Symmetric functions and characters of the symmetric group

From symmetric functions to qubits Lecture II Symmetric functions and characters of the symmetric group Jean-Gabriel Luque Representations of finite group Consider a finite group G and a finite dimension space V. A representation of G in V is a morphism That is Equivalently, ρ defines an action of G on V ( ) The dimension of the representation ρ is the dimension of V. Representations of finite groups: Example 1 Trivial representation Alternated representation of Representations of finite groups: Example 2 Regular representation We consider a space V whose a basis is indexed by the elements of G. The regular representation of G, is defined by Proof: Consider the product of the two matrices There is at most one term y' in the sum, and if it exists it verifies or equivalently Hence, This show that ρ is a representation Representations of finite groups: Example 3 Regular representation of It suffices to construct the representation for the two generators [213] and [132] One verifies that Representations of finite groups: Example 4 Permutation representation We suppose that the group G is a subgroup of a symmetric group and we replace each permutation by the associated permutation matrix. For example consider the symmetric group A permutation representation of the cyclic group G={[123],[231],[312]} is given by the matrices [123] [231] [312] Representations of finite groups: Example 5 Representations of dimension 1 of the cyclic groups Consider the cyclic group Representations of finite groups: Example 6 (1) Specht modules Consider a partition λ of n and the set of the set partitions of n with shape λ. For example, if λ=21 one has We to each set partition, one associates a tableau: We consider the space spanned by these generators. We select only the standard tableaux Consider the natural action of on the integers {1,...,n} Representations of finite groups: Example 6 (2) Specht modules For each tableau, we consider the permutation which does not change the set of elements of each column: And we sort the lines From a standard tableau, we construct the formal alternated sums of the tableaux obtained by this process Induced representations(2) Example Characters of representations The character χ(ρ) of a representation ρ is defined by Remark that But Hence, a character is invariant on the conjugacy class Induced characters Induced character= character of induced representation Let Consider the decomposition of G into cosets One has where Central functions (1) Definition and example The central functions of a group G are the complex functions which are invariant on the conjugacy classes. Examples The characters are central functions If denotes the set of the conjugacy classes of G, the functions are central. Central functions (2) A scalar product and a natural basis One defines a scalar product on the space of the central functions as the following summation over the orbit of G: The functions are a natural orthogonal basis Orthogonality of irreducible characters Irreducible characters = characters of irreducible representations. We denote by Irrep G the set of all inequivalent irreductible representations of G, and by the set of the conjucacy classes of G. Two orthogonality relations: 1- If The set of irreducible characters is an orthonormal basis with respect to the scalar product. 2- for each pair (i,j): Furthermore a representation ρ is irreducible iff Conjugacy classes in the symmetric group A cycle c of length k is a permutation such there exist disctinct elements such that Example: Each permutation is a product of disjoint cycles. Cycle type: the partition of the lengths of the cycles Fact: Two permutations are conjugate iff they have the same cycle type. Table of characters: an example Consider the table of characters of the group Conjugacy classes Trivial representation Specht module 21 (Example 6) Alternated representation First orthogonality relations Second orthogonality relations Central functions of the symmetric group Recall that the space of central functions is spanned by the functions: In the case of the symmetric group , the conjugacy classes are indexed by the partitions of n. The cardinal of the conjugacy classes is known to be Recall that if . Hence, the space of central functions of the symmetric group is spanned by About symmetric functions Compare and where if and denotes the multiplicative basis of power sums symmetric functions. The polynomial encodes the function . Remark: But The complete function encodes the trivial representation. Conclusion We have shown the following correspondancen: (Frobenius map) Central functions Symmetric functions Irreducible characters Schur functions Other correspondances Characters of induced of irreducible rep of Young subgroups Product of Schur functions Kronecker product Inner product Thank you