arXiv subjects

Brendan Pawlowski

Publications and source records attributed to Brendan Pawlowski.

At least 19 recordsLinked to original sources

arXiv2026

Large products of double cosets for symmetric subgroups

We consider the problem of classifying pairs $x,y \in G$ such that $K x K y K = G$ where $G$ is a simple compact connected Lie group and $K$ is a symmetric subgroup. We give a necessary condition on $x,y$ for all simply connected $G$, and a complete classification when $G = \operatorname{SU}(n)$ and any symmetric $K \subseteq G$ except the type AIII case $K \simeq \operatorname{S}(\operatorname{U}(p) \times \operatorname{U}(n-p))$ with $p \neq n/2$. We also present some applications of these results to gate decompositions in quantum computing.

math.GR↗

arXiv2026

Real 3-qubit gate decompositions via triality

We show that any unimodular real 3-qubit gate can be expressed as the product of at most 14 CNOT gates plus single-qubit gates, improving on the bound of 16 CNOTs due to Wei and Di. Our method uses the exotic triality symmetry of $\operatorname{PSO}(8)$, and we explore some of the useful properties of this map in relation to the study of real 3-qubit gates.

quant-ph↗

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

Introduction to the Cohomology of the Flag Variety

One hundred years ago, Hilbert gave a list of important open problems in mathematics. His 15th problem asked for the development of a rigorous calculus explaining Schubert's enumerative results for intersecting varieties defined by rank conditions on vector spaces. Today by way of many contributions in algebraic topology, geometry, and combinatorics, we consider this solved. Yet, deep questions remain about the subtleties of actually carrying out the process. In this chapter, we hope to summarize the rigorous development of what has become known as Schubert calculus, with an eye toward computation. We discuss Grassmannians and flag varieties and their cohomology rings, following Monk's constructive algebraic approach. We derive formulas for Schur and Schubert polynomials, which represent cohomology classes of Schubert varieties. We hint at the vast literature in this area and point to the other references in the Handbook for more information. Finally, we identify open problems that remain a challenge even with modern tools at our fingertips in hopes of inspiring further contributions in this fascinating field. This is intended as the first chapter of a book entitled "Handbook of Combinatorial Algebraic Geometry: Subvarieties of the Flag Variety", a compendium of topics in the area. The book is being edited by Erik Insko, Martha Precup, and Ed Richmond. In addition to this introductory chapter, others will cover more advanced topics such as Kazhdan-Lusztig varieties, generalized smooth Schubert varieties, Richardson varieties and positroid varieties, spherical and torus orbit closures, spanning line configurations, different types of Hessenberg varieties, and generalizations to Kac-Moody flag varieties, each written by experts in those areas. We hope you enjoy this chapter enough to seek out the others, and that you send us any comments or corrections you find as you read this article!

math.CO↗

arXiv2024

Chromatic symmetric functions via the group algebra of $S_n$

We prove some Schur positivity results for the chromatic symmetric function $X_G$ of a (hyper)graph $G$, using connections to the group algebra of the symmetric group. The first such connection works for (hyper)forests $F$: we describe the Schur coefficients of $X_F$ in terms of eigenvalues of a product of Hermitian idempotents in the group algebra, one factor for each edge (a more general formula of similar shape holds for all chordal graphs). Our main application of this technique is to prove a conjecture of Taylor on the Schur positivity of certain $X_F$, which implies Schur positivity of the formal group laws associated to various combinatorial generating functions. We also introduce the pointed chromatic symmetric function $X_{G,v}$ associated to a rooted graph $(G,v)$. We prove that if $X_{G,v}$ and $X_{H,w}$ are positive in the generalized Schur basis of Strahov, then the chromatic symmetric function of the wedge sum of $(G,v)$ and $(H,w)$ is Schur positive.

math.CO↗

arXiv2022

Gröbner geometry for skew-symmetric matrix Schubert varieties

Matrix Schubert varieties are the closures of the orbits of $B\times B$ acting on all $n\times n$ matrices, where $B$ is the group of invertible lower triangular matrices. Extending work of Fulton, Knutson and Miller identified a Gröbner basis for the prime ideals of these varieties. They also showed that the corresponding initial ideals are Stanley-Reisner ideals of shellable simplicial complexes, and derived a related primary decomposition in terms of reduced pipe dreams. These results lead to a geometric proof of the Billey-Jockusch-Stanley formula for a Schubert polynomial, among many other applications. We define skew-symmetric matrix Schubert varieties to be the nonempty intersections of matrix Schubert varieties with the subspace of skew-symmetric matrices. In analogy with Knutson and Miller's work, we describe a natural generating set for the prime ideals of these varieties. We then compute a related Gröbner basis. Using these results, we identify a primary decomposition for the corresponding initial ideals involving certain fpf-involution pipe dreams. We show that these initial ideals are likewise the Stanley-Reisner ideals of shellable simplicial complexes. As an application, we give a geometric proof of an explicit generating function for symplectic Grothendieck polynomials. Our methods differ from Knutson and Miller's and can be used to give new proofs of some of their results, as we explain at the end of this article.

