Schubert polynomials

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#Schubert polynomials

Schubert polynomials were introduced by A. Lascoux and M.-P. Schützenberger in [LS82]. The Schubert polynomials generalize the Schur polynomials and represent Schubert cycles in flag varieties; this is one of the polynomial models behind Schubert calculus. The Schubert polynomials are generalized by the Grothendieck polynomials, which play the same role in K-theory. Kursat Aker and Nesrin Tutas define rational Schubert, rational Grothendieck, and rational key polynomials [AT15]. Their motivation is to view A. Molev’s dual Schur functions through Lascoux-style divided-difference constructions. Neil J. Y. Fan and P. L. Guo give an elementary divided-difference proof of the Buch–Rimanyi specialization formula for double Grothendieck polynomials [FG18]. Z. Hamaker, O. Pechenik, D. E. Speyer, and A. Weigandt study a differential operator on Schubert polynomials [HPSW20]. Their application gives a short proof of an identity of I. G. Macdonald and proves a determinant conjecture of R. P. Stanley. C. Robichaux, H. Yadav, and A. Yong survey edge-labeled Young tableaux as models for equivariant Schubert calculus of Grassmannians [RYY22]. They also report on a shifted analogue connected to isotropic Grassmannians.

The Schubert polynomials are indexed by permutations. For fixed $n,$ the set \[\{ \schubert_\omega(x_1,\dotsc,x_n) \}_{\omega \in \symS_n}\] is a basis for the quotient $\setZ[x_1,\dotsc,x_n]/I_n$ where $I_n = \langle \elementaryE_1, \elementaryE_2,\dotsc, \elementaryE_n \rangle,$ that is, the ideal generated by nonconstant symmetric functions. The Schubert polynomials $\schubert_\omega$ with $\omega \in \symS_\infty$ form an integral basis for $\setZ[x_1,x_2,\dotsc].$

#Operator definition

Consider the ring $\setR[x_1,\dotsc,x_n]$ and let $s_i$ act by permuting $x_i$ and $x_{i+1}.$ Define \[\partial_i(f)=\frac{f-s_i(f)}{x_i-x_{i+1}}, \qquad 1 \leq i \leq n-1.\] These divided difference operators satisfy the braid relations, and we let $\partial_\omega = \partial_{a_1} \partial_{a_2} \dotsm \partial_{a_k}$ where $(a_1,a_2,\dotsc,a_k)$ is a reduced word for $\omega.$

The Schubert polynomials are then defined as \[\schubert_\omega(x_1,\dotsc,x_n) \coloneqq \partial_{\omega^{-1}\omega_0}(x_1^{n-1}x_2^{n-2}\dotsm x_{n-1}).\]

#Orthodontia formula

P. Magyar gives a Demazure-operator formula for Schubert polynomials using Bott–Samelson varieties [Mag98]. More precisely, a family of subsets is contained in a chamber family exactly when it is strongly separated; the dual character of its flagged Weyl module then has a nested isobaric-Demazure formula, and the dual character associated with a Rothe diagram is the corresponding Schubert polynomial [Prop. 5, Prop. 10, Thm. 14, Mag98]. In later combinatorial work, this is often phrased as Magyar’s orthodontia formula: one applies an algorithm to the diagram and records a word of Demazure operators whose product recovers the Schubert polynomial. The Grothendieck orthodontia formula is the $K$-theoretic analogue.

#Reduced word formula

Let $RW(\omega)$ be the set of reduced words of $\omega.$

We say that a $k$-tuple $\alpha_1,\dotsc,\alpha_k$ is $a$-compatible if

$\alpha_1 \leq \alpha_2 \leq \dotsb \leq \alpha_k$
$\alpha_j \leq a_j$ for $1 \leq j \leq k$
$\alpha_j \lt \alpha_{j+1}$ whenever $a_j \lt a_{j+1}.$

Let $C(a)$ denote the set of $a$-compatible sequences. Then \[\schubert_\omega(x_1,\dotsc,x_n) = \sum_{a \in RW(\omega)} \sum_{\alpha \in C(a)} x^\alpha .\]

#Pipe dream formula

The set of reduced words and compatible sequences can be described in a more combinatorial manner using rc-graphs. Suppose $(\alpha_1,\dotsc,\alpha_k)$ is an $a$-compatible sequence, for $a\in RW(\omega).$ Let \[D(a,\alpha) = \{ (\alpha_j, a_j - \alpha_j + 1) \text{ for } j=1,\dotsc,k \}.\] This set of points is illustrated in an rc-graph, also known as a pipe dream.

