Schubert polynomial variations

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#Schubert polynomial variations

This page collects variants and extensions of Schubert polynomials. The basic type $A$ theory, divided-difference definition, pipe-dream models, multiplication rules, and complexity results are covered on the main Schubert polynomial page.

#Type B/C/D Schubert polynomials

In type $B,$ there are combinatorial analogues of Schubert polynomials introduced in [FK96]. They also introduce type $B$ Stanley symmetric functions.

Divided difference formulas for Schubert polynomials in types $B,$ $C$ and $D$ were introduced by S. Billey and M. Haiman [BH95]. See also S. Billey’s PhD thesis on Schubert polynomials for classical groups [Bil94].

Type $B,$ $C$ and $D$ double Schubert polynomials are defined in [IMN11]. T. Matsumura gives a tableau formula for vexillary Schubert $P$-polynomials in type $C$ [Mat23]. The formula uses flagged factorial Schur $Q$-functions and a tableau model adapted to vexillary signed permutations. E. Smirnov and A. Tutubalina construct pipe-dream models for Schubert polynomials of the classical groups [ST23], complementing the divided-difference definitions in types $B,$ $C,$ and $D.$

#Skew Schubert polynomials

In [LS03], a skew version of Schubert polynomials is defined. These polynomials expand positively into Schubert polynomials, and a formula in the monomial basis is given. H. Tamvakis gives tableau formulas for skew Schubert polynomials [Tam23], including formulas in classical types.

#Dual Schubert polynomials

The dual Schubert polynomials of A. Postnikov and R. P. Stanley are defined by a weighted-chain formula in Bruhat order [PS08]. Z. Hamaker gives a combinatorial proof that this formula is equivalent to the classical Cauchy identity for Schubert polynomials [Ham23]. This also gives a natural interpretation of the insertion algorithms of D. Huang and P. Pylyavskyy.

#Double Schubert polynomials

The double Schubert polynomials were introduced by A. Lascoux and M.-P. Schützenberger [LS82], and by I. Macdonald [Mac91].

They are defined as \[\schubert_\omega(x_1,\dotsc,x_n,y_1,\dotsc,y_n) = \partial_{\omega^{-1}\omega_0} \prod_{i + j \leq n} (x_i-y_j),\] where the divided difference operator acts on the $\xvec$-variables. Setting $y_i=0$ recovers the usual Schubert polynomials. The double Schubert polynomials generalize the double Schur polynomials, and the factorial Schur polynomials are obtained from the Grassmannian double Schur subfamily by specializing the coefficient alphabet [CLL02].

D. Anderson and W. Fulton construct enriched type $A$ Schubert polynomials with coefficients in a polynomial ring [AF21]. Specializations recover double Schubert polynomials, back-stable double Schubert polynomials, and type $C$ double Schubert polynomials. D. Anderson also studies Schubert polynomials through infinite-dimensional flag varieties and degeneracy loci [And25], including positivity in equivariant coproduct and back-stable/enriched Schubert expansions.

Example

Anderson and Fulton use the convention $\schubert^{\mathrm{en}}_w(0,x,\eta)=\schubert_w(x,-\eta)$ for the double Schubert specialization [Thm. 1.1, AF21]. Thus, to match the double Schubert convention of this page, we set $\eta_j=-y_j$ after specializing the enriched variables $c_i$ to zero. For instance, their Examples 1 give \[\schubert^{\mathrm{en}}_{s_k}(c,x,\eta) = c_1+\sum_{i=1}^k (x_i+\eta_i), \qquad k\gt{}0.\] Hence \[\schubert^{\mathrm{en}}_{s_k}(0,x,-y) = \sum_{i=1}^k (x_i-y_i) = \schubert_{s_k}(x,y),\] the familiar double Schubert polynomial for a simple transposition.

There is a pipe-dream formula, expressing $\schubert_\omega(\xvec,\yvec)$ as a sum over pipe dreams: \[\schubert_\omega(\xvec,\yvec) = \sum_{D \in RC(\omega) } \prod_{(i,j) \in D} \left(x_{i} - y_{j} \right).\] See also the survey by N. Bergeron and S. Billey [BB93].

A. Knutson [Knu19] provides new proofs of several combinatorial formulas for the double Schubert polynomials.

M. J. Samuel proves a Molev–Sagan-type formula for products of double Schubert polynomials under descent hypotheses [Sam24]. The same paper gives a Pieri rule for multiplying a double Schubert polynomial by a factorial elementary symmetric polynomial, and gives positivity results after specializing the two equivariant alphabets to be equal. Y. Gao and R. Xiong prove Graham positivity results for triple Schubert calculus [GX25]. As a corollary, they prove Kirillov’s conjectured positivity for skew divided difference operators applied to Schubert polynomials. S. An, K. Tung, and Y. Zhang prove that Postnikov–Stanley polynomials are Lorentzian [ATZ24]. These polynomials generalize skew dual Schubert polynomials to arbitrary Weyl groups, and the proof realizes them as degree polynomials of Richardson varieties.

