Schubert Polynomials Research Articles

Combinatorics of Double Grothendieck Polynomials

We give a thorough combinatorial treatment of double Grothendieck polynomials that is accessible to anyone with a basic knowledge of algebra and combinatorics. The text includes many new combinatorial models for these and related polynomials and provides detailed combinatorial proofs. We generalize the Giambelli formula for double Schubert polynomials to a $k$-theoretic version and our proof specializes to a new combinatorial proof of the former.

Upper bounds of dual flagged Weyl characters

For a subset D of boxes in an n×n square grid, let χD(x) denote the dual character of the flagged Weyl module associated to D. It is known that χD(x) specifies to a Schubert polynomial (resp., a key polynomial) in the case when D is the Rothe diagram of a permutation (resp., the skyline diagram of a composition). One can naturally define a lower and an upper bound of χD(x). Mészáros, St. Dizier and Tanjaya conjectured that χD(x) attains the upper bound if and only if D avoids a certain single subdiagram. We provide a proof of this conjecture.

Combinatorics of semi-toric degenerations of Schubert varieties in type C

An approach to Schubert calculus is to realize Schubert classes as concrete combinatorial objects such as Schubert polynomials. Using the polytope ring of the Gelfand–Tsetlin polytopes, Kiritchenko–Smirnov–Timorin realized each Schubert class as a sum of reduced Kogan faces. The first named author introduced a generalization of reduced Kogan faces to symplectic Gelfand–Tsetlin polytopes using a semi-toric degeneration of a Schubert variety, and extended the result of Kiritchenko–Smirnov–Timorin to type C case. In this paper, we introduce a combinatorial model to this type C generalization using a kind of pipe dream with self-crossings. As an application, we prove that the type C generalization can be constructed by skew mitosis operators.

James reduced product schemes and double quasisymmetric functions

Symmetric function theory is a key ingredient in the Schubert calculus of Grassmannians. Quasisymmetric functions are analogues that are similarly central to algebraic combinatorics, but for which the associated geometry is poorly developed. Baker and Richter (2008) showed that QSym manifests topologically as the cohomology ring of the loop suspension of infinite projective space or equivalently of its combinatorial homotopy model, the James reduced product JCP∞. In recent work, we used this viewpoint to develop topologically-motivated bases of QSym and initiate a Schubert calculus for JCP∞ in both cohomology and K-theory.Here, we study the torus-equivariant cohomology of JCP∞. We identify a cellular basis and introduce double monomial quasisymmetric functions as combinatorial representatives, analogous to the factorial Schur functions and double Schubert polynomials of classical Schubert calculus. We also provide a combinatorial Littlewood–Richardson-type rule for the structure coefficients of this basis.Furthermore, we introduce an algebro-geometric analogue of the James reduced product construction. In particular, we prove that the James reduced product of a complex (quasi-)projective variety also carries the structure of a (quasi-)projective variety.

Complements of Schubert polynomials

Let Sw(x) be the Schubert polynomial for a permutation w of {1,2,…,n}. For any given composition μ, we say that xμSw(x−1) is the complement of Sw(x) with respect to μ. When each part of μ is equal to n−1, Huh, Matherne, Mészáros and St. Dizier proved that the normalization of xμSw(x−1) is a Lorentzian polynomial. They further conjectured that the normalization of Sw(x) is Lorentzian. It can be shown that if there exists a composition μ such that xμSw(x−1) is a Schubert polynomial, then the normalization of Sw(x) will be Lorentzian. This motivates us to investigate the problem of when xμSw(x−1) is a Schubert polynomial. We show that if xμSw(x−1) is a Schubert polynomial, then μ must be a partition. We also consider the case when μ is the staircase partition δn=(n−1,…,1,0), and obtain that xδnSw(x−1) is a Schubert polynomial if and only if w avoids the patterns 132 and 312. A conjectured characterization of when xμSw(x−1) is a Schubert polynomial is proposed.

A Molev-Sagan type formula for double Schubert polynomials

We give a Molev-Sagan type formula for computing the product Su(x;y)Sv(x;z) of two double Schubert polynomials in different sets of coefficient variables where the descents of u and v satisfy certain conditions that encompass Molev and Sagan's original case and conjecture positivity in the general case. Additionally, we provide a Pieri formula for multiplying an arbitrary double Schubert polynomial Su(x;y) by a factorial elementary symmetric polynomial Ep,k(x;z). Both formulas remain positive in terms of the negative roots when we set y=z, so in particular this gives a new equivariant Littlewood-Richardson rule for the Grassmannian, and more generally a positive formula for multiplying a factorial Schur polynomial sλ(x1,…,xm;y) by a double Schubert polynomial Sv(x1,…,xp;y) such that m≥p. An additional new result we present is a combinatorial proof of a conjecture of Kirillov of nonnegativity of the coefficients of skew Schubert polynomials, and we conjecture a weight-preserving bijection between a modification of certain diagrams used in our formulas and RC-graphs/pipe dreams arising in formulas for double Schubert polynomials.