Example

Let $\omega = 314652.$ Then $a = 521345$ is in $RW(\omega),$ and $\alpha = 111235$ is $a$-compatible. We mark the corresponding entries in $D(a,\alpha)$ with a cross, $+,$ in the rc-graph:

A pipe dream for the permutation 314652, with rows and columns indexed from
the upper-left corner.

Here \[D(a,\alpha)=\{(1,5),(1,2),(1,1),(2,2),(3,2),(5,1)\},\] so this pipe dream contributes the monomial \[\xvec_D=x_1^3x_2x_3x_5\] to $\schubert_{314652}.$

Let $RC(\omega)$ denote the set of all such diagrams. Let $\xvec_D = \prod_{(i,j) \in D} x_i.$ Then the Schubert polynomial is given as \[\schubert_\omega(x_1,\dotsc,x_n) = \sum_{D \in RC(\omega)} \xvec_D.\]

#Bumpless pipe dreams

The notion of bumpless pipe dreams was introduced by T. Lam, S. J. Lee, and M. Shimozono in [LLS18]. They prove the following formula for the double Schubert polynomials: \[\schubert_\omega(\xvec,\yvec) = \sum_{P \in RBPD(\omega)} \prod_{(i,j)\in D(P)} (x_i - y_j)\] where the sum runs over reduced bumpless pipe dreams. This formula can be lifted to a formula for the double Grothendieck polynomials.

A. Weigandt uses bumpless pipe dreams to give explicit change-of-basis rules between Grothendieck and Schubert polynomials [Wei25]. In one direction, Grothendieck polynomials expand as signed sums of Schubert polynomials; in the other, Schubert polynomials expand positively in the Grothendieck basis. The bumpless pipe dream model also extends to a formula for expanding back stable Grothendieck polynomials in the back stable Schubert basis [Thm. 8.1, Wei25]. T. Lam, S. J. Lee, and M. Shimozono develop the corresponding back stable $K$-theoretic Schubert calculus [LLS22]. This includes back stable Grothendieck polynomials and their role in the $K$-theory Schubert basis. The ordinary cohomological theory, its stable-limit ring, and its standard monomial bases are summarized on the page about back stable Schubert polynomials. P. Klein and A. Weigandt show that bumpless pipe dreams index irreducible components of diagonal Gröbner degenerations of matrix Schubert varieties [KW21]. Thus bumpless pipe dreams have a geometric role parallel to the classical anti-diagonal pipe-dream theory. Z. Hamaker, O. Pechenik, and A. Weigandt study the Gröbner geometry of Schubert polynomials through the six-vertex ice model [HPW22]. Their work relates diagonal Gröbner degenerations to Lascoux’s ice formula, now also viewed through bumpless pipe dreams.

D. Huang and P. Pylyavskyy study Knuth moves for the Schubert RSK correspondence [HP25]. In this setting biwords insert to bumpless pipe dreams, while recording objects are decorated Bruhat chains. Their plactic biwords with the same insertion bumpless pipe dream are connected by Schubert Knuth moves. D. Huang and S. Nguyen later give a growth-diagram version for plactic biwords [HN23], recovering the Gao–Huang bijection between pipe dreams and bumpless pipe dreams. Y. Gao and D. Huang give a direct bijection between reduced pipe dreams and reduced bumpless pipe dreams [GH23]. The bijection is defined through reduced compatible sequences on bumpless pipe dreams and preserves Monk’s formula. D. Huang and P. Pylyavskyy had earlier introduced left and right Schubert RSK insertion into bumpless pipe dreams, with decorated Bruhat chains as recording objects [HP22]. Their growth-diagram framework gives a combinatorial rule for Schubert structure constants in the separated-descent case. A. Gregory and Z. Hamaker show that, for vexillary permutations, the Gao–Huang bijection preserves the associated semistandard tableaux [GH22]. T. Yu embeds bumpless pipe dreams as Bruhat chains [Yu24], giving another bridge between the bumpless model and the Bruhat-chain viewpoint on Schubert calculus. C. Gaetz and Y. Gao study weighted enumeration of Bruhat chains in the symmetric group [GG20]; this is a useful source for Schubert formulas that are naturally phrased in terms of Bruhat chains.

#Representation theory

This subsection follows the introduction in [FG21].