#Giambelli formula

The following Giambelli, or splitting, formula expresses a double Schubert polynomial in terms of ordinary Schubert polynomials in the two alphabets; see [Mac91]. It states that \[\schubert_w(\xvec;\yvec) = \sum_{\substack{ v^{-1}u=w \\ \length(u)+\length(v) = \length(w) }} (-1)^{\length(v)}\schubert_u(\xvec) \schubert_v(\yvec).\] The Cauchy identity follows by applying this formula to the longest permutation.

#Affine Schubert polynomials

In [Lee15], an affine extension of Schubert polynomials (indexed by affine permutations) is defined via divided difference operators after being introduced by T. Lam in [Lam06].

Fix $n$ and set \[R_n = \setQ[p_1,\dotsc,p_{n-1}] \otimes_{\setQ} \setQ[x_1,\dotsc,x_n]/\langle e_1(x),\dotsc,e_n(x)\rangle .\] Here the $p_j$ variables come from the affine Grassmannian factor, while the $x_i$ variables come from the ordinary flag-variety factor. Lee defines affine divided difference operators $\partial_i,$ indexed by $i\in \setZ/n\setZ,$ on $R_n.$ For $i\neq 0,$ these operators fix each $p_m$ and restrict to the usual divided differences on the $x$-variables. The affine operator is determined by \[\partial_0(p_m) = \sum_{j=0}^{m-1} x_1^{m-1-j}x_0^j, \qquad \partial_i(x_j)=\delta_{ij}-\delta_{i,j+1},\] with $x$-indices read modulo $n.$

The affine Schubert polynomial $\widetilde{\schubert}_w$ is the unique homogeneous element of $R_n$ satisfying \[\partial_i \widetilde{\schubert}_w = \begin{cases} \widetilde{\schubert}_{ws_i}, & \text{if } \length(ws_i)=\length(w)-1,\\ 0, & \text{otherwise,} \end{cases} \qquad \widetilde{\schubert}_{\mathrm{id}}=1\] for all $i\in\setZ/n\setZ$ [Thm. 1.2, Lee15]. The projection $x_i\mapsto 0$ gives the affine Stanley symmetric function, while the projection $p_j\mapsto 0$ gives the ordinary Schubert polynomial for finite permutations [Thm. 1.3, Lee15].

Example

For $n=3,$ Lee’s first examples are \[\widetilde{\schubert}_{\mathrm{id}}=1,\qquad \widetilde{\schubert}_{s_0}=p_1,\qquad \widetilde{\schubert}_{s_1}=p_1+x_1,\qquad \widetilde{\schubert}_{s_2}=p_1+x_1+x_2 .\] Setting $p_1=0$ recovers the ordinary Schubert polynomials $\schubert_{s_1}=x_1$ and $\schubert_{s_2}=x_1+x_2.$

In [Thm. 6.1, LLS19], the authors prove that the affine Schubert polynomials expand positively in the monomial basis.

#Involution Schubert polynomials

The involution Schubert polynomials were introduced in [WY16]. Additional properties are proved in [HMP18], where the authors introduce the name of the family of polynomials. The following definition is from that paper.

Let $y \in I_n,$ the set of involutions in $\symS_n,$ and let $A(y)$ denote the set of minimal-length permutations $w$ such that $y = w^{-1} \circ w.$ That is, we consider all minimal factorizations of $y$ into simple transpositions of the form $(s_{i_1}\dotsc s_{i_k})\circ (s_{i_k}\dotsc s_{i_1}).$ Then let \[\ischubert_y(x_1,\dotsc,x_n) \coloneqq \sum_{w \in A(y)} \schubert_w(x_1,\dotsc,x_n)\] These polynomials can also be produced recursively via divided difference operators.

The stable limit of these yields the involution Stanley symmetric functions.

A combinatorial model using involution pipe dreams for these polynomials is given by Z. Hamaker, E. Marberg, and B. Pawlowski in [HMP19].

Example (A small involution Schubert polynomial).

For the involution $321 \in \symS_3,$ Hamaker–Marberg–Pawlowski give \[\hat{R}(321)=\{(s_1,s_2),(s_2,s_1)\}, \qquad A(321)=\{231,312\}\] [Ex. 2.7, HMP18]. Hence \[\ischubert_{321} = \schubert_{231}+\schubert_{312} = x_1x_2+x_1^2.\] The involution pipe-dream model gives a graphical refinement of the same positive expansion.

#Quantum Schubert polynomials

The quantum Schubert polynomials $\schubert^q_\omega(\xvec)$ are deformations of the Schubert polynomials by a vector $q=(q_1,\dotsc,q_{n-1}).$ These were introduced in [FGP97].

