Mathematical Foundations of Quantum Information Workshop (2009. Sevilla)
URI permanente para esta colecciónhttps://hdl.handle.net/11441/33997
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Ponencia Vertex operators, Kronecker products, and Hilbert series(2009-11) Thibon, Jean-YvesPonencia Symmetric functions and characters of the symmetric group(2009-11) Luque, Jean-GabrielPonencia Introduction to symmetric functions(2009-11) Luque, Jean-GabrielPonencia Kronecker coefficients, convexity, and generating functions(2009-11) Zeier, RobertWe give definitions for Kronecker coefficients and discuss possible applications in algebraic complexity theory and quantum information. We describe methods to compute Kronecker coefficients and convex polytopes corresponding to limits of Kronecker coefficients. We explore how generating functions can help to answer questions on Kronecker coefficients. We present low-dimensional examples.Ponencia Tensor algebras, words, and invariants of polynomials in non-commutative variables(2009-11) Zabrocki, MikeConsider a vector space V for which we specify a basis, then the tensor algebra T(V) corresponds to the non-commutative polynomials expressed in that basis. If V has an S_n module structure (more generally, for a finite group) then identifying the invariants of the non-commutative polynomials corresponds to calculating the multiplicity of the trivial representation in the repeated Kronecker product of the Frobenius image of the module V. We consider a general method of arriving at a combinatorial interpretation for the Kronecker coefficients by embedding the representation ring within a group algebra. This is joint work with Anouk Bergeron-Brlek and Christophe Hohlweg.Ponencia Discrete tomography, RSK correspondence and Kronecker products(2009-11) Vallejo Ruiz, Ernesto; Avella Alaminos, DianaPonencia Some remarks on characters of symmetric groups, Schur functions, Littlewood-Richardson and Kronecker coefficients(2009-11) King, Ronald C.The evaluation of characters of symmetric groups as polynomials in class numbers will be discussed, along with their use in evaluating Kronecker coefficients of both conventional and reduced type. Some properties of the polynomial and quasi-polynomial behaviour of stretched Littlewood-Richardson coefficients and stretched Kronecker coefficients will be pointed out. It will be shown that as an alternative to the Kronecker tableaux that have been used to provide a combinatorial model of a certain restricted class of Kronecker coefficients it is possible to use the hive model that was originally used in the study of Littlewood-Richardson coefficients.Ponencia A max-flow algorithm for positivity of Littlewood-Richardson coefficients(2009-11) Bürgisser, Peter; Ikenmeyer, ChristianLittlewood-Richardson coefficients appear as limits of certain families of Kronecker coefficients. They have a wide variety of interpretations in combinatorics, representation theory and geometry. Mulmuley and Sohoni pointed out that it is possible to decide the positivity of Littlewood-Richardson coefficients in polynomial time. This follows by combining the saturation property of Littlewood-Richardson coefficients (shown by Knutson and Tao 1999) with the well-known fact that linear optimization is solvable in polynomial time. We design an explicit *combinatorial* polynomial time algorithm for deciding the positivity of Littlewood-Richardson coefficients. This algorithm is highly adapted to the problem and it is based on ideas from the theory of optimizing flows in networks.Ponencia Kronecher powers and character polynomials(2009-11) Goupil, Alain; Chauve, Cedric; Garsia, AdrianoIn this talk, I will present joint works with Cedric Chauve and Adriano Garsia. With C. Chauve, we studied Kronecker powers of the irreducible representation of Sn indexed with (n-1,1). We gave a combinatorial interpretation and a generating function for the coefficients of any irreducible representation in a k-th Kronecker power ( χ(n-1,1) )⊗k. With A. Garsia, we studied character polynomials qλ(x1,…,xn) which are polynomials in several variables with the fundamental property that their evaluation on the multiplicities (m1,m2, …,mn) of a partition µ of n gives the value of the irreducible character χ( n- | λ | , λ ) of the symmetric group Sn on the conjugacy class Cµ . Character polynomials are closely related to the problem of decomposition of Kronecker product of representations of Sn. They were defined by Specht in 1960. Since then they received little attention from the combinatorics community. I will show how character polyomials are related to Kronecker products, how to produce them, their algebraic structure and show some applications.Ponencia Computational aspects of invariants of multipartite quantum systems(2009-11) Grassl, MarkusWe consider the problem of finding complete sets of polynomial invariants of multipartite quantum system with respect to local unitary operations. One of the main tools in that respect are both univariate and multivariate Hilbert series which are already hard to compute. We will also discuss various techniques for showing completeness of a set of invariants. Furthermore, new results for three qubit systems will be presented.Ponencia Invariant theory of projective reflection groups, and their Kronecker coefficients(2009-11) Caselli, FabrizioWe introduce the class of projective reflection groups which includes all complex reflection groups. We show that several aspects involving the relationship between the combinatorics and the (invariant) representation theory of all non exceptional irreducible complex reflection groups find a natural description in this wider setting.