# <   AlCoVE: an Algebraic Combinatorics Virtual Expedition   >

## Michela Ceria

In the theory of classical matroids, there are several known equivalent axiomatic systems that define a matroid, which are described as matroid cryptomorphisms. A $$q$$-matroid is a $$q$$-analogue of a matroid where subspaces play the role of the subsets in the classical theory. In this paper, that is a joint work with E. Byrne (University College Dublin) and R. Jurrius (Netherlands Defence Academy), we establish cryptomorphisms of $$q$$-matroids. In doing so we highlight the difference between classical theory and its $$q$$-analogue. We introduce a comprehensive set of $$q$$-matroid axiom systems and show cryptomorphisms between them and existing axiom systems of a $$q$$-matroid. These axioms are described as the rank, closure, basis, independence, dependence, circuit, hyperplane, flat, open space, spanning space, non-spanning space, and bi-colouring axioms.

## Sunita Chepuri

LP algebras, introduced by Lam and Pylyavskyy, are a generalization of cluster algebras. LP algebras are known to have the Laurent phenomenon, but positivity remains conjectural. Graph LP algebras are finite LP algebras encoded by a graph. For the graph LP algebra defined by a tree, we define a family of clusters called centered clusters. We prove positivity for these clusters by giving explicit formulas for each cluster variable. We also conjecture a combinatorial interpretation for these expansions.

## Oliver Clarke

Matching fields were introduced by Sturmfels and Zelevinsky to study certain Newton polytopes and more recently have been shown to give rise to toric degenerations of various families of varieties. Whenever a matching field gives rise to a toric degeneration, the associated polytope of the toric variety coincides with the matching field polytope. We study combinatorial mutations, which are analogues of cluster mutations for polytopes, of matching field polytopes and show that the property of giving rise to a toric degeneration of the Grassmannians, is preserved by mutation. Moreover the polytopes arising through mutations are Newton-Okounkov bodies for the Grassmannians with respect to certain full-rank valuations. We produce a large family of such polytopes, extending the family of so-called block diagonal matching fields.

## Mark Curiel

The enumeration of vector partitions is difficult, and, in many cases, a succinct closed form solution is effectively out-of-reach. As an alternative to enumerating vector partitions, we can seek other objects with some additional structure that agree in number. For instance, Kostant's partition function counts the number of ways to express a vector as a nonnegative integer linear combination of the positive roots of a Lie algebra. It has been shown that the number of juggling sequences with n balls gives a way to enumerate Kostant’s partition function of n times the highest root of a Lie algebra. For this poster presentation, we shall define another object called an $$n$$-star, which also enumerates Kostant’s partition function of $$n$$ times the highest root of a Lie algebra, and we provide a correspondence between these two objects through an example. Further, we shall give an upper bound for 2-stars up to certain symmetries and, hence, an upper bound for Kostant's partition function of the highest root of a Lie algebra multiplied by 2.

## Will Dana

Inspired by the infinite families of finite and affine root systems, we consider a "stretching" operation on general Coxeter diagrams which replaces a vertex with a path. We embed the associated root system into those of the stretched diagrams using a similar operation on individual roots. For a fixed root, we consider the growth of two structures as we lengthen the stretched path: the downset in the root poset and the arrangement of shards. We show that both eventually admit a uniform description, and deduce enumerative consequences: the size of the downset is eventually a polynomial, and the number of shards grows exponentially.

## Colin Defant

Let $$W$$ be an irreducible Coxeter group. We define the Coxeter pop-stack-sorting operator $$\mathsf{Pop}_W:W\to W$$ to be the map that fixes the identity element and sends each nonidentity element $$w$$ to the meet of the elements covered by $$w$$ in the right weak order. When $$W$$ is the symmetric group $$S_n$$, $$\mathsf{Pop}_W$$ coincides with the previously-studied pop-stack-sorting map. Generalizing a theorem about the pop-stack-sorting map due to Ungar, we show that $\sup\limits_{w\in W}\left|O_{\mathsf{Pop}_W}(w)\right|=h,$ where $$h$$ is the Coxeter number of $$W$$ (with $$h=\infty$$ if $$W$$ is infinite) and $$O_f(w)$$ denotes the forward orbit of $$w$$ under a map $$f$$. When $$W$$ is finite, this result is equivalent to the statement that the maximum number of terms appearing in the Brieskorn normal form of an element of $$W$$ is $$h-1$$. More generally, we define a map $$f:W\to W$$ to be compulsive if for every $$w\in W$$, $$f(w)$$ is less than or equal to $$\mathsf{Pop}_W(w)$$ in the right weak order. We show that if $$f$$ is compulsive, then $$\sup\limits_{w\in W}|O_f(w)|\leq h$$. This result is new even for symmetric groups. Claesson and Guðmundsson proved that for each fixed nonnegative integer $$t$$, the generating function that counts $$t$$-pop-stack-sortable permutations in symmetric groups is rational; we establish analogous results in types $$B$$ and $$\widetilde A$$. We also discuss a notion of pop-stack-sorting for an arbitrary complete meet-semilattice.