Recall the formula that expresses the Schubert polynomials as sums of products of elementary symmetric functions: \[\schubert_\omega(\xvec) = \sum a_{k_1 \dotsc k_n} \elementaryE_{k_1}(1)\elementaryE_{k_2}(2) \dotsm \elementaryE_{k_n}(n)\] The quantum Schubert polynomials are then defined as \[\schubert^q_\omega(\xvec) = \sum a_{k_1 \dotsc k_n} \elementaryE^{q}_{k_1}(1) \elementaryE^{q}_{k_2}(2) \dotsm \elementaryE^{q}_{k_n}(n)\] where \[\sum_{i=0}^k \elementaryE^{q}_{i}(k)t^i = \det(I+tG_k), \qquad G_k= \begin{pmatrix} x_1 & q_1 & 0 & \dotsc & 0 \\ -1 & x_2 & q_2 & \dotsc & 0 \\ 0 & -1 & x_3 & \ddots & 0 \\ \vdots & \vdots & \ddots & \ddots & q_{k-1} \\ 0 & 0 & 0 & -1 & x_k \end{pmatrix}\] Setting $q_i=0$ recovers the classical Schubert polynomials.

There is a bumpless pipe-dream model for the quantum double Schubert polynomials, see [Thm. 3.3, LOTR+24]. L. Colmenarejo and N. Mayers study the quantum $k$-Bruhat order that appears in quantum Schubert multiplication [CM26]. They encode maximal chains in intervals by operators from a free monoid acting on a $q$-extension of $\symS_n,$ and formulate a completeness conjecture for the resulting operator equivalences.

#Quantum Monk’s rule

The following refinement of Monk’s rule is given in [Thm. 7.1, FGP97]. Let $s_r$ be a simple transposition. Then \[\schubert^q_{s_r}(\xvec) \schubert^q_\omega(\xvec) = (x_1+x_2+\dotsb + x_r) \schubert^q_\omega(\xvec) = \sum \schubert^q_{\omega t_{ab}}(\xvec) + \sum q_{cd} \schubert^q_{\omega t_{cd}}(\xvec)\] where the first sum is over all transpositions $t_{ab}$ such that $a\leq r \lt b$ and $\length(\omega t_{ab}) = \length(\omega)+1,$ and the second sum runs over all transpositions $t_{cd}$ such that $c\leq r \lt d$ and $\length(\omega t_{cd}) = \length(\omega)-\length(t_{cd}) = \length(\omega)-2(d-c)+1.$

Here, $q_{cd} = q_c q_{c+1} \dotsm q_{d-1}.$

#Quantum Pieri rule

S. Fomin, S. Gelfand, and A. Postnikov derive quantum Pieri-type formulas as part of their construction of quantum Schubert polynomials [FGP97]. For the Grassmannian subcase, the corresponding quantum multiplication of Schur polynomials is described by A. Bertram, Ionuț Ciocan-Fontanine, and W. Fulton [BCF99].

#Quantum Sottile rule

C. B. Velásquez, N. Bergeron, L. Colmenarejo, F. Saliola, and F. Sottile prove a quantum hook Schur multiplication rule in the small quantum cohomology ring of the flag manifold [Thm. 3.2, VBCS+25]. This is the quantum analogue of the hook/Sottile rule and is the main input for their quantum Murnaghan–Nakayama rule below.

#Quantum Murnaghan–Nakayama

C. B. Velásquez, N. Bergeron, L. Colmenarejo, F. Saliola, and F. Sottile prove a quantum Murnaghan–Nakayama rule for the flag manifold [VBCS+25]. The result describes multiplication by a tautological class in small quantum cohomology and is obtained from a hook quantum Schur multiplication formula together with a detailed analysis of the quantum Bruhat order.

#Quantum double Schubert polynomials

The double quantum Schubert polynomials were introduced by A. Kirillov and T. Maeno [KM00].

#Cauchy identity

In [KM00], the following Cauchy identity for quantum double Schubert polynomials is given: \[\sum_{\omega \in \symS_n} \schubert^q_\omega(\xvec) \schubert^q_{\omega \omega_0}(\yvec) = \schubert^q_{\omega_0}(\xvec;\yvec)\]

#Universal Schubert polynomials

The universal Schubert polynomials were introduced by W. Fulton [Ful99] and generalize the (double) quantum Schubert polynomials.

The double universal Schubert polynomials are defined via \[\schubert_w(c;d) = \sum_{u,v} (-1)^{\length(v)}\schubert_u(c) \schubert_v(d)\] as for double Schubert polynomials.

#Twisted Schubert polynomials

The twisted Schubert polynomials are defined via $\tilde{\schubert}_{\omega_0}=x_1^{n-1} x_2^{n-2}\dotsm x_{n-1},$ and the recursion $\tilde{\schubert}_{\omega s_i}= (s_i + \partial_i)\tilde{\schubert}_{\omega}.$

It was proved in [Liu19] that these are positive in the monomial basis. This appears to be an early reference that explicitly defines these polynomials. They are implicitly studied in earlier works, and are related to the Chern–Schwartz–MacPherson classes of Schubert cells in flag varieties.

#Macdonald’s seventh-variation Schur polynomials

Macdonald’s seventh variation is a finite-field analogue of Schur functions attached to subspaces of a vector space. D. Grinberg proves and generalizes Macdonald’s conjectured $V/L$ recursion, including a skew version [Gri26]. The recursion separates subspaces according to whether they contain a distinguished line, giving a finite-field counterpart of branching recurrences for Schur polynomials.

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