math.AG↗

arXiv2022

Principal specializations of Schubert polynomials in classical types

There is a remarkable formula for the principal specialization of a type A Schubert polynomial as a weighted sum over reduced words. Taking appropriate limits transforms this to an identity for the backstable Schubert polynomials recently introduced by Lam, Lee, and Shimozono. This note identifies some analogues of the latter formula for principal specializations of Schubert polynomials in classical types B, C, and D. We also describe some more general identities for Grothendieck polynomials. As a related application, we derive a simple proof of a pipe dream formula for involution Grothendieck polynomials.

math.CO↗

arXiv2021

Involution pipe dreams

Involution Schubert polynomials represent cohomology classes of $K$-orbit closures in the complete flag variety, where $K$ is the orthogonal or symplectic group. We show they also represent $T$-equivariant cohomology classes of subvarieties defined by upper-left rank conditions in the spaces of symmetric or skew-symmetric matrices. This geometry implies that these polynomials are positive combinations of monomials in the variables $x_i + x_j$, and we give explicit formulas of this kind as sums over new objects called involution pipe dreams. Our formulas are analogues of the Billey-Jockusch-Stanley formula for Schubert polynomials. In Knutson and Miller's approach to matrix Schubert varieties, pipe dream formulas reflect Gröbner degenerations of the ideals of those varieties, and we conjecturally identify analogous degenerations in our setting.

math.CO↗

arXiv2020

K-theory formulas for orthogonal and symplectic orbit closures

The complex orthogonal and symplectic groups both act on the complete flag variety with finitely many orbits. We study two families of polynomials introduced by Wyser and Yong representing the $K$-theory classes of the closures of these orbits. Our polynomials are analogous to the Grothendieck polynomials representing $K$-classes of Schubert varieties, and we show that like Grothendieck polynomials, they are uniquely characterized among all polynomials representing the relevant classes by a certain stability property. We show that the same polynomials represent the equivariant $K$-classes of symmetric and skew-symmetric analogues of Knutson and Miller's matrix Schubert varieties. We derive explicit expressions for these polynomials in special cases, including a Pfaffian formula relying on a more general degeneracy locus formula of Anderson. Finally, we show that taking an appropriate limit of our representatives recovers the $K$-theoretic Schur $Q$-functions of Ikeda and Naruse.

math.CO↗

arXiv2020

On some properties of symplectic Grothendieck polynomials

Grothendieck polynomials, introduced by Lascoux and Schützenberger, are certain $K$-theory representatives for Schubert varieties. Symplectic Grothendieck polynomials, described more recently by Wyser and Yong, represent the $K$-theory classes of orbit closures for the complex symplectic group acting on the complete flag variety. We prove a transition formula for symplectic Grothendieck polynomials and study their stable limits. We show that each of the $K$-theoretic Schur $P$-functions of Ikeda and Naruse arises from a limiting procedure applied to symplectic Grothendieck polynomials representing certain "Grassmannian" orbit closures.

math.CO↗

arXiv2020

The fraction of an $S_n$-orbit on a hyperplane

Huang, McKinnon, and Satriano conjectured that if $v \in \mathbb{R}^n$ has distinct coordinates and $n \geq 3$, then a hyperplane through the origin other than $\sum_i x_i = 0$ contains at most $2\lfloor n/2 \rfloor (n-2)!$ of the vectors obtained by permuting the coordinates of $v$. We prove this conjecture.

math.CO↗

arXiv2019

Spanning subspace configurations and representation stability

Let $V_1, V_2, V_3, \dots $ be a sequence of $\mathbb{Q}$-vector spaces where $V_n$ carries an action of $\mathfrak{S}_n$ for each $n$. {\em Representation stability} and {\em multiplicity stability} are two related notions of when the sequence $V_n$ has a limit. An important source of stability phenomena arises in the case where $V_n$ is the $d^{th}$ homology group (for fixed $d$) of the configuration space of $n$ distinct points in some fixed topological space $X$. We replace these configuration spaces with the variety $X_{n,k}$ of {\em spanning configurations} of $n$-tuples $(\ell_1, \dots, \ell_n)$ of lines in $\mathbb{C}^k$ which satisfy $\ell_1 + \cdots + \ell_n = \mathbb{C}^k$ as vector spaces. We study stability phenomena for the homology groups $H_d(X_{n,k})$ as the parameter $(n,k)$ grows.

math.CO↗

arXiv2019

Stanley symmetric functions for signed involutions

An involution in a Coxeter group has an associated set of involution words, a variation on reduced words. These words are saturated chains in a partial order first considered by Richardson and Springer in their study of symmetric varieties. In the symmetric group, involution words can be enumerated in terms of tableaux using appropriate analogues of the symmetric functions introduced by Stanley to accomplish the same task for reduced words. We adapt this approach to the group of signed permutations. We show that involution words for the longest element in the Coxeter group $C_n$ are in bijection with reduced words for the longest element in $A_n = S_{n+1}$, which are known to be in bijection with standard tableaux of shape $(n, n-1, \ldots, 2, 1)$.

