arXiv subjects

Eric Marberg

Publications and source records attributed to Eric Marberg.

At least 19 recordsLinked to original sources

arXiv2026

Square root crystals and the square root of $B(\infty)$

We introduce a general monoidal category of $\mathbf{N}$-root crystals and then study the special case of square root $\mathfrak{gl}_n$-crystals. The latter objects include Yu's crystals on semistandard set-valued tableaux. Prior work of the first author, Tong, and Yu showed that regular square root $\mathfrak{gl}_n$-crystals can be a useful tool for proving Grothendieck positivity results. The objects studied here go beyond the regular case and allow us to construct a square root analog of the direct limit crystal $B(\infty)$. We give several descriptions of our square root of $B(\infty)$, using marginally large tableaux, the Lusztig or PBW parameterization, and the Nakashima--Zelevinsky polyhedral model. We show that this crystal has a simple character formula, exhibits a nontrivial Demazure filtration, and recovers Yu's semistandard set-valued tableau crystals after taking appropriate tensor products. We also investigate a number of differences between square root crystals and classical crystal constructions.

math.RT↗

arXiv2026

On some Grothendieck expansions

The orthogonal and symplectic groups act on the complete flag variety with finitely many orbits. The orthogonal Grothendieck polynomials $\mathfrak{G}^{\mathsf{O}}_z$ and symplectic Grothendieck polynomials $\mathfrak{G}^{\mathsf{Sp}}_z$ are distinguished representatives for the $K$-theory classes of the corresponding orbit closures. There is a simple formula to expand $\mathfrak{G}^{\mathsf{Sp}}_z$ as a linear combination of Grothendieck polynomials $\mathfrak{G}^{(β)}_w$, which represent the $K$-theory classes of Schubert varieties. Although the constructions of $\mathfrak{G}^{\mathsf{Sp}}_z$ and $\mathfrak{G}^{\mathsf{O}}_z$ are similar, finding the $\mathfrak{G}^{(β)}$-expansion of $\mathfrak{G}^{\mathsf{O}}_z$ or even computing $\mathfrak{G}^{\mathsf{O}}_z$ is much harder. If $z$ is vexillary then $\mathfrak{G}^{\mathsf{O}}_z$ has a nonnegative $\mathfrak{G}^{(β)}$-expansion, but the associated coefficients are mostly unknown. This paper derives several new formulas for $\mathfrak{G}^{\mathsf{O}}_z$ and its $\mathfrak{G}^{(β)}$-expansion when $z$ is vexillary. Among other applications, we prove that the latter expansion has a nontrivial stability property.

math.CO↗

arXiv2026

Brion atoms for classical types

Let $G$ be a classical group defined over the complex numbers with a Borel subgroup $B$. Choose a holomorphic involution of $G$ and let $K$ be its set of fixed points. The group $K$ acts on the flag variety $G/B$ with finitely many orbits and Brion has derived a general formula for the cohomology classes of the corresponding orbit closures as linear combinations of Schubert classes. This article provides a uniform description of the sets of Weyl group elements (which we refer to as Brion atoms) that index the terms in this formula. This builds on prior work addressing types A, B, and C. The main novelty of our results is a thorough treatment of type D. As one application, we introduce a notion of involution Schubert polynomials for all classical types and present several conjectures related to these objects.

math.RT↗

arXiv2026

Key and Lascoux polynomials for symmetric orbit closures

We introduce shifted analogues of key polynomials related to symplectic and orthogonal orbit closures in the complete flag variety. Our definitions are given by applying isobaric divided difference operators to the analogues of Schubert polynomials for orbit closures that correspond to dominant involutions. We show that our shifted key polynomials are linear combinations of key polynomials with nonnegative integer coefficients. We also prove that they are partial versions of the classical Schur $P$- and $Q$-polynomials. Finally, we examine $K$-theoretic generalizations of these functions, which give shifted forms of Lascoux polynomials. In the symplectic case, these generalizations are partial versions of the $GP$-polynomials introduced by Ikeda and Naruse. Besides developing basic properties, we identify a number of conjectures and open problems.

math.CO↗

arXiv2026

Grothendieck positivity for normal square root crystals

Normal crystals (also known as Stembridge crystals) are commonly used to establish the Schur positivity of symmetric functions, as their characters are sums of Schur polynomials. In this paper, we develop a combinatorial framework for a novel family of objects called normal square root crystals, which are closely related to symmetric Grothendieck functions, the $K$-theoretic analogue of Schur functions. Among other applications, this tool leads to a new proof of Buch's combinatorial rule for the multiplication of symmetric Grothendieck functions. The definition of a normal square root crystal, originally formulated by the first two authors, largely mirrors that of normal crystals. Our main result is to show that the character of such a crystal is always a sum of symmetric Grothendieck polynomials. The proof relies on an unexpected connection between the raising operators for our crystals and the Hecke insertion algorithm developed by Buch, Kresch, Shimozono, Tamvakis, and Yong.