Monk rules for type GLn Macdonald polynomials

In this paper we give Monk rules for Macdonald polynomials which are analogous to the Monk rules for Schubert polynomials. These formulas are similar to the formulas given by Baratta [6], but our method of derivation is to use Cherednik's intertwiners. Deriving Monk rules by this technique addresses the relationship between the work of Baratta and the product formulas of Yip [19]. Specializations of the Monk formula's at q=0 and/or t=0 provide Monk rules for Iwahori-spherical polynomials and for finite and affine key polynomials.

Bijective Proofs of Monk's rule for Schubert and Double Schubert Polynomials with Bumpless Pipe Dreams

We give bijective proofs of Monk's rule for Schubert and double Schubert polynomials computed with bumpless pipe dreams. In particular, they specialize to bijective proofs of transition and cotransition formulas of Schubert and double Schubert polynomials, which can be used to establish bijections with ordinary pipe dreams.

A representation-theoretic interpretation of positroid classes

A positroid is the matroid of a real matrix with nonnegative maximal minors, a positroid variety is the closure of the locus of points in a complex Grassmannian whose matroid is a fixed positroid, and a positroid class is the cohomology class Poincaré dual to a positroid variety. We define a family of representations of general linear groups whose characters are symmetric polynomials representing positroid classes. These representations are certain diagram Schur modules in the sense of James and Peel. This gives a new algebraic interpretation of the Schubert structure constants for the product of a Schubert polynomial and Schur polynomial, and of the 3-point Gromov-Witten invariants for Grassmannians, proving a conjecture of Postnikov. As a byproduct, we obtain an effective algorithm for decomposing positroid classes into Schubert classes.

Grothendieck-to-Lascoux expansions

We establish the conjecture of Reiner and Yong for an explicit combinatorial formula for the expansion of a Grothendieck polynomial into the basis of Lascoux polynomials. This expansion is a subtle refinement of its symmetric function version due to Buch, Kresch, Shimozono, Tamvakis, and Yong, which gives the expansion of stable Grothendieck polynomials indexed by permutations into Grassmannian stable Grothendieck polynomials. Our expansion is the K K -theoretic analogue of that of a Schubert polynomial into Demazure characters, whose symmetric analogue is the expansion of a Stanley symmetric function into Schur functions. Our expansions extend to flagged Grothendieck polynomials.

Tableau formulas for skew Schubert polynomials

AbstractThe skew Schubert polynomials are those that are indexed by skew elements of the Weyl group, in the sense of Tamvakis [J. reine angew. Math. 652 (2011), 207–244]. We obtain tableau formulas for the double versions of these polynomials in all four classical Lie types, where the tableaux used are fillings of the associated skew Young diagram. These are the first such theorems for symplectic and orthogonal Schubert polynomials, even in the single case. We also deduce tableau formulas for double Schur, double theta, and double eta polynomials, in their specializations as double Grassmannian Schubert polynomials. The latter results generalize the tableau formulas for symmetric (and single) Schubert polynomials due to Littlewood (in type A) and the author (in types B, C, and D).

A Tableau Formula for Vexillary Schubert Polynomials in Type C

Ikeda-Mihalcea-Naruse's double Schubert polynomials [Adv. Math. 226 (2011)] represent the equivariant cohomology classes of Schubert varieties in the type C flag varieties. The goal of this paper is to obtain a new tableau formula of these polynomials associated to vexillary signed permutations introduced by Anderson-Fulton. To achieve that goal, we introduce flagged factorial (Schur) $Q$-functions, combinatorially defined functions in terms of marked shifted tableaux for flagged strict partitions, and prove their Schur-Pfaffian formula. As an application, we also obtain a new combinatorial formula of factorial $Q$-functions of Ivanov in which monomials bijectively corresponds to flagged marked shifted tableaux.

Double Schubert polynomials do have saturated Newton polytopes

Abstract We prove that double Schubert polynomials have the saturated Newton polytope property. This settles a conjecture by Monical, Tokcan and Yong. Our ideas are motivated by the theory of multidegrees. We introduce a notion of standardization of ideals that enables us to study nonstandard multigradings. This allows us to show that the support of the multidegree polynomial of each Cohen–Macaulay prime ideal in a nonstandard multigrading, and in particular, that of each Schubert determinantal ideal is a discrete polymatroid.

Kohnert's rule for flagged Schur modules

Flagged Schur modules generalize the irreducible representations of the general linear group under the action of the Borel subalgebra. Their characters include many important generalizations of Schur polynomials, such as Demazure characters, flagged skew Schur polynomials, and Schubert polynomials. In this paper, we prove the characters of flagged Schur modules can be computed using a simple combinatorial algorithm due to Kohnert if and only if the indexing diagram is northwest. This gives a new proof that characters of flagged Schur modules are nonnegative sums of Demazure characters and gives a representation theoretic interpretation for Kohnert polynomials.