math.CO↗

arXiv2019

Universal graph Schubert varieties

We consider the loci of invertible linear maps $f : \mathbb{C}^n \to {(\mathbb{C}^n)}^*$ together with pairs of flags $(E_\bullet, F_\bullet)$ in $\mathbb{C}^n$ such that the various restrictions $f : F_j \to E_i^*$ have specified ranks. Identifying an invertible linear map with its graph viewed as a point in a Grassmannian, we show that the closures of these loci have cohomology classes represented by the back-stable Schubert polynomials of Lam, Lee, and Shimozono. As a special case, we recover the result of Knutson, Lam, and Speyer that Stanley symmetric functions represent the classes of graph Schubert varieties. We consider similar loci where $f$ is restricted to be symmetric or skew-symmetric. Their classes are now given by back-stable versions of the polynomials introduced by Wyser and Yong to represent classes of orbit closures for the orthogonal and symplectic groups acting on the type A flag variety. Using degeneracy locus formulas of Kazarian and of Anderson and Fulton, we obtain new Pfaffian formulas for these polynomials in the vexillary case. We also give a geometric interpretation of the involution Stanley symmetric functions of Hamaker, Marberg, and the author: they represent classes of involution graph Schubert varieties in isotropic Grassmannians.

math.CO↗

arXiv2018

Pattern avoidance and quasisymmetric functions

Given a set of permutations Pi, let S_n(Pi) denote the set of permutations in the symmetric group S_n that avoid every element of Pi in the sense of pattern avoidance. Given a subset S of {1,...,n-1}, let F_S be the fundamental quasisymmetric function indexed by S. Our object of study is the generating function Q_n(Pi) = sum F_{Des sigma} where the sum is over all sigma in S_n(Pi) and Des sigma is the descent set of sigma. We characterize those Pi contained in S_3 such that Q_n(Pi) is symmetric or Schur nonnegative for all n. In the process, we show how each of the resulting Pi can be obtained from a theorem or conjecture involving more general sets of patterns. In particular, we prove results concerning symmetries, shuffles, and Knuth classes, as well as pointing out a relationship with the arc permutations of Elizalde and Roichman. Various conjectures and questions are mentioned throughout.

math.CO↗

arXiv2018

Fixed-point-free involutions and Schur P-positivity

The orbits of the symplectic group acting on the type A flag variety are indexed by the fixed-point-free involutions in a finite symmetric group. The cohomology classes of the closures of these orbits have polynomial representatives $\hat{\mathfrak{S}}^{\tt{FPF}}_z$ akin to Schubert polynomials. We show that the fixed-point-free involution Stanley symmetric functions $\hat{F}^{\tt{FPF}}_z$, which are stable limits of the polynomials $\hat{\mathfrak{S}}^{\tt{FPF}}_z$, are Schur $P$-positive. To do so, we construct an analogue of the Lascoux-Schützenberger tree, an algebraic recurrence that computes Schubert polynomials. As a byproduct of our proof, we obtain a Pfaffian formula of geometric interest for $\hat{\mathfrak{S}}^{\tt{FPF}}_z$ when $z$ is a fixed-point-free version of a Grassmannian permutation. We also classify the fixed-point-free involution Stanley symmetric functions that are single Schur $P$-functions, and show that the decomposition of $\hat{F}^{\tt{FPF}}_z$ into Schur $P$-functions is unitriangular with respect to dominance order on strict partitions. These results and proofs mirror previous work by the authors related to the orthogonal group action on the type A flag variety.

math.CO↗

arXiv2018

Cohomology classes of interval positroid varieties and a conjecture of Liu

To each finite subset of $\mathbb{Z}^2$ (a diagram), one can associate a subvariety of a complex Grassmannian (a diagram variety), and a representation of a symmetric group (a Specht module). Liu has conjectured that the cohomology class of a diagram variety is represented by the Frobenius characteristic of the corresponding Specht module. We give a counterexample to this conjecture. However, we show that for the diagram variety of a permutation diagram, Liu's conjectured cohomology class $σ$ is at least an upper bound on the actual class $τ$, in the sense that $σ- τ$ is a nonnegative linear combination of Schubert classes. To do this, we exhibit the appropriate diagram variety as a component in a degeneration of one of Knutson's interval positroid varieties (up to Grassmann duality). A priori, the cohomology classes of these interval positroid varieties are represented by affine Stanley symmetric functions. We give a different formula for these classes as ordinary Stanley symmetric functions, one with the advantage of being Schur-positive and compatible with inclusions between Grassmannians.

math.CO↗

arXiv2018

Catalan matroid decompositions of certain positroids

A positroid is the matroid of a matrix whose maximal minors are all nonnegative. Given a permutation $w$ in $S_n$, the matroid of a generic $n \times n$ matrix whose non-zero entries in row $i$ lie in columns $w(i)$ through $n+i$ is an example of a positroid. We enumerate the bases of such a positroid as a sum of certain products of Catalan numbers, each term indexed by the $123$-avoiding permutations above $w$ in Bruhat order. We also give a similar sum formula for their Tutte polynomials. These are both avatars of a structural result writing such a positroid as a disjoint union of matroids, each isomorphic to a direct sum of Catalan matroids and a matroid with one basis.

math.CO↗