Let $D=(D_1,\dotsc,D_n)$ be an $n$-tuple of subsets of $[n].$ Then $D$ encodes a subset of $[n]\times [n],$ where $D_j$ is considered as a set of row indices (seen as a strictly increasing list). For example, the following diagram encodes $(23,14,\emptyset,124)$

$\; $ $ \; $ $ \;$ $ \; $ $ \;$ $ \; $ $ \; $ $ \;$ $ \; $ $ \; $ $ \;$

We define a partial order on such diagrams, where $C \leq D$ if for every $j \in [n],$ we have $|C_j| = |D_j|$ and $C_{jk} \leq D_{jk}$ for all $k.$

Let $B \subset GL_n(\setC)$ be the set of invertible upper-triangular $n\times n$ matrices. Let $Y$ denote the upper-triangular matrix with indeterminate entries $\{y_{ij}\}_{i\leq j},$ and let $\setC[Y]$ be the polynomial ring with these indeterminates. A matrix $M \in GL_n(\setC)$ acts on $f \in \setC[Y]$ on the right by $f \cdot M \coloneqq f(M^{-1} Y).$

Given a diagram $D,$ the flagged Weyl module $\mathcal{M}_D$ is a $B$-module, defined as \[\mathcal{M}_D \coloneqq \mathrm{Span} \left\{ \prod_{j=1}^n \det( Y_{D_j}^{C_j} ) : C \leq D \right\} \subseteq \setC[Y],\] where $Y_{D_j}^{C_j}$ denotes the submatrix of $Y$ with row indices from $C_j$ and column indices from $D_j.$

The character, $\mathrm{char}(\mathcal{M}_D)(x_1,\dotsc,x_n),$ is defined as the trace of the map $X : \mathcal{M}_D \to \mathcal{M}_D,$ where $X = (x_1,\dotsc,x_n)$ is a diagonal matrix acting on $\setC[Y].$ Note that for $C \leq D,$ \[\left( \prod_{j=1}^n \det( Y_{D_j}^{C_j} ) \right) \cdot X = \prod_{j=1}^n \prod_{i \in C_j} x_i^{-1} \det( Y_{D_j}^{C_j} )\] so all polynomials $\det( Y_{D_j}^{C_j} )$ with $C \leq D$ are eigenvectors, and they span $\mathcal{M}_D.$ It was shown in [KP87, KP04] that the Schubert polynomial indexed by $\omega$ is related to the character of $\mathcal{M}_{D(\omega)},$ via \[\schubert_\omega(x_1,\dotsc,x_n) = \mathrm{char}(\mathcal{M}_{D(\omega)})(x^{-1}_1,\dotsc,x^{-1}_n).\]

Z. Lin, Simon C. Y. Peng, and Sophie C. C. Sun prove the K. Mészáros–Avery St. Dizier– S. Tanjaya upper-bound conjecture for dual flagged Weyl characters [LPS23]. The criterion says that the natural upper bound is attained exactly for diagrams avoiding a specified forbidden subdiagram. Rothe diagrams and skyline diagrams give the Schubert and key-polynomial special cases.

Y. Gao, R. Hodges, and A. Yong classify Levi-spherical Schubert varieties [GHY23]. Their proof uses type $A$ Demazure characters, i.e. key polynomials, and gives a formulation in terms of standard Coxeter elements.

P. L. Guo, Z. Lin, and Simon C. Y. Peng prove the zero-one criterion for dual characters of flagged Weyl modules [GLP25]. A diagram $D$ has zero-one dual character if and only if it avoids the twelve multiplicitous configurations of [Fig. 1.1, GLP25]. Since Schubert and key polynomials are special cases of dual flagged Weyl characters, this gives a unified proof of the known zero-one criteria for Schubert and key polynomials. For Schubert polynomials, A. Fink, K. Mészáros, and Avery St. Dizier prove that the zero-one property is closed under permutation pattern containment and characterize it by avoidance of twelve patterns [FMD19]. A. Weigandt gives a lower bound for the sum of coefficients in terms of $132$-pattern containment [Wei18], while Fan and Guo characterize when Schubert polynomials attain the upper bound coming from the Rothe diagram by avoidance of the two patterns $1432$ and $1423$ [FG21].

D. Anderson gives a representation-theoretic interpretation of the P. Nadeau–H. Spink–V. Tewari recursion for Schubert polynomials [And24]. Using filtrations of Kraskiewicz–Pragacz Schubert modules, Anderson proves that the recursion determines the characters of these modules, and extends the same mechanism to flagged Schur modules coming from transparent and translucent diagrams.

