New PDF release: q-Clan Geometries in Characteristic 2 (Frontiers in

By Ilaria Cardinali

ISBN-10: 3764385073

ISBN-13: 9783764385071

A q-clan with q an influence of two is reminiscent of a undeniable generalized quadrangle with a family members of subquadrangles each one linked to an oval within the Desarguesian airplane of order 2. it's also reminiscent of a flock of a quadratic cone, and accordingly to a line-spread of three-dimensional projective area and therefore to a translation airplane, and extra. those geometric gadgets are tied jointly via the so-called basic Theorem of q-Clan Geometry. The ebook offers a complete evidence of this theorem, through an in depth learn of the recognized examples. The collineation teams of the linked generalized quadrangles and the stabilizers in their linked ovals are labored out thoroughly.

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23) 26 Chapter 2. 1 (The Fundamental Theorem). Let C = {At ≡ C = {At ≡ so that A0 = xt 0 ( 00 yt zt 0) 0 xt yt 0 zt : t ∈ F } and : t ∈ F } be two (not necessarily distinct) q-clans normalized = A0 . Then the following are equivalent: (i) C ∼ C . (ii) The flocks F (C) and F (C) are projectively equivalent. (iii) GQ(C) and GQ(C) are isomorphic by an isomorphism mapping (∞) → (∞), [A(∞)] → [A (∞)], and (0, 0, 0) → (0, 0, 0). (iv) The associated spreads S(C) and S(C) are equivalent by a semilinear transformation leaving L∞ fixed and mapping the special reguli of S(C) to the special reguli of S(C) .

10 , the most general special automorphism of G⊗ mapping J (C) to J (C is ) and A(∞) to Ais (∞) is θ¯ = θ(σ, 1 0 1 ¯ 02 λ ⊗ B), where Ait¯s + Ai¯0s ≡ λB −1 Aσt B −T for all t ∈ F , with 0 = λ ∈ F , σ ∈ Aut(F ), B ∈ SL(2, q), and π : t → t¯ = λ2 tσ + ¯ 0. And using Eq. 59) for all t ∈ F . In Fig. 6 replace s with ∞ and t with s to see that the most general automorphism θ of GQ(C) mapping [A(∞)] to [A(s)] is θ = θ¯ ◦ i−1 s = θ(σ, ¯ 12 1 0 0 λ ⊗ B) ◦ θ(id, = θ(σ, 0 1 1 1 s2 ⊗ I) ¯0 21 λ 1 + (¯0s) 2 1 λs 2 1 ⊗ B).

Then the following hold: (i) C 0 D E = (I ⊗ B) a24 A(a2 /a4 )τ 0 a2 a3 P a23 A(a1 /a3 )τ (I ⊗ B)T . (ii) θ(σ, A ⊗ B, C, E) = θ(σ, A ⊗ B) : ((α, β), c) → T ((ασ , β σ )(A ⊗ B), λ∆cσ + a24 ασ (BA(a2 /a4 )τ B )(ασ )T T + ∆a2 a3 (α ◦ β)σ + a23 β σ (BA(a1 /a3 )τ B )(β σ )T ). (iii) θ : A(t) → A (t¯), where t¯ = a1 tσ/τ +a2 a3 tσ/τ +a4 τ . Note. In determining the specific form of C and E we used only the effect of θ = θ(σ, A ⊗ B, C, E) on A(0) and on A(∞). 4 are necessary but in general not sufficient for θ to map J (C) to J (C ).

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q-Clan Geometries in Characteristic 2 (Frontiers in Mathematics) by Ilaria Cardinali


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