Representation Theory of the Symmetric Groups: The by Tullio Ceccherini-Silberstein

By Tullio Ceccherini-Silberstein

The illustration concept of the symmetric teams is a classical subject that, because the pioneering paintings of Frobenius, Schur and younger, has grown right into a large physique of thought, with many very important connections to different components of arithmetic and physics. This self-contained ebook offers a close advent to the topic, overlaying classical themes resembling the Littlewood-Richardson rule and the Schur-Weyl duality. Importantly the authors additionally current many contemporary advances within the quarter, together with Lassalle's personality formulation, the speculation of partition algebras, and an exhaustive exposition of the strategy constructed by means of A. M. Vershik and A. Okounkov. A wealth of examples and routines makes this an awesome textbook for graduate scholars. it is going to additionally function an invaluable reference for more matured researchers throughout a number of parts, together with algebra, computing device technology, statistical mechanics and theoretical physics.

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Additional resources for Representation Theory of the Symmetric Groups: The Okounkov-Vershik Approach, Character Formulas, and Partition Algebras

Example text

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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