math.CO↗

arXiv2026

Ideal transition systems

We study an inductive method of computing initial ideals and Gröbner bases for families of ideals in a polynomial ring. This method starts from a given set of pairs $(I,J)$ where $I$ is any ideal and $J$ is a monomial ideal contained in the initial ideal of $I$. These containments become a system of equalities if one can establish a particular transition recurrence among the chosen ideals. We describe explicit constructions of such systems in two motivating cases -- namely, for the ideals of matrix Schubert varieties and their skew-symmetric analogues. Despite many formal similarities with these examples, for the symmetric versions of matrix Schubert varieties, it is an open problem to construct the same kind of transition system. We present several conjectures that would follow from such a construction, while also discussing the special obstructions arising in the symmetric case.

math.AC↗

arXiv2025

Positive specializations of K-theoretic Schur P- and Q-functions

Yeliussizov has classified the positive specializations of symmetric Grothendieck functions, defined in several different ways, providing a K-theoretic lift of the classical Edrei-Thoma theorem. This note studies the analogous classification problem for Ikeda and Naruse's K-theoretic Schur P- and Q-functions, which are the shifted versions of symmetric Grothendieck functions. Our results extend a shifted variant of the Edrei-Thoma theorem due to Nazarov. We also discuss an application to the problem of determining the extreme harmonic functions on a filtered version of the shifted Young lattice.

math.CO↗

arXiv2025

Classical double Grothendieck transitions

Kirillov and Naruse have constructed double Grothendieck polynomials to represent the equivariant K-theory classes of Schubert varieties in the complete flag manifolds of types B, C, and D. We derive a recursive formula for these polynomials, extending certain K-theoretic transition equations known in type A to all classical types. As an application, we obtain an identity that expands the K-Stanley symmetric functions in types B, C, and D into positive linear combinations of K-theoretic Schur P- and Q-functions. We also resolve several positivity conjectures related to the skew generalizations of the latter functions.

math.RT↗

arXiv2025

Atoms for signed permutations

There is a natural analogue of weak Bruhat order on the involutions in any Coxeter group. The saturated chains of intervals in this order correspond to reduced words for a certain set of group elements called atoms. Brion gives a general formula for the cohomology class of a $K$-orbit closure in an arbitrary flag variety, where $K$ is a symmetric subgroup of a complex algebraic group. In type A, the terms in this formula are indexed by atoms for permutations. We study the combinatorics of atoms for involutions in the group of signed permutations. In particular, we give a compact description of the atom set for any signed involution and endow it with the structure of a graded poset. Our main result, as an application, is to identify explicitly the terms in Brion's cohomology formula in types B and C. These descriptions apply to all $K$-orbits in these types and are the first of their kind outside of type A.

math.CO↗

arXiv2025

Crystals for shifted key polynomials

This article continues our study of $P$- and $Q$-key polynomials, which are (non-symmetric) "partial" Schur $P$- and $Q$-functions as well as "shifted" versions of key polynomials. Our main results provide a crystal interpretation of $P$- and $Q$-key polynomials, namely, as the characters of certain connected subcrystals of normal crystals associated to the queer Lie superalgebra $\mathfrak{q}_n$. In the $P$-key case, the ambient normal crystals are the $\mathfrak{q}_n$-crystals studied by Grantcharov et al., while in the $Q$-key case, these are replaced by the extended $\mathfrak{q}_n$-crystals recently introduced by the first author and Tong. Using these constructions, we propose a crystal-theoretic lift of several conjectures about the decomposition of involution Schubert polynomials into $P$- and $Q$-key polynomials. We verify these generalized conjectures in a few special cases. Along the way, we establish some miscellaneous results about normal $\mathfrak{q}_n$-crystals and Demazure $\mathfrak{gl}_n$-crystals.

math.CO↗

arXiv2025

Primed decomposition tableaux and extended queer crystals

Our previous work introduced a category of extended queer crystals, whose connected normal objects have unique highest weight elements and characters that are Schur $Q$-polynomials. The initial models for such crystals were based on semistandard shifted tableaux. Here, we introduce a simpler construction using certain "primed" decomposition tableaux, which slightly generalize the decomposition tableaux used in work of Grantcharov et al. This leads to a new, shorter proof of the highest weight properties of the normal subcategory of extended queer crystals. Along the way, we analyze a primed extension of Grantcharov et al.'s insertion scheme for decomposition tableaux.

math.CO↗

arXiv2025

Kromatic quasisymmetric functions

We provide a construction for the kromatic symmetric function $\overline{X}_G$ of a graph introduced by Crew, Pechenik, and Spirkl using combinatorial (linearly compact) Hopf algebras. As an application, we show that $\overline{X}_G$ has a positive expansion into multifundamental quasisymmetric functions. We also study two related quasisymmetric $q$-analogues of $\overline{X}_G$, which are $K$-theoretic generalizations of the quasisymmetric chromatic function of Shareshian and Wachs. We classify exactly when one of these analogues is symmetric. For the other, we derive a positive expansion into symmetric Grothendieck functions when $G$ is the incomparability graph of a natural unit interval order.

