arXiv subjects

Stephanie van Willigenburg

Publications and source records attributed to Stephanie van Willigenburg.

At least 19 recordsLinked to original sources

arXiv2026

Equality of Dual Immaculate Functions Under Automorphisms

The dual immaculate functions are an example of a Schur-like basis in the algebra of quasisymmetric functions. We classify when the image of a dual immaculate function under one of the involutions $ρ, ψ, ω$ is equal to a dual immaculate function. As well, the leading term of the transition matrix is identified, and sufficient and necessary conditions for the existence of immaculate tableaux are determined. As a consequence, new maps and canonical tableaux associated with compositions are discovered.

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arXiv2026

A Schur-positivity classification for complete multipartite graphs

A graph is Schur-positive if its chromatic symmetric function expands non-negatively in the Schur basis. We determine a full Schur-positivity classification for complete multipartite graphs by showing that a complete multipartite graph $K_λ$ is Schur-positive if and only if either $λ_i\in \{1,2\}$ for all $i$ or $λ=(3,2^β)$ for some $β\ge 1$. These results extend earlier classifications for complete bipartite and complete tripartite graphs to full generality. Our proofs combine structural arguments ruling out most cases, with a combinatorial analysis of Schur coefficients for the remaining family $K_{(3,2^β)}$ via special rim hook $G$-tabloids. Along the way, we establish a simpler formula for Schur coefficients of incomparability graphs, which we then apply to compute the coefficients of interest in terms of non-increasing sequences.

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arXiv2026

Symmetric quasisymmetric Schur-like functions

In this paper we classify when (row-strict) dual immaculate functions and (row-strict) extended Schur functions, as well as their skew generalizations, are symmetric. We also classify when their natural variants, termed advanced functions, are symmetric. In every case our classification recovers classical skew Schur functions.

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arXiv2025

Minimal elements in the skew extended 0-Hecke poset

The row-strict 0-Hecke action on standard immaculate skew tableaux was studied by the present authors, who showed that it gives rise to a bounded poset, called the \emph{skew immaculate Hecke poset}, and consequently to a cyclic 0-Hecke module. It was further shown that the subposet of skew standard extended immaculate tableaux always has a unique maximal element, but may have multiple minimal elements. In this paper we focus on these minimal elements, completely classifying them for a family of skew shapes that we call \emph{lobsters}. Moreover, we prove that when the skew shape is connected, the skew extended Hecke poset does have a unique minimal element, thereby showing that the associated 0-Hecke module is cyclic for both the row-strict and the dual immaculate actions.

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arXiv2025

Chromatic quasisymmetric functions of the path graph

We show that the chromatic quasisymmetric function (CQF) of a labeled path graph on $n$ vertices is not symmetric unless the labeling is the natural labeling $1, 2, ..., n$ or its reverse $n, ..., 2, 1$. We also show that the star graph $K_{1, n-1}$ with $n\geq 3$ has a nonsymmetric CQF for all labelings.

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arXiv2025

Equality of skew Schur functions in noncommuting variables

The question of classifying when two skew Schur functions are equal is a substantial open problem, which remains unsolved for over a century. In 2022, Aliniaeifard, Li and van Willigenburg introduced skew Schur functions in noncommuting variables, $s_{(δ,D)}$, where $D$ is a connected skew diagram with $n$ boxes and $δ$ is a permutation in the symmetric group $S_n$. In this paper, we combine these two and classify when two skew Schur functions in noncommuting variables are equal: $s_{(δ,D)} = s_{(τ,T)}$ such that $D\ne T$ if and only if $D$ is a nonsymmetric ribbon, $T$ is the antipodal rotation of $D$ and $\overline{τ^{-1}δ}$ is an explicit bijection between two set partitions determined by $D$.

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arXiv2024

The chromatic symmetric function of a graph centred at a vertex

We discover new linear relations between the chromatic symmetric functions of certain sequences of graphs and apply these relations to find new families of e-positive unit interval graphs. Motivated by the results of Gebhard and Sagan, we revisit their ideas and reinterpret their equivalence relation in terms of a new quotient algebra of NCSym. We investigate the projection of the chromatic symmetric function $Y_G$ in noncommuting variables in this quotient algebra, which defines $y_{G : v}$, the chromatic symmetric function of a graph G centred at a vertex v. We then apply our methods to $y_{G :v}$ and find new families of unit interval graphs that are (e)-positive, a stronger condition than classical e-positivity, thus confirming new cases of the (3+1)-free conjecture of Stanley and Stembridge. In our study of $y_{G : v}$, we also describe methods of constructing new e-positive graphs from given $(e)$-positive graphs and classify the (e)-positivity of trees and cut vertices. We moreover construct a related quotient algebra of NCQSym to prove theorems relating the coefficients of $y_{G : v}$ to acyclic orientations of graphs, including a noncommutative refinement of Stanley's sink theorem.

