Introduction to symmetric functions

From symmetric functions to qubits From symmetric functions to qubits Our purpose: Explain how to compute the Hilbert series of the algebras of covariants for (pure) qubits systems. Why these algebras are relevant? The (pure) qubit systems are regarded as multilinear forms on which acts a product of linear groups (SLOCC: Stochastic Local Operations and the Classical Communication). The knowledge of the covariants allows: - (In principle) to describe the structure of the orbits. - Construct entanglement monotones and measure of entanglement. From symmetric functions to qubits Our plan: Lecture I: Introduction to symmetric functions (Jean-Gabriel Luque) We will present the main tool: the symmetric functions. Lecture II: Symmetric functions and characters of the symmetric groups.(JGL) We will explain why the symmetric functions encodes the characters of the symmetric groups. Lecture III: Vertex operator and Kronecker coefficients (Jean-Yves Thibon) We will explain the links between the Kronecker coefficients and the symmetric functions. Lecture IV: Hilbert series of the algebras of invariants for a k-qubits (pure) system. (JYT) We will explain how to use these tools to compute the Hilbert series. From symmetric functions to qubits Lecture I Introduction to symmetric functions Jean-Gabriel Luque Symmetric functions Definition: Polynomials in several variables (alphabet X={x1,...,xn,...}) which are invariant under permutations of the variables. Sym(X) : algebra of symmetric polynomials for the alphabet X Example The monomial functions: Symmetric functions Definition: Polynomials in several variables (alphabet X={x1,...,xn,...} which are invariant under permutations of the variables. Sym(X) : algebra of symmetric polynomials for the alphabet X Example The complete function hn is the sum of all the monomials of degree n Symmetric functions Definition: Polynomials in several variables (alphabet X={x1,...,xn,...} which are invariant under permutations of the variables. Sym(X) : algebra of symmetric polynomials for the alphabet X Example The elementary functions Symmetric functions Definition: Polynomials in several variables (alphabet X={x1,...,xn,...} which are invariants under permutation if the variables. Sym(X) : algebra of symmetric polynomials for the alphabet X Example The power sums Bases Suppose that X={x1,...,xn,...} is an infinite alphabet. Multiplicative bases: Non multiplicative bases: Complete functions Power sums Elementary functions Monomial functions Schur functions Differences of alphabets Product of alphabets A power sum of a product of alphabets is the product of the power sums. Hence, Scalar product and reproducing Kernel (1) Consider a scalar product: With a pair of bases in duality The reproducing kernel associated to { , } is a multivariate series on two alphabets defined by Why this series is called a reproducing kernel? Scalar product and reproducing Kernel (2) As a consequence, if one has For any pair of bases in duality: Usual scalar product and inner product (1) We set Since, This is the reproducing kernel of the usual scalar product defined by Or equivalently, The usual inner product is defined by Usual scalar product and inner product (2) Write Since The coefficient of in is More generaly Orthogonalisation of complete functions One apply the Gramm-Schmidt orthogonalisation process to the complete functions for the inverse order of dominance Example for the degree 4: ..... By definition the are orthogonal. This basis is called the Schur basis. An orthonormal basis (1): Semi-standard tableaux Semi-standard (Young) tableaux of shape We fill up the nodes of the shape with non-negative integers such that the entries are strictly increasing along each column and just non-degreasing along each row. no no ok An orthonormal basis (2): Schensted algorithm Schensted algorithm: construct a semi-standard tableau from a sequence of integers Iteration of the insertion: Insertion of an integer n into a tableau t 1) First try to insert on the first line. ...... ...... ...........m n If m< n the result is obtain by glueing n at the end of the line 2) otherwise, let p the smallest integer of the line strictly greater that n ........... ........... .......p........m n Replace the first occurrence of p by n and try to add p into the next line ........... ........... p .......n........m An orthonormal basis (3): Schensted algorithm, example Consider the sequence 42214163311 4 4 42 2 4 22 4 4 42 2 221 12 12 4 2 124 4 4 2 2 2 1241 1 1 4 4 22 1146 4 4 22 224 11463 1136 4 4 224 2246 11363 1133 4 4 44 2246 22463 2236 11331 1113 1113 44 44 446 2236 22363 2233 11131 1111 1111 Schur to m Hence where are the Kostka numbers Example: h to Schur Since are in duality The Schur basis being orthonormal, one has implies Conclusion + We have shown how to use generating function for + Computing changes of bases. + Defining the scalar standard scalar product and the standard inner product. + We have defined Schur function by orthogonalizing the complete functions. + We have proved that the Schur functions are orthonormal for the standard scalar product.