Commutative Algebra II by O. Zariski, P. Samuel

By O. Zariski, P. Samuel

Covers subject matters akin to valuation thought; conception of polynomial and tool sequence earrings; and native algebra. This quantity contains the algebro-geometric connections and functions of the merely algebraic fabric.

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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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