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arXiv2024

Schur-positivity for generalized nets

A graph is Schur-positive if its chromatic symmetric function expands nonnegatively in the Schur basis. All claw-free graphs are conjectured to be Schur-positive. We introduce a combinatorial object corresponding to a graph G, called a special rim hook G-tabloid, which is a variation on the special rim hook tabloid. These objects can be employed to compute any Schur coefficient of the chromatic symmetric function of a graph. We construct sign-reversing maps on these special rim hook G-tabloids to obtain a recurrence relation for the Schur coefficients of a family of claw-free graphs called generalized nets, then we prove the entire family is Schur-positive. We subsequently determine an analogous recurrence relation for another, similar family of claw-free graphs. Thus, we demonstrate a new method for proving Schur-positivity of chromatic symmetric functions, which has the potential to be applied to make further progress toward the aforementioned conjecture.

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arXiv2024

The skew immaculate Hecke poset and 0-Hecke modules

The immaculate Hecke poset was introduced and investigated by Niese, Sundaram, van Willigenburg, Vega and Wang, who established the full poset structure, and determined modules for the 0-Hecke algebra action on immaculate and row-strict immaculate tableaux. In this paper, we extend their results by introducing the skew immaculate Hecke poset. We investigate the poset structure, and construct modules for the 0-Hecke algebra action on skew immaculate and skew row-strict immaculate tableaux, thus showing that the skew immaculate Hecke poset captures representation-theoretic information analogous to the immaculate Hecke poset. We also describe branching rules for the resulting skew modules.

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arXiv2024

The extra basis in noncommuting variables

We answer a question of Bergeron, Hohlweg, Rosas, and Zabrocki from 2006 to give a combinatorial description for the coproduct of the x-basis in the Hopf algebra of symmetric functions in noncommuting variables, NCSym, which arises in the theory of Grothendieck bialgebras. We achieve this using the theory of Hopf monoids and the Fock functor. We also determine combinatorial expansions of this basis in terms of the monomial and power sum symmetric functions in NCSym, and by taking the commutative image of the x-basis we discover a new multiplicative basis for the algebra of symmetric functions.

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arXiv2023

Generalized chromatic functions

We define vertex-colourings for edge-partitioned digraphs, which unify the theory of P-partitions and proper vertex-colourings of graphs. We use our vertex-colourings to define generalized chromatic functions, which merge the chromatic symmetric and quasisymmetric functions of graphs and generating functions of P-partitions. Moreover, numerous classical bases of symmetric and quasisymmetric functions, both in commuting and noncommuting variables, can be realized as special cases of our generalized chromatic functions. We also establish product and coproduct formulas for our functions. Additionally, we construct the new Hopf algebra of r-quasisymmetric functions in noncommuting variables, and apply our functions to confirm its Hopf structure, and establish natural bases for it.

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arXiv2023

0-Hecke modules for row-strict dual immaculate functions

We introduce a new basis of quasisymmetric functions, the row-strict dual immaculate functions. We construct a cyclic, indecomposable 0-Hecke algebra module for these functions. Our row-strict immaculate functions are related to the dual immaculate functions of Berg-Bergeron-Saliola-Serrano-Zabrocki (2014-15) by the involution $ψ$ on the ring of quasisymmetric functions. We give an explicit description of the effect of $ψ$ on the associated 0-Hecke modules, via the poset induced by the 0-Hecke action on standard immaculate tableaux. This remarkable poset reveals other 0-Hecke submodules and quotient modules, often cyclic and indecomposable, notably for a row-strict analogue of the extended Schur functions studied in Assaf-Searles (2019). Like the dual immaculate function, the row-strict dual immaculate function is the generating function of a suitable set of tableaux, corresponding to a specific descent set. We give a complete combinatorial and representation-theoretic picture by constructing 0-Hecke modules for the remaining variations on descent sets, and showing that \emph{all} the possible variations for generating functions of tableaux occur as characteristics of the 0-Hecke modules determined by these descent sets.