## An Introduction to Parking Functions

In 1966, Alan G. Konheim and Benjamin Weiss defined “parking functions” as follows: We have a one-way, one-lane street with $$n$$ parking spaces, numbered in consecutive order from 1 to $$n$$, and we have $$n$$ cars in line waiting to park. Each driver has a favorite (not necessarily distinct) parking spot, which we call its preference. We order these preferences in a preference vector. As each car parks, it drives to its preferred spot. If that spot is open, the car parks there; if not, it parks in the next available spot. If a preference vector allows all cars to park, we call it a parking function. In 1974, Henry O. Pollak proved the total number of parking functions of length $$n$$, meaning there are $$n$$ parking spots and $$n$$ cars, to be $$(n+1)^{(n-1)}$$. In this presentation, we describe a recursive formula, expound Pollak's succinct six-sentence proof of an explicit formula, and conclude with a discussion of other parking function generalizations.

## Max Hlavacek

In geometric, algebraic, and topological combinatorics, we often study the unimodality of combinatorial generating polynomials, which in turn follows when the polynomial is (real) stable, a property often deduced via the theory of interlacing polynomials. Motivated by a conjecture of Brenti and Welker on the real-rootedness of the $$h$$-polynomial of the barycentric subdivision of the boundary complex of a convex polytope, we introduce a framework for proving real-rootedness of $$h$$-polynomials for subdivisions of polytopal complexes by relating interlacing polynomials to shellability via the existence of so-called stable shellings. We show that any shellable cubical, or simplicial, complex admitting a stable shelling has barycentric and edgewise (when well-defined) subdivisions with real-rooted $$h$$-polynomials. Such shellings are shown to exist for well-studied families of cubical polytopes, giving a positive answer to the conjecture of Brenti and Welker in these cases. The framework of stable shellings is also applied to answer to a conjecture of Mohammadi and Welker on edgewise subdivisions in the case of shellable simplicial complexes

## Michael Joseph

The Lalanne–Kreweras involution (LK) on Dyck paths yields a bijective proof of the symmetry of two statistics: the number of valleys and the major index. When equivalently considered on the antichains of the type A root poset, the LK involution is closely related to an action called rowmotion. Specifically, there is a dihedral action generated by rowmotion and the LK involution. A birational lifting of rowmotion, first defined by Einstein and Propp, has been of recent interest. This motivated us to search for a birational liftings of the LK involution. In addition to finding previously undiscovered properties of the LK involution, we discovered that the key properties of the LK involution, including the symmetry of the number of valleys and major index, extend to the birational realm. This is joint work with Sam Hopkins.

## Harshit J Motwani

We study the connection between probability distributions satisfying certain conditional independence (CI) constraints, and point and line arrangements in incidence geometry. To a family of CI statements, we associate a polynomial ideal whose algebraic invariants are encoded in a hypergraph. The primary decompositions of these ideals give a characterisation of the distributions satisfying the original CI statements. Classically, these ideals are generated by 2-minors of a matrix of variables, however, in the presence of hidden variables, they contain higher degree minors. This leads to the study of the structure of determinantal hypergraph ideals whose decompositions can be understood in terms of point and line configurations in the projective space.

This is a joint work with Fatemeh Mohammadi and Oliver Clarke. Arxiv link

## Henri Mühle

For any northeast path $$\nu$$, we define two bivariate polynomials associated with the $$\nu$$-associahedron: the $$F$$- and the $$H$$-triangle. We prove combinatorially that we can obtain one from the other by an invertible transformation of variables. These polynomials generalize the classical $$F$$- and $$H$$-triangles of F. Chapoton in type $$A$$. Our proof is completely new and has the advantage of providing a combinatorial explanation of the nature of the relation between the $$F$$- and $$H$$-triangle.

## Sridhar Narayanan

We present a combinatorial interpretation for the multiplicity of the sign representation of the symmetric group $$S_n$$ in the restriction of irreducible representations of $$GL_n$$ corresponding to hook partitions by defining a sign-reversing involution on the terms of the generating function associated to the signed moment of character polynomials.

