arXiv subjects

Shu Xiao Li

Publications and source records attributed to Shu Xiao Li.

11 recordsLinked to original sources

arXiv2026

Towards equivalent thickened ribbon Schur functions

Two skew diagrams are defined to be equivalent if their corresponding skew Schur functions are equal. The equivalence classes of ribbons (edgewise connected skew diagrams without a $2\times 2$ block of boxes) have been classified by Billera, Thomas and van Willigenburg in 2006. In this paper, we provide a complete characterization of the equivalence classes of connected skew diagrams with exactly one inclusion-maximal $2\times m$ or $m\times 2$ block of boxes for every $m\ge 2$. In particular, the possible sizes of such equivalence classes are one, two, or four, demonstrating that a single $2\times m$ or $m\times 2$ block dramatically reduces the sizes of equivalence classes. Our result confirms special cases of the elusive conjecture on equivalent connected skew diagrams proposed by McNamara and van Willigenburg in 2009.

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arXiv2026

How to Use Deep Learning to Identify Sufficient Conditions: A Case Study on Stanley's $e$-Positivity

In a study, published in Nature, researchers from DeepMind and mathematicians demonstrated a general framework using machine learning to make conjectures in pure mathematics. Here, we build upon this framework to develop a method for identifying sufficient conditions that imply a given mathematical statement. As a demonstration, we apply this process to Stanley's problem of $e$-positivity of graphs, one of the problems that has been at the center of algebraic combinatorics for the past three decades. Guided by AI, we rediscover that one sufficient condition for a graph to be $e$-positive is that it is co-triangle-free. Based on Saliency Map analysis, we suggest that the classification of $e$-positive graphs is more related to continuous graph invariants rather than the discrete ones, which we support it with three conjectures. Furthermore, we show that the claw-free and claw-contractible-free graphs with 10 and 11 vertices are $e$-positive, resolving a conjecture by Dahlberg, Foley, and van Willigenburg.

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arXiv2026

Extending the descent-to-peak map and its applications

The descent-to-peak map serves as a bridge between algebra and combinatorics. We use it as a tool for proving the equidistribution of peak and valley sets of standard Young tableaux with a very short argument. We also introduce a new shuffle basis of quasisymmetric functions whose elements are eigenvectors of the descent-to-peak map. Using this basis, we then extend the notion of the peak algebra and of the descent-to-peak map to shuffle, tensor, and symmetric algebras.

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arXiv2025

The peak algebra in noncommuting variables

The well-known descent-to-peak map $Θ_{\mathrm{QSym}}$ for the Hopf algebra of quasisymmetric functions, $\mathrm{QSym}$, and the peak algebra $Π$ were originally defined by Stembridge in 1997. We introduce their noncommutative analogues, the labelled descent-to-peak map $Θ_{\mathrm{NCQSym}}$ for the Hopf algebra of quasisymmetric functions in noncommuting variables, $\mathrm{NCQSym}$, and the peak algebra in noncommuting variables $\mathrm{NC}Π$. Then, we define the Hopf algebra of Schur $Q$-functions in noncommuting variables. We show that our generalizations possess many properties analogous to their classical counterparts. Furthermore, we show that the coefficients in the expansion of certain elements of $\mathrm{NC}Π$ in the monomial basis of $\mathrm{NCQSym}$ satisfy the generalized Dehn-Sommerville equation of Bayer and Billera. In the end, we give representation-theoretic interpretations of the descent-to-peak map for the Hopf algebras of symmetric functions and noncommutative 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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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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arXiv2021

Stable set polytopes and their 1-skeleta

We characterize the edges of two classes of $0/1$-polytopes. The first class corresponds to the stable set polytope of a graph $G$ and includes chain polytopes of posets, some instances of matroid independence polytopes, as well as newly-defined polytopes whose vertices correspond to noncrossing set partitions. In analogy with matroid basis polytopes, the second class is obtained by considering the stable sets of maximal cardinality. We investigate how the class of $0/1$-polytopes whose edges satisfy our characterization is situated within the hierarchy of $0/1$-polytopes. This includes the class of matroid polytopes. We also study the diameter of these classes of polytopes and improve slightly on the Hirsch bound.