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arXiv2023

Row-strict dual immaculate functions

We define a new basis of quasisymmetric functions, the row-strict dual immaculate functions, as the generating function of a particular set of tableaux. We establish that this definition gives a function that can also be obtained by applying the $ψ$ involution to the dual immaculate functions of Berg, Bergeron, Saliola, Serrano, and Zabrocki (2014) and establish numerous combinatorial properties for our functions. We give an equivalent formulation of our functions via Bernstein-like operators, in a similar fashion to Berg et. al (2014). We conclude the paper by defining skew dual immaculate functions and hook dual immaculate functions and establishing combinatorial properties for them.

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arXiv2023

P-partition power sums

We develop the theory of weighted P-partitions, which generalises the theory of P-partitions from labelled posets to weighted labelled posets. We define the related generating functions in the natural way and compute their product, coproduct and other properties. As an application we introduce the basis of combinatorial power sums for the Hopf algebra of quasisymmetric functions and the reverse basis, both of which refine the power sum symmetric functions. These bases share many properties with the type 1 and type 2 quasisymmetric power sums introduced by Ballantine, Daugherty, Hicks, Mason and Niese, and moreover expand into the monomial basis of quasisymmetric functions with nonnegative integer coefficients. We prove formulas for products, coproducts and classical quasisymmetric involutions via the combinatorics of P-partitions, and give combinatorial interpretations for the coefficients when expanded into the monomial and fundamental bases.

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arXiv2022

Pieri rules for skew dual immaculate functions

In this paper we give Pieri rules for skew dual immaculate functions and their recently discovered row-strict counterparts. We establish our rules using a right-action analogue of the skew Littlewood-Richardson rule for Hopf algebras of Lam-Lauve-Sottile. We also obtain Pieri rules for row-strict (dual) immaculate functions.

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arXiv2022

Schur functions in noncommuting variables

In 2004 Rosas and Sagan asked whether there was a way to define a basis in the algebra of symmetric functions in noncommuting variables, NCSym, having properties analogous to the classical Schur functions. This was because they had constructed a partial such set that was not a basis. We answer their question by defining Schur functions in noncommuting variables using a noncommutative analogue of the Jacobi-Trudi determinant. Our Schur functions in NCSym map to classical Schur functions under commutation, and a subset of them indexed by set partitions forms a basis for NCSym. Amongst other properties, Schur functions in NCSym also satisfy a noncommutative analogue of the product rule for classical Schur functions in terms of skew Schur functions. We also show how Schur functions in NCSym are related to Specht modules, and naturally refine the Rosas-Sagan Schur functions. Moreover, by generalizing Rosas-Sagan Schur functions to skew Schur functions in the natural way, we prove noncommutative analogues of the Littlewood-Richardson rule and coproduct rule for them. Finally, we relate our functions to noncommutative symmetric functions by proving a subset of our functions are natural extensions of noncommutative ribbon Schur functions, and immaculate functions indexed by integer partitions.

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arXiv2022

Skew key polynomials and a generalized Littlewood-Richardson rule

Young's lattice is a partial order on integer partitions whose saturated chains correspond to standard Young tableaux, one type of combinatorial object that generates the Schur basis for symmetric functions. Generalizing Young's lattice, we introduce a new partial order on weak compositions that we call the key poset. Saturated chains in this poset correspond to standard key tableaux, the combinatorial objects that generate the key polynomials, a nonsymmetric polynomial generalization of the Schur basis. Generalizing skew Schur functions, we define skew key polynomials in terms of this new poset. Using weak dual equivalence, we give a nonnegative weak composition Littlewood-Richardson rule for the key expansion of skew key polynomials, generalizing the flagged Littlewood-Richardson rule of Reiner and Shimozono.

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arXiv2022

Slide multiplicity free key polynomials

Schubert polynomials are refined by the key polynomials of Lascoux-Schützenberger, which in turn are refined by the fundamental slide polynomials of Assaf-Searles. In this paper we determine which fundamental slide polynomial refinements of key polynomials, indexed by strong compositions, are multiplicity free. We also give a recursive algorithm to determine all terms in the fundamental slide polynomial refinement of a key polynomial indexed by a strong composition. From here, we apply our results to begin to classify which fundamental slide polynomial refinements, indexed by weak compositions, are multiplicity free. We completely resolve the cases when the weak composition has at most two nonzero parts or the sum has at most two nonzero terms.

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