, 2, . . , n} \ A and the second is the symmetric group on A. As the action is transitive, we may write n−k,k = Sn /(Sn−k × Sk ). Note that (A, B) and (A , B ) in n−k,k × n−k,k are in the same Sn -orbit if and only if |A ∩ B| = |A ∩ B |.

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16-dimensional compact projective planes with a collineation group of dimension >= 35 by Salzmann H.


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