Unit:
Department of MathematicsLibrary of the School of Science
Author:
Moustakas Vasileios - Dionysios
Dissertation committee:
Χρίστος Αθανασιάδης, Καθηγητής, Τμήμα Μαθηματικών, ΕΚΠΑ
Ιωάννης Εμμανουήλ, Καθηγητής, Τμήμα Μαθηματικών, ΕΚΠΑ
Αριστείδης Κοντογεώργης, Καθηγητής, Τμήμα Μαθηματικών, ΕΚΠΑ
Μιχάλης Μαλιάκας, Καθηγητής, Τμήμα Μαθηματικών, ΕΚΠΑ
Ιωάννης Ντόκας, Επίκουρος Καθηγητής, Τμήμα Μαθηματικών, ΕΚΠΑ.
Ελένη Τζανάκη, Επίκουρη Καθηγήτρια, Τμήμα Μαθηματικών, Πανεπιστήμιο Κρήτης
Δημήτρης Χελιώτης, Αναπληρωτής Καθηγητής, Τμήμα Μαθηματικών, ΕΚΠΑ
Original Title:
Enumerative combinatorics, representations and quasisymmetric functions
Translated title:
Enumerative combinatorics, representations and quasisymmetric functions
Summary:
The present thesis consists of two parts whose main protagonists are colored quasisymmetric functions. In 1984, Gessel introduced quasisymmetric functions, a generalization of symmetric functions. In 1993, together with Reutenauer they studied specializations of families of quasisymmetric functions associated to subsets of the symmetric group, which have many desirable properties, such as symmetry and Schur-positivity. In 1998, Poirier introduced colored quasisymmetric functions, a colored analogue of quasisymmetric functions. In the first part, we develop a general theory of specializations of colored quasisymmetric functions in the spirit of Gessel and Reutenauer's work. This allows us to systematically prove refined Euler--Mahonian identities on colored permutation groups and subsets of these, such as derangements and involutions. In 2017, Elizalde and Roichman proved that the quasisymmetric function of the product of a collection of permutations whose quasisymmetric generating function equals the Frobenius characteristic of some character χ of the symmetric group and an inverse descent class equals the Frobenius characteristic of the character of the tensor product of χ and the corresponding descent representation of the symmetric group. The second part deals with proving a colored analogue of Elizalde and Roichman's result. More precisely, we introduce a notion of colored ribbons and prove that the (colored) Frobenius characteristic of the descent representation of colored permutation groups equals the colored quasisymmetric generating function of colored ribbon shaped tableaux. This provides a colored analogue of Gessel's zig-zag shape approach to descent representations of the symmetric group. In addition, exploiting Hsiao--Petersen's theory of colored P-partitions and the method developed in the first part, we prove a colored analogue of Stanley's shuffling theorem.
Main subject category:
Science
Keywords:
generating function, symmetric function, quasisymmetric function, symmetric group, colored permutation group, descent set, eulerian distribution, mahonian distribution, P-partitions, shuffle, descent representation, ribbon, Schur-positivity
