By Komarchev I.A.

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For later use, it is interesting to note that the multiplication of an element of Γ (P) from the left by ρ j corresponds to passing from a suitable ﬂag to its j-adjacent ﬂag. Note further that, by Proposition 2B4, such automorphisms ρ j exist for one ﬂag Φ if and only if they exist for each ﬂag of P. More precisely, if Φ and Ψ are ﬂags and Φϕ = Ψ with ϕ ∈ Γ (P), then for each j = 0, . . , n − 1, we have Φρ j = Φ j if and only if Ψ ϕ −1 ρ j ϕ = Ψ i . That is, the involutory generators of Γ (P) corresponding to Ψ are the conjugates ϕ −1 ρ j ϕ of those corresponding to Φ.

For more general kinds of posets or geometries, the homogeneity parameter 2 in (P4) must be replaced by other values (if it exists at all). However, for abstract polytopes it is crucial that this value is 2. Note that (P4) can be rephrased by saying that all 1-sections of P are of diamond shape; see Figure 2A1(b). For later use, we introduce the following notation. By (P4), if n 1 then, for each j = 0, . . , n − 1 and each ﬂag Φ of P, there exists precisely one adjacent ﬂag differing from Φ in the j-face; we shall denote this ﬂag by Φ j .

Theorem 2B14 has important consequences. In effect, it says that, as is familiar from similar situations in the theory of transitive permutation groups, we may identify a face F j ϕ of P with the right coset Γ j ϕ of the stabilizer Γ j = Γ (P, F j ) = ρi | i = j of F j in Γ (P). Then Theorem 2B14 tells us when two such cosets must be regarded as “incident”. 4]); it was ﬁrst discovered by Tits [416]. In Section 2E, this approach will be explored further. By Propositions 2B9 and 2B11, if P is a regular n-polytope of type { p1 , .

### 2-Abolutely summable oeprators in certain Banach spaces by Komarchev I.A.

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