Combinatorial links between quasisymmetric functions and tableaux for Coxeter groups.

Abstract : The algebra of symmetric functions is a major tool in algebraic combinatorics that plays a central role in the representation theory of the symmetric group. This thesis deals with quasisymmetric functions, a powerful generalisation introduced by Gessel in 1984, with significant applications in the enumeration of major combinatorial objects as permutations, Young tableaux and P-partitions. More specifically we find a new connection between Chow's extension of quasisymmetric functions to Coxeter groups of type B and domino tableaux. It allows us to contribute new results to various fields including the structure constants of type B Solomon's descent algebra, the extension of the theory of Schur-positivity to signed permutations and the study a $q$-deformed type B Cauchy formula with important implications regarding statistics for domino tableaux.Among the remarkable bases of the algebra of symmetric functions, Schur functions received a particular attention as they are strongly related to the irreducible characters of the general linear group and Young diagrams. The Schur symmetric function is the generating function for semistandard Young tableaux. This result extends to skew shapes and allows to write any (skew-) Schur function as the sum of Gessel's fundamental quasisymmetric functions indexed by the descent set of all standard Young tableaux of a given shape. Furthermore the celebrated Cauchy formula for Schur functions gives an algebraic proof of the Robinson-Schensted-Knuth correspondence. Finally, the structure constants for the outer product and inner product of Schur polynomials are respectively the Littlewood-Richardson and Kronecker coefficients, two important families of coefficients with various combinatorial and algebraic applications. Using known results about Gessel's fundamental quasisymmetric functions we show that these properties imply directly and in a pure algebraic fashion, various results for the structure constants of the Solomon descent algebra of a finite Coxeter group of type A and the descent preserving property of the Robinson-Schensted correspondence, an essential tool to identify Schur-positive sets, i.e. sets of permutations whose associated quasisymmetric function is symmetric and can be written as a non-negative sum of Schur symmetric functions.To extend these results to Coxeter groups of type B we introduced a family of modified generating functions for domino tableaux and relate it to Chow's type B fundamental quasisymmetric functions. Thanks to this relation we derive new formulas relating the structure constants of the type B Solomon's descent algebra with type B Kronecker and Littlewood-Richardson coefficients.It further allows us to introduce a new type B extension of Schur-positivity based on a definition of descent for signed permutations that is conform to the abstract definition of Solomon for any Coxeter groups. We design descent preserving bijections between signed arc permutations and sets of domino tableaux to show that they are indeed type B Schur-positive.Finally, we introduce a $q$-deformation of the modified generating functions for domino tableaux to extend a type B Cauchy identity by Lam and link it with Chow's quasisymmetric functions. We apply this result to a new framework of type B $q$-Schur positivity and to prove new equidistribution results for some sets of domino tableaux.
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Alina Mayorova. Combinatorial links between quasisymmetric functions and tableaux for Coxeter groups.. Combinatorics [math.CO]. Université Paris-Saclay; Moskovskij gosudarstvennyj universitet imeni M. V. Lomonosova, 2019. English. ⟨NNT : 2019SACLX023⟩. ⟨tel-02271384⟩

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