math.CO↗

arXiv2024

Crystals for set-valued decomposition tableaux

We describe two crystal structures on set-valued decomposition tableaux. These provide the first examples of interesting "$K$-theoretic" crystals on shifted tableaux. Our first crystal is modeled on a similar construction of Monical, Pechenik, and Scrimshaw for semistandard (unshifted) set-valued tableaux. Our second crystal is adapted from the "square root" operators introduced by Yu on the same set. Neither of our shifted crystals is normal, but we conjecture that our second construction is connected with a unique highest weight element. These results lead to partial progress on a conjectural formula of Cho--Ikeda for $K$-theoretic Schur $P$-functions. We also study a new category of "square root crystals" that includes our second construction and Yu's set-valued tableau crystals as examples. We observe that Buch's formula for the coefficients expanding products of symmetric Grothendieck functions has a simple description in terms of the tensor product for this category.

math.CO↗

arXiv2024

Insertion algorithms for Gelfand $S_n$-graphs

The two tableaux assigned by the Robinson--Schensted correspondence are equal if and only if the input permutation is an involution, so the RS algorithm restricts to a bijection between involutions in the symmetric group and standard tableaux. Beissinger found a concise way of formulating this restricted map, which involves adding an extra cell at the end of a row after a Schensted insertion process. We show that by changing this algorithm slightly to add cells at the end of columns rather than rows, one obtains a different bijection from involutions to standard tableaux. Both maps have an interesting connection to representation theory. Specifically, our insertion algorithms classify the molecules (and conjecturally the cells) in the pair of $W$-graphs associated to the unique equivalence class of perfect models for a generic symmetric group.

math.CO↗

arXiv2023

Shifted insertion algorithms for primed words

This article studies some new insertion algorithms that associate pairs of shifted tableaux to finite integer sequences in which certain terms may be primed. When primes are ignored in the input word these algorithms reduce to known correspondences, namely, a shifted form of Edelman-Greene insertion, Sagan-Worley insertion, and Haiman's shifted mixed insertion. These maps have the property that when the input word varies such that one output tableau is fixed, the other output tableau ranges over all (semi)standard tableaux of a given shape with no primed diagonal entries. Our algorithms have the same feature, but now with primes allowed on the main diagonal. One application of this is to give another Littlewood-Richardson rule for products of Schur $Q$-functions. It is hoped that there will exist set-valued generalizations of our bijections that can be used to understand products of $K$-theoretic Schur $Q$-functions.

math.CO↗

arXiv2023

Highest weight crystals for Schur Q-functions

Work of Grantcharov et al. develops a theory of abstract crystals for the queer Lie superalgebra $\mathfrak{q}_n$. Such $\mathfrak{q}_n$-crystals form a monoidal category in which the connected normal objects have unique highest weight elements and characters that are Schur $P$-polynomials. This article studies a modified form of this category, whose connected normal objects again have unique highest weight elements but now possess characters that are Schur $Q$-polynomials. The crystals in this category have some interesting features not present for ordinary $\mathfrak{q}_n$-crystals. For example, there is an extra crystal operator, a different tensor product, and an action of the hyperoctahedral group exchanging highest and lowest weight elements. There are natural examples of $\mathfrak{q}_n$-crystal structures on certain families of shifted tableaux and factorized reduced words. We describe extended forms of these structures that give similar examples in our new category.

math.RT↗

arXiv2023

A symplectic refinement of shifted Hecke insertion

Buch, Kresch, Shimozono, Tamvakis, and Yong defined Hecke insertion to formulate a combinatorial rule for the expansion of the stable Grothendieck polynomials $G_π$ indexed by permutations in the basis of stable Grothendieck polynomials $G_λ$ indexed by partitions. Patrias and Pylyavskyy introduced a shifted analogue of Hecke insertion whose natural domain is the set of maximal chains in a weak order on orbit closures of the orthogonal group acting on the complete flag variety. We construct a generalization of shifted Hecke insertion for maximal chains in an analogous weak order on orbit closures of the symplectic group. As an application, we identify a combinatorial rule for the expansion of "orthogonal" and "symplectic" shifted analogues of $G_π$ in Ikeda and Naruse's basis of $K$-theoretic Schur $P$-functions.

math.CO↗

arXiv2023

Shifted combinatorial Hopf algebras from $K$-theory

In prior joint work with Lewis, we developed a theory of enriched set-valued $P$-partitions to construct a $K$-theoretic generalization of the Hopf algebra of peak quasisymmetric functions. Here, we situate this object in a diagram of six Hopf algebras, providing a shifted version of the diagram of $K$-theoretic combinatorial Hopf algebras studied by Lam and Pylyavskyy. This allows us to describe new $K$-theoretic analogues of the classical peak algebra. We also study the Hopf algebras generated by Ikeda and Naruse's $K$-theoretic Schur $P$- and $Q$-functions, as well as their duals. Along the way, we derive several product, coproduct, and antipode formulas and outline a number of open problems and conjectures.

math.CO↗