Multiplicity Free Schur, Skew Schur, and Quasisymmetric Schur Functions

In this paper we classify all Schur functions and skew Schur functions that are multiplicity free when expanded in the basis of fundamental quasisymmetric functions, termed F-multiplicity free. Combinatorially, this is equivalent to classifying all skew shapes whose standard Young tableaux have distinct descent sets. We then generalize our setting, and classify all F-multiplicity free quasisymmetric Schur functions with one or two terms in the expansion, or one or two parts in the indexing composition. This identifies composition shapes such that all standard composition tableaux of that shape have distinct descent sets. We conclude by providing such a classification for quasisymmetric Schur function families, giving a classification of Schur functions that are in some sense almost F-multiplicity free.     

Quasisymmetric Schur functions

We introduce a new basis for quasisymmetric functions, which arise from a specialization of nonsymmetric Macdonald polynomials to standard bases, also known as Demazure atoms. Our new basis is called the basis of quasisymmetric Schur functions, since the basis elements refine Schur functions in a natural way. We derive expansions for quasisymmetric Schur functions in terms of monomial and fundamental quasisymmetric functions, which give rise to quasisymmetric refinements of Kostka numbers and standard (reverse) tableaux. From here we derive a Pieri rule for quasisymmetric Schur functions that naturally refines the Pieri rule for Schur functions. After surveying combinatorial formulas for Macdonald polynomials, including an expansion of Macdonald polynomials into fundamental quasisymmetric functions, we show how some of our results can be extended to include the t parameter from Hall-Littlewood theory.     

Quasisymmetric and Schur expansions of cycle index polynomials

Given a subgroup $G$ of the symmetric group $S_n$, the cycle index polynomial ${\mathrm{cyc}}_G$ is the average of the power-sum symmetric polynomials indexed by the cycle types of permutations in $G$. By Pólya’s Theorem, the monomial expansion of ${\mathrm{cyc}}_G$ is the generating function for weighted colorings of $n$ objects, where we identify colorings related by one of the symmetries in $G$. This paper develops combinatorial formulas for the fundamental quasisymmetric expansions and Schur expansions of certain cycle index polynomials. We give explicit bijective proofs based on standardization algorithms applied to equivalence classes of colorings. Subgroups studied here include Young subgroups of $S_n$, the alternating groups $A_n$, direct products, conjugate subgroups, and certain cyclic subgroups of $S_n$ generated by $(1,2,\dots ,k)$. The analysis of these cyclic subgroups when $k$ is prime reveals an unexpected connection to perfect matchings on a hypercube with certain vertices identified.     

Colored Posets and Colored Quasisymmetric Functions

The colored quasisymmetric functions, like the classic quasisymmetric functions, are known to form a Hopf algebra with a natural peak subalgebra. We show how these algebras arise as the image of the algebra of colored posets. To effect this approach, we introduce colored analogs of P-partitions and enriched P-partitions. We also frame our results in terms of Aguiar, Bergeron, and Sottile’s theory of combinatorial Hopf algebras and its colored analog.     

Ribbon Schur Functions

We present a new determinantal expression for Schur functions. Previous expressions were due to Jacobi, Trudi, Giambelli and others (see [7]) and involved elementary symmetric functions or hook functions. We give, in Theorem 1.1, a decomposition of a Schur function into ribbon functions (also called skew hook functions, new functions by MacMahon, and MacMahon functions by others). We provide two different proofs of this result in Sections 2 and 3. In Section 2, we use Bazin's formula for the minors of a general matrix, as we already did in [6], to decompose a skew Schur function into hooks. In Section 3, we show how to pass from hooks to ribbons and conversely. In Section 4, we generalize to skew Schur functions. In Section 5, we give some applications, and show how such constructions, in the case of staircase partitions, generalize the classical continued fraction for the tangent function due to Euler.     

Symmetric Functions, Noncommutative Symmetric Functions, and Quasisymmetric Functions

This paper is concerned with two generalizations of the Hopf algebra of symmetric functions that have more or less recently appeared. The Hopf algebra of noncommutative symmetric functions and its dual, the Hopf algebra of quasisymmetric functions. The focus is on the incredibly rich structure of the Hopf algebra of symmetric functions and the question of which structures and properties have good analogues for the noncommutative symmetric functions and/or the quasisymmetric functions. This paper attempts to survey the ongoing investigations in this topic as dictated by the knowledge and interests of its author. There are many open questions that are discussed.     

Equality of Schur and Skew Schur Functions

We determine the precise conditions under which any skew Schur function is equal to a Schur function over both infinitely and finitely many variables. Received May 29, 2004     

On Odd Symplectic Schur Functions

Proctor defined combinatorially a family of Laurent Polynomials, called odd symplectic Schur functions, indexed by pairs (λ, c), where λ is partition andcis a column length of λ. A conjecture of Proctor (Invent. Math.92,1988, 307–332) includes the statement that the odd symplectic Schur functions are actually characters ofSp(2n + 1, C). The purpose of the present note is to prove this.     

Multiplicity-Free Products of Schur Functions

. We classify all multiplicity-free products of Schur functions and all multiplicity-free products of characters of SL(n, C).     

Lagrange Inversion and Schur Functions

Macdonald defined an involution on symmetric functions by considering the Lagrange inverse of the generating function of the complete homogeneous symmetric functions. The main result we prove in this note is that the images of skew Schur functions under this involution are either Schur positive or Schur negative symmetric functions. The proof relies on the combinatorics of Lagrange inversion. We also present a q-analogue of this result, which is related to the q-Lagrange inversion formula of Andrews, Garsia, and Gessel, as well as the operator of Bergeron and Garsia.     

The Schur Algorithm for Generalized Schur Functions II: Jordan Chains and Transformations of Characteristic Functions

 In the first paper of this series (Daniel Alpay, Tomas Azizov, Aad Dijksma, and Heinz Langer: The Schur algorithm for generalized Schur functions I: coisometric realizations, Operator Theory: Advances and Applications 129 (2001), pp. 1–36) it was shown that for a generalized Schur function s(z), which is the characteristic function of a coisometric colligation V with state space being a Pontryagin space, the Schur transformation corresponds to a finite-dimensional reduction of the state space, and a finite-dimensional perturbation and compression of its main operator. In the present paper we show that these formulas can be explained using simple relations between V and the colligation of the reciprocal s(z)⁻¹ of the characteristic function s(z) and general factorization results for characteristic functions. Received October 31, 2001; in revised form August 21, 2002 RID="a" ID="a" Dedicated to Professor Edmund Hlawka on the occasion of his 85th birthday     

Interpolation, Schur Functions and Moment Problems

No Abstract.  .     

Unimodality of Differences of Specialized Schur Functions

The centered difference of principally specialized Schur functions

Quasisymmetric groups

One of the first problems in the theory of quasisymmetric and convergence groups was to investigate whether every quasisymmetric group that acts on the sphere , , is a quasisymmetric conjugate of a Möbius group that acts on . This was shown to be true for by Sullivan and Tukia, and it was shown to be wrong for by Tukia. It also follows from the work of Martin and of Freedman and Skora. In this paper we settle the case of by showing that any -quasisymmetric group is -quasisymmetrically conjugated to a Möbius group. The constant is a function .

     

Using Hook Schur Functions to Compute Matrix Cocharacters

We develop a new integration method based on hook Schur functions instead of Schur functions to compute the cocharacters of matrices. We then use this method to compute some of the multiplicities in the cocharacter sequence of 3 × 3 matrices.