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arXiv2021

Theta maps for combinatorial Hopf algebras

There is a very natural and well-behaved Hopf algebra morphism from quasisymmetric functions to peak algebra, which we call it Theta map. This paper focuses on generalizing the peak algebra by constructing generalized Theta maps for an arbitrary combinatorial Hopf algebra. The image of Theta maps lies in the odd Hopf subalgebras, so we present a strategy to find odd Hopf subalgebra of any combinatorial Hopf algebra. We also give a combinatorial description of a family of Theta maps for Malvenuto-Reutenauer Hopf algebra of permutations $\operatorname{\mathsf{\mathfrak{S}Sym}}$ whose images are generalizations of the peak algebra. We also indicate a criterion to check whether a map is a Theta map. Moreover, precise descriptions of the Theta maps for the following Hopf algebras will be presented, Hopf subalgebras of quasisymmetric functions, commutative and co-commutative Hopf algebras, and theta maps for a Hopf algebra $\mathcal{V}$ on permutations.

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arXiv2021

Hopf algebras of parking functions and decorated planar trees

We construct three new combinatorial Hopf algebras based on the Loday-Ronco operations on planar binary trees. The first and second algebras are defined on planar trees and labeled planar trees extending the Loday-Ronco and Malvenuto-Reutenauer Hopf algebras respectively. We show that the latter is bidendriform which implies that is also free, cofree, and self-dual. The third algebra involves a new visualization of parking functions as decorated binary trees; it is also bidendriform, free, cofree, and self-dual, and therefore abstractly isomorphic to the algebra PQSym of Novelli and Thibon. We define partial orders on the objects indexing each of these three Hopf algebras, one of which, when restricting to (m+1)-ary trees, coarsens the m-Tamari order of Bergeron and Préville-Ratelle. We show that multiplication of dual fundamental basis elements are given by intervals in each of these orders. Finally, we use an axiomatized version of the techniques of Aguiar and Sottile on the Malvenuto-Reutenauer Hopf algebra to define a monomial basis on each of our Hopf algebras, and to show that comultiplication is cofree on the monomial elements. This in particular, implies the cofreeness of the Hopf algebra on planar trees. We also find explicit positive formulas for the multiplication on monomial basis and a cancellation-free and grouping-free formula for the antipode of monomial elements.

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arXiv2016

Ideals and quotients of diagonally quasi-symmetric functions

In 2004, J-C. Aval, F. Bergeron and N. Bergeron studied the algebra of diagonally quasi-symmetric functions $\operatorname{\mathsf{DQSym}}$ in the ring $\mathbb{Q}[\mathbf{x},\mathbf{y}]$ with two sets of variables. They made conjectures on the structure of the quotient $\mathbb{Q}[\mathbf{x},\mathbf{y}]/\langle\operatorname{\mathsf{DQSym}}^+\rangle$, which is a quasi-symmetric analogue of the diagonal harmonic polynomials. In this paper, we construct a Hilbert basis for this quotient when there are infinitely many variables i.e. $\mathbf{x}=x_1,x_2,\dots$ and $\mathbf{y}=y_1,y_2,\dots$. Then we apply this construction to the case where there are finitely many variables, and compute the second column of its Hilbert matrix.

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arXiv2016

Structure Constants for Immaculate Functions

The immaculate functions, $\mathfrak{S}_α$, were introduced as a Schur-like basis for $\operatorname{\mathsf{Nsym}}$. We investigate facts about their structure constants. These are analogues of Littlewood-Richardson coefficents. We will give a new proof of the left Pieri rule for the $\mathfrak{S}_α$, a translation invariance property for the structure coefficients of the $\mathfrak{S}_α$, and a counterexample to an $\mathfrak{S}_α$-analogue of the saturation conjecture.

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