## Matteo Parisi

We discover a striking duality, T-duality, between two seemingly unrelated objects. The hypersimplex $$\Delta_{k+1,n}$$ is a polytope obtained as the image of the positive Grassmannian $$\mbox{Gr}^{\geq 0}_{k+1,n}$$ under the well-known moment map. Meanwhile, the amplituhedron $$\mathcal{A}_{n,k,2}$$ is the projection from the positive Grassmannian $$\mbox{Gr}^{\geq 0}_{k,n}$$ into the Grassmannian $$\mbox{Gr}_{k,k+2}$$ under the amplituhedron map. Introduced in the context of the physics of scattering amplitudes, it is not a polytope and has full dimension inside $$\mbox{Gr}_{k,k+2}$$. We draw novel connections between the two objects and prove many new properties of them. We exploit T-duality to relate their triangulations and generalised triangles—maximal cells in a triangulation. We subdivide $$\mathcal{A}_{n,k,2}$$ into chambers as $$\Delta_{k+1,n}$$ can be subdivided into simplices—both enumerated by Eulerian numbers. Moreover, we prove a main result about the hypersimplex and the positive tropical Grassmannian $$\mbox{Trop}^+ \mbox{Gr}_{k+1, n}$$, several conjectures on the amplituhedron, and find novel cluster-algebraic structures. This is based on recent joint work with M. Sherman-Bennett and L. K. Williams (Preprint, arXiv: 2104.08254), and a previous work with T. Lukowski and L. K. Williams (Preprint, arXiv:2002.06164).

## Stephan Pfannerer

Lusztig's fake degree is the generating polynomial for the major index of standard Young tableaux of a given shape. Results of Springer and James & Kerber imply that, mysteriously, its evaluation at a $$d$$-th primitive root of unity yields the number of border strip tableaux with all strips of size $$d$$, up to sign. This is essentially the special case of the Murnaghan-Nakayama rule for rectangular partitions as cycle type. We refine this result to standard Young tableaux and border strip tableaux with a given number of descents. To do so, we introduce a new descent statistic for border strip tableaux, extending the classical definition for standard Young tableaux.

## João Miguel Santos

The right and left key maps for Kashiwara-Nakashima tableaux are used to describe type C Demazure and opposite Demazure crystals, respectively. For instance, Fu-Lascoux non-symmetric Cauchy kernels expand into products of opposite Demazure characters and Demazure atoms. These key maps are related via the Lusztig involution and it is known that each type C key map can be computed using the Lecouvey-Sheats symplectic jeu de taquin. Motivated by Willis' direct way for computing type A right and left keys, we also give a way of computing symplectic right and left keys without the use of jeu de taquin.

## Miruna-Stefana Sorea

Our work is motivated by the theory of rough paths in stochastic analysis, where information from a path is usually encoded in a sequence of tensors with real entries, called the path signature. Using tools from representation theory, we prove an intriguing combinatorial identity in the shuffle algebra, that has a close connection to de Bruijn’s formula. This talk is based on joint work with Laura Colmenarejo and Joscha Diehl.

## Foster Tom

In recent years, Alexandersson and others proved combinatorial formulas for the Schur function expansion of the horizontal-strip LLT polynomial $$G_\lambda(x;q)$$ in some special cases. We associate a weighted graph $$\Pi$$ to $$\lambda$$ and we use it to express a linear relation among LLT polynomials. We apply this relation to prove an explicit combinatorial Schur-positive expansion of $$G_\lambda(x;q)$$ whenever $$\Pi$$ is triangle-free. We also prove that the largest power of $$q$$ in the LLT polynomial is the total edge weight of our graph.

## Lorenzo Venturello

Gorenstein algebras form an important subclass of Cohen-Macaulay algebras and they often arise from combinatorial objects. For instance, the Stanley-Reisner ring of a triangulated sphere is a Gorenstein algebra whose defining ideal is monomial. In this work we study a construction which associates to every pure simplicial complex $$D$$ a graded Gorenstein algebra $$R$$, whose defining ideal is generated by monomials and binomials. We turn our attention to the minimal free resolution of the residue field as an $$R$$-module. While the general behaviour of this infinite resolution is known to be wild, we show that in our setting a lot be read off from the simplicial complex $$D$$. As a main result we prove that $$R$$ is Koszul if and only if the simplicial complex $$D$$ is flag and Cohen-Macaulay over the residue field of $$R$$. One can use this correspondence to construct Gorenstein graded algebras which are Koszul if and only if the characteristic of the field is not in any prescribed finite list of primes. Moreover, we show that $$R$$ has a quadratic Gröbner basis if and only if $$D$$ is flag and shellable. We conclude by observing an unexpected behaviour of the gamma-polynomial associated to the numerator of the Hilbert series of $$R$$, which answers a question of Peeva and Stillman. This is joint work with Alessio D'Alì.