#Gelfand–Tsetlin polytopes

In [LMD19], the authors express the Schubert polynomials as a projection of a Minkowski sum of Gelfand–Tsetlin polytopes. This generalizes the way Schur polynomials are expressible as an integer point transform of a single Gelfand–Tsetlin polytope.

F. Castillo, Y. Cid-Ruiz, F. Mohammadi, and J. Montaño prove that double Schubert polynomials have saturated Newton polytopes [CCMM23]. Thus every lattice point in the Newton polytope occurs as an exponent vector.

See also a similar result in [FMD18], using the integer point transforms of generalized permutohedra.

Top-degree components of Lascoux polynomials provide another bridge to Schubert polynomials. In [Yu25], T. Yu shows that top Lascoux polynomials are reverse-complement transforms of Schubert polynomials and that their structure constants coincide with Schubert structure constants.

S. W. Zhang relates back stable Schubert calculus to the Heisenberg algebra [Zha24]. The Hilbert space with basis indexed by infinite permutations and the ring of back symmetric functions both carry faithful Heisenberg actions, and the map sending an infinite permutation to its back stable Schubert polynomial is an isomorphism of representations. The same framework interprets the pipe dream model as Hamiltonian time evolution of two-dimensional fermions.

#Block decomposition formula

If $\sigma \in \symS_j$ and $\tau \in \symS_{k},$ we can form the skew sum $\sigma \ominus \tau \in \symS_{j+k}.$ We then have the factorization (see [BJS93]) \[\schubert_{\sigma \ominus \tau} = (x_1 \dotsm x_j)^k \schubert_{\sigma}\;\uparrow^j\!\schubert_{\tau}\] where $\uparrow^j x_i \coloneqq x_{i+j}$ is the operator which increases the variable index by $j$ on all variables.

#Grassmannian permutations

A permutation $\omega \in \symS_n$ is called Grassmannian (of descent $k$) if $\omega_i \lt \omega_{i+1}$ for all $i\neq k.$ All Grassmannian permutations are vexillary, meaning that they avoid the permutation pattern $2143.$

A Grassmannian permutation determines a partition: \[\lambda(\omega) = (\omega_k - k, \omega_{k-1} - k + 1, \dotsc, \omega_1 - 1 ).\] The same partitions index Schubert cells in the Grassmannian $\mathrm{Gr}(k,n).$ Fix a complete flag $0=F_0\subset F_1\subset \dotsb \subset F_n=\setC^n.$ For $\lambda$ contained in the $k\times(n-k)$ rectangle, the Schubert variety $\Omega_\lambda(F_\bullet)$ is cut out by the rank conditions \[\dim(V\cap F_{n-k+i-\lambda_i}) \geq i,\qquad 1\leq i\leq k.\] Its open cell $\Omega^\circ_\lambda(F_\bullet)$ has complex codimension $|\lambda|,$ and these cells form the standard cell decomposition of the Grassmannian. The corresponding Schubert class is represented by the Schur polynomial $\schurS_\lambda$ in the cohomology ring of the Grassmannian; see [Ch. 9, Ful97]. Thus \[\schurS_{\lambda(\omega)}(x_1,\dotsc,x_k) = \schubert_{\omega}(\xvec).\]

Furthermore, the Stanley symmetric functions are obtained as a stable limit of the Schubert polynomial.

#Skew Schur polynomials as Schubert specializations

The flagged-skew-Schur description of $321$-avoiding Schubert polynomials has the following useful converse [BJS93, Zha26].

Lemma

For every nonempty skew shape $D,$ there is a $321$-avoiding permutation $w$ with associated skew shape $\Sigma_w$ such that \[\schurS_{\Sigma_w}(x_1,\dotsc,x_n) = \schurS_D(x_1,\dotsc,x_n)\] for every $n\geq1.$

After adding sufficiently many initial fixed points, the equality becomes an actual specialization of a Schubert polynomial.

Proposition

Let $D$ be a skew shape and $n\geq1.$ There are integers $p,m_0\geq1$ and a $321$-avoiding permutation $w\in\symS_p$ such that, for every $m\geq m_0,$ \[\schurS_D(x_1,\dotsc,x_n) = \schubert_{\operatorname{id}_m\times w} (x_1,\dotsc,x_n,0,\dotsc,0).\]

This bridge transfers properties preserved by nonnegative linear specializations from Schubert polynomials to finite-variable skew Schur polynomials. In particular, because Schubert polynomials are dually Lorentzian, it is the main input in the proof that normalized skew Schur polynomials are Lorentzian.