Order recognition by Schubert polynomials generated by optical near-field statistics via nanometre-scale photochromism

Irregular spatial distribution of photon transmission through a photochromic crystal photoisomerized by a local optical near-field excitation was previously reported, which manifested complex branching processes via the interplay of material deformation and near-field photon transfer therein. Furthermore, by combining such naturally constructed complex photon transmission with a simple photon detection protocol, Schubert polynomials, the foundation of versatile permutation operations in mathematics, have been generated. In this study, we demonstrated an order recognition algorithm inspired by Schubert calculus using optical near-field statistics via nanometre-scale photochromism. More specifically, by utilizing Schubert polynomials generated via optical near-field patterns, we showed that the order of slot machines with initially unknown reward probability was successfully recognized. We emphasized that, unlike conventional algorithms, the proposed principle does not estimate the reward probabilities but exploits the inversion relations contained in the Schubert polynomials. To quantitatively evaluate the impact of Schubert polynomials generated from an optical near-field pattern, order recognition performances were compared with uniformly distributed and spatially strongly skewed probability distributions, where the optical near-field pattern outperformed the others. We found that the number of singularities contained in Schubert polynomials and that of the given problem or considered environment exhibited a clear correspondence, indicating that superior order recognition is attained when the singularity of the given situations is presupposed. This study paves way for physical computing through the interplay of complex natural processes and mathematical insights gained by Schubert calculus.

Schubert Products for Permutations with Separated Descents

Abstract We say that two permutations $\pi $ and $\rho $ have separated descents at position $k$ if $\pi $ has no descents before position $k$ and $\rho $ has no descents after position $k$. We give a counting formula, in terms of reduced word tableaux, for computing the structure constants of products of Schubert polynomials indexed by permutations with separated descents, and recognize that these structure constants are certain Edelman–Greene coefficients. Our approach uses generalizations of Schützenberger’s jeu de taquin algorithm and the Edelman–Greene correspondence via bumpless pipe dreams.

Schubert polynomials as projections of Minkowski sums of Gelfand-Tsetlin polytopes

Gelfand-Tsetlin polytopes are classical objects in algebraic combinatorics arising in the representation theory of \(\mathfrak{gl}_n(\mathbb{C})\). The integer point transform of the Gelfand-Tsetlin polytope \(\mathrm{GT}(\lambda)\) projects to the Schur function \(s_{\lambda}\). Schur functions form a distinguished basis of the ring of symmetric functions; they are also special cases of Schubert polynomials \(\mathfrak{S}_{w}\) corresponding to Grassmannian permutations. For any permutation \(w \in S_n\) with column-convex Rothe diagram, we construct a polytope \(\mathcal{P}_{w}\) whose integer point transform projects to the Schubert polynomial \(\mathfrak{S}_{w}\). Such a construction has been sought after at least since the construction of twisted cubes by Grossberg and Karshon in 1994, whose integer point transforms project to Schubert polynomials \(\mathfrak{S}_{w}\) for all \(w \in S_n\). However, twisted cubes are not honest polytopes; rather one can think of them as signed polytopal complexes. Our polytope \(\mathcal{P}_{w}\) is a convex polytope, namely it is a Minkowski sum of Gelfand-Tsetlin polytopes of varying sizes. When the permutation \(w\) is Grassmannian, the Gelfand-Tsetlin polytope is recovered. We conclude by showing that the Gelfand-Tsetlin polytope is a flow polytope.Mathematics Subject Classifications: 05E05Keywords: Schubert polynomials, Gelfand-Tsetlin polytopes, flow polytopes

Twisted Schubert polynomials

We prove that twisted versions of Schubert polynomials defined by $$\widetilde{\mathfrak S}_{w_0} = x_1^{n-1}x_2^{n-2} \cdots x_{n-1}$$ and $$\widetilde{\mathfrak S}_{ws_i} = (s_i+\partial _i)\widetilde{\mathfrak S}_w$$ are monomial positive and give a combinatorial formula for their coefficients. In doing so, we reprove and extend a previous result about positivity of skew divided difference operators and show how it implies the Pieri rule for Schubert polynomials. We also give positive formulas for double versions of the $$\widetilde{\mathfrak S}_w$$ as well as their localizations.

The "Young" and "Reverse" Dichotomy of Polynomials

A ``flip-and-reversal" involution arising in the study of quasisymmetric Schur functions provides a passage between what we term ``Young" and ``reverse" variants of bases of polynomials or quasisymmetric functions. Building on this perspective, which has found recent application in the study of q-analogues of combinatorial Hopf algebras and generalizations of dual immaculate functions, we develop and explore Young analogues of well-known bases for polynomials. We prove several combinatorial formulas for the Young analogue of the key polynomials, show that they form the generating functions for left keys, and provide a representation-theoretic interpretation of Young key polynomials as traces on certain modules. We also give combinatorial formulas for the Young analogues of Schubert polynomials, including their crystal graph structure. We moreover determine the intersections of (reverse) bases and their Young counterparts, further clarifying their relationships to one another.

Pipe dreams for Schubert polynomials of the classical groups

Schubert polynomials for the classical groups were defined by S.Billey and M.Haiman in 1995; they are polynomial representatives of Schubert classes in a full flag variety over a classical group. We provide a combinatorial description for these polynomials, as well as their double versions, by introducing analogues of pipe dreams, or RC-graphs, for Weyl groups of classical types.