Automorphisms of First-Order Structures by Richard Kaye, Dugald Macpherson

By Richard Kaye, Dugald Macpherson

This remarkable survey of the examine of mathematical buildings info how either version theoretic equipment and permutation theoretic tools are worthy in describing such constructions. moreover, the ebook offers an advent to present study in regards to the connections among version conception and permutation staff conception. made from a set of articles--some introductory, a few extra in-depth, and a few containing formerly unpublished research--the booklet will end up worthy to graduate scholars assembly the topic for the 1st time in addition to to energetic researchers learning mathematical good judgment and permutation team theory.

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N} = (A ∪ B)C = (A ∪ B )C [A \ (A ∩ B)] [B \ (A ∩ B)] [A \ (A ∩ B )] (A ∩ B) [B \ (A ∩ B )] (A ∩ B ) we can construct π ∈ Sn such that • π(A ∩ B) = A ∩ B • π[A \ (A ∩ B)] = A \ (A ∩ B ) • π[B \ (A ∩ B)] = B \ (A ∩ B ) so that π(A, B) = (A , B ). Setting j = {(A, B) ∈ n−k,k × n−k,k : |A ∩ B| = j } we have that k n−k,k × n−k,k = j j =0 is the decomposition of n−k,k × n−k,k into Sn -orbits. Observe that every orbit j is symmetric: |A ∩ B| = |B ∩ A|, so that (Sn , Sn−k × Sk ) is a symmetric Gelfand pair.

Hint. ] Suppose that (ρ, W ) is an irreducible representation of G. Set dρ = dimW and suppose that W K (the subspace of K-invariant vectors in W ) is non-trivial. 24) W for all g ∈ G and u ∈ W . Since G is transitive on X, this is defined for all x ∈ X. Moreover, if g1 , g2 ∈ G and g1 x0 = g2 x0 , then g1−1 g2 ∈ K and therefore (v is K-invariant) (Tv u)(g2 x0 ) = dρ u, ρ(g1 )ρ(g1−1 g2 )v |X| W = (Tv u)(g1 x0 ). This shows that Tv u is well defined. 12 (Frobenius reciprocity for a permutation representation) With the above notation we have the following.

We now show that it is also bijective. Suppose that T ∈ HomG (W, L(X)). Then W u → (T u)(x0 ) ∈ C is a linear map, and therefore there exists v ∈ W such that (T u)(x0 ) = u, v W , for all u ∈ W . 29) W W, that is, T = |X| Tv . dρ Clearly, v ∈ W K : if k ∈ K then u, ρ(k)v W = (T u)(kx0 ) = (T u)(x0 ) = u, v W for all u ∈ W , and therefore ρ(k)v = v. 25) is a bijection. 5) and therefore to dimW K . 13 (G, K) is a Gelfand pair if and only if dimW K ≤ 1 for every irreducible G-representation W . In particular, dimW K = 1 if and only if W is a sub-representation of L(X).

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