#Murnaghan–Nakayama rule

In [MS18], K. N. Morrison and F. Sottile prove a Murnaghan–Nakayama rule for multiplying a Schubert polynomial by $\powerSum_r(x_1,\dotsc,x_k).$ The expansion is indexed by certain cycles in the $k$-Bruhat order, with signs determined by a descent statistic on the cycle. In the Grassmannian case this recovers the classical border-strip rule for Schur functions.

#Monk’s rule

The geometric version of this result was given by D. Monk in [Mon59].

Let $s_r$ be a simple transposition. Then \[\schubert_{s_r}(\xvec) \schubert_\omega(\xvec) = \sum \schubert_{\omega t_{ij}}(\xvec)\] where the sum is over all transpositions $t_{ij}$ such that $i\leq r \lt j$ and $\length(\omega t_{ij}) = \length(\omega)+1.$ Notice that \[\schubert_{s_r}(\xvec) = x_1+x_2+\dotsb + x_r.\]

A combinatorial proof is given in [CT18], and a bijective proof using bumpless pipe dreams can be found in [Hua20]. Huang also proves Monk’s rule for the double Schubert polynomials with this model.

#Pieri rule

F. Sottile provides the following Pieri-type formulas for Schubert polynomials [Sot96]. Let $\omega \in \symS_n,$ then \[\completeH_{m}(x_1,\dotsc,x_k) \schubert_\omega(\xvec) = \sum_{\omega'} \schubert_{\omega'}(\xvec)\] where the sum runs over all $\omega' = \omega t_{a_1b_1}\dotsm t_{a_mb_m}$ such that $a_i \leq k \leq b_i,$ $\length(\omega t_{a_1b_1}\dotsm t_{a_ib_i}) = \length(\omega)+i$ and all $b_i$ distinct.

Similarly, \[\elementaryE_{m}(x_1,\dotsc,x_k) \schubert_\omega(\xvec) = \sum_{\omega'} \schubert_{\omega'}(\xvec)\] with the same sum as above, but now the $a_i$ are distinct.

#Sottile rule

Sottile also gives a rule for multiplying $\schubert_\omega(\xvec)$ by a Schur polynomial in the first $k$ variables when the Schur shape is a hook [Sot96]. This sits between Monk’s rule and the Schubert Littlewood–Richardson problem: it is positive and is controlled by chains in the $k$-Bruhat order.

#Cauchy identity

Setting $w=\omega_0$ in the Giambelli formula for double Schubert polynomials gives the following Cauchy identity for Schubert polynomials: \[\prod_{i+j \leq n} (x_i - y_j) = \sum_{\omega} \schubert_{\omega}(\xvec) \schubert_{\omega_0 \omega}(-\yvec).\]

#Littlewood–Richardson rule

Problem

Give a combinatorial interpretation of the coefficients in the product \[\schubert_u(\xvec) \schubert_v(\xvec) = \sum_{w} c^w_{uv} \schubert_w(\xvec).\]

Because of the geometric interpretation of Schubert polynomials, it is known that the coefficients $c^w_{uv}$ are nonnegative integers.

Note that the coefficients $c^w_{uv}$ generalize the classical Littlewood–Richardson coefficients in the Littlewood–Richardson rule for Schur polynomials. Finding a positive combinatorial rule is a major open problem in Schubert calculus and algebraic combinatorics. N. Bergeron and F. Sottile relate these structure constants to the Bruhat order and the geometry of flag manifolds [BS98]. They prove identities among Schubert structure constants, derive formulas for several cases, obtain chain-enumeration results in Bruhat order, and introduce a graded partial order on the symmetric group containing Young’s lattice. I. Pak and C. Robichaux disprove the proposed saturation property for Schubert coefficients [PR26]. They show that the failure occurs for a large family of instances, also refute saturation under bit scaling, and discuss consequences for computational complexity.

This problem is closely related to the Fomin–Kirillov conjecture, see [Conjecture 8.1 and Problem 8.3, FK99]. It concerns the evaluation of Schubert polynomials at Dunkl elements.

There are many special cases proved for this problem. One such case, concerning products of two permutations with separated descents, is treated in [Hua22].

O. Pechenik and A. Weigandt introduce backstable clans and prove an inverse Grassmannian Littlewood–Richardson rule [PW24]. Their work also gives new linear relations among Schubert structure constants, extending the range of products accessible by positive combinatorial formulas.

M. Samuel maintains schubmult, a Python package for computing products of Schubert polynomials.

Problem

Can one give an efficient algorithm to decide if $c^w_{uv} = 0$ or not?

This problem is discussed in [PR25], where it is shown that, assuming GRH, this problem lies in $AM \cap coAM,$ where $AM$ denotes the Arthur–Merlin probabilistic proof class.

#Expansion in key polynomials

There are several proofs of the fact that Schubert polynomials expand positively into key polynomials, the first proof appearing in [LS90].

One recent proof is by using crystal bases, see S. Assaf and A. Schilling [AS18]. A crystal structure on RC-graphs where the key expansion is obtained is given by S. Gold, E. Milićević, and Y. Sun in [GMS24].

#Sum of elementary

Let $\omega \in \symS_{n+1},$ and let $\elementaryE_{j}(m) = \elementaryE_{j}(x_1,\dotsc,x_m)$ denote the elementary symmetric functions of degree $j$ in $m$ variables. Then there is a unique way to express $\schubert_\omega(\xvec)$ as \[\schubert_\omega(\xvec) = \sum a_{k_1 \dotsc k_n} \elementaryE_{k_1}(1) \elementaryE_{k_2}(2) \dotsm \elementaryE_{k_n}(n)\] where the sum ranges over integers such that $0 \leq k_i \leq i$ and $k_1+\dotsb + k_n = \length(\omega).$ This expansion is the starting point for defining the quantum Schubert polynomials. The coefficients $a_{k_1 \dotsc k_n}$ are in general not always positive. These coefficients are studied in [Win98] but many questions are left unanswered. Their eventual behavior under adjoining fixed points is encoded by the back stable standard elementary basis.

D. Woodruff characterizes the permutations $w$ for which the Schubert polynomial $\schubert_w$ is a single standard elementary monomial [Woo25]. The characterization is by pattern avoidance and also has an analogue for complete homogeneous monomials.

O. Makhija studies when a Schubert polynomial factors as a product of elementary symmetric polynomials [Mak25]. The paper conjectures a pattern-avoidance characterization, proves one direction, and gives obstructions coming from rectangular arrays of crosses in bottom pipe dreams.

Problem (See [MS15]).

For each $n,$ find a permutation $w\in\symS_n$ which maximizes $\schubert_w(1,1,\dotsc,1)$ and find the maximal value.

The sequence is given as A284661. See also [Sta17].

P. Nadeau and V. Tewari relate the permutahedral variety, mixed Eulerian numbers, and principal specializations of Schubert polynomials [NT21]. H. Dennin proves pattern lower bounds for principal specializations of Schubert polynomials in the $1243$-avoiding case and extends the result to $\beta$-Grothendieck polynomials for vexillary $1243$-avoiding permutations [Den25]. The proof gives bijective interpretations of the coefficients appearing in these bounds.

K. Mészáros and A. Tanjaya prove nonnegativity of an inclusion–exclusion expression for Schubert polynomials of permutations avoiding $1432$ and $1423$ [MT21]. This gives a partial answer to a conjecture of Y. Gao on principal specializations of Schubert polynomials. G. Nenashev studies differential operators on Schur and Schubert polynomials [Nen20]. H. Dennin gives a pipe-dream proof of the Gaetz–Tung positivity result for a dual differential operator acting on padded Schubert polynomials [Den25]. P. L. Guo and Z. Lin give lower bounds for the number of support monomials of a Schubert polynomial, equivalently lattice points in its Newton polytope, in terms of permutation-pattern occurrences [GL24]. Their framework is the more general setting of dual characters of flagged Weyl modules.

#Complexity

Deciding whether a monomial coefficient, or a Schubert structure constant coefficient $c^w_{uv},$ is nonzero can be done in probabilistic polynomial time; see [ARY21, PR25]. Avery St. Dizier and A. Yong connect generalized permutahedra with Schubert calculus [DY20]. They obtain sufficient vanishing criteria for Schubert intersection numbers, using Schubitopes as Newton polytopes of Schubert polynomials, and give a polynomial-time tableau test. I. Pak and C. Robichaux prove that the vanishing problem for Schubert coefficients lies in $\mathrm{coAM}$ assuming the generalized Riemann hypothesis [PR24]. They also formulate a conditional positive rule for deciding Schubert coefficient positivity [PR24]. I. Pak and C. Robichaux also give a signed puzzle rule for Schubert coefficients based on Knutson’s recurrence [PR25]. They use it to prove polynomiality for sums of Schubert coefficients with bounded number of inversions.

For type B/C/D, skew, double, affine, involution, quantum, universal, and twisted variants, see the companion page on Schubert polynomial variations.

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