Schur polynomials (flagged)

2026-09-08 Improve this page

#Schur polynomials (flagged)

The flagged Schur polynomials generalize the classical Schur polynomials, but they are no longer symmetric in general. They were first considered in [LS82], in the study of Schubert polynomials.

We follow the introduction given in [MS15].

Let $m \leq n$ be fixed integers and let $\lambda$ be a partition with $m$ parts. Let $1 \leq b_1 \leq b_2 \leq \dotsb \leq b_m \leq n$ be a weakly increasing sequence of integers. Let $T(\lambda,b)$ denote the set of semistandard tableaux of shape $\lambda$ such that the entries in row $i$ do not exceed $b_i.$

Then the flagged Schur polynomial is defined as \[\schurS_{\lambda,b}(x_1,\dotsc,x_n) \coloneqq \sum_{T \in T(\lambda,b)} \xvec^T.\] Note that if $b_1=\dotsb = b_m = n,$ then $\schurS_{\lambda,b}(\xvec)$ is the usual Schur polynomial in $n$ variables.

Example

Let $\lambda=(2,1)$ and $b=(1,2).$ The first row must contain only $1$’s, while the second row may contain a $2.$ Thus the only tableau is

$1$ $1$ $2$

and \[\schurS_{(2,1),(1,2)}(x_1,x_2)=x_1^2x_2.\]

Every flagged Schur polynomial $\schurS_{\lambda,b}(\xvec)$ is a key polynomial, see [RS95, Thm. 23]. Moreover, a Schubert polynomial $\schubert_w(\xvec)$ is a flagged Schur function if and only if $w$ is a vexillary permutation (avoids 2143). S. Armon, S. Assaf, G. Bowling, and H. Ehrhard prove a Kohnert rule for flagged Schur modules [AABE23]. Their module characters include flagged skew Schur polynomials, Demazure characters, and Schubert polynomials, and Kohnert’s algorithm applies exactly to northwest indexing diagrams.

#$h$-flagged Schur polynomials

Whenever $b = (h,h+1,\dotsc,h+m-1)$ for some $h,$ we say that $\schurS_{\lambda,b}(\xvec)$ is $h$-flagged, and use the shorthand notation $\schurS^{(h)}_{\lambda}(\xvec).$

#Divided differences

In [Wac85], it is proven that flagged Schur polynomials can be obtained via divided difference operators: \[\begin{aligned} \schurS_{\lambda,b}(\xvec) = \partial_w( x_1^{a_1} \dotsm x_m^{a_m}) \end{aligned}\] where $a_i = \lambda_i + b_i -i$ and \[w = (m,m+1,\dotsc, b_m-1,\; m-1,m,\dotsc, b_{m-1} - 1, \dotsc 1,2,\dotsc, b_1 -1 )\] and the entries in the word are applied from left to right.

#Jacobi–Trudi identity

Define the complete homogeneous polynomials in $k$ variables as \[\completeH_d(\xvec_k) \coloneqq \sum_{1 \leq i_1 \leq \dotsb \leq i_d \leq k} x_{i_1}\dotsm x_{i_d}.\] Here $\completeH_0(\xvec_k)=1$ and $\completeH_d(\xvec_k)=0$ for $d\lt{}0.$ Then, using the Lindström–Gessel–Viennot lemma, one can prove that \[\schurS_{\lambda,b}(\xvec) = \det( \completeH_{\lambda_i -i+j}(\xvec_{b_i}) )_{1\leq i,j \leq m}.\] See also [Wac85] or [Man01, Cor. 2.6.3]. This generalizes the classical Jacobi–Trudi identity for Schur polynomials. R. A. Proctor and M. J. Willis refine the row-bound conditions needed for such Gessel–Viennot determinants [PW21]. They drop the weak-increase condition on the row bounds, give a necessary and sufficient nonpermutability criterion, and identify efficient representatives among row-bound tuples.

The flagged skew Schur polynomials , $\schurS_{\lambda/\mu,a,b}(\avec,\bvec),$ are now defined as the sum over all skew semistandard Young tableaux of shape $\lambda/\mu$ such that entries in row $i$ must be from the interval $\{a_i,a_i+1,\dotsb,b_i\}.$ We still require that the corresponding flags are weakly increasing. Taking $a_i=1$ and $b_i=n$ for every row recovers the ordinary skew Schur polynomial in the variables $x_1,\dotsc,x_n.$

With $\completeH_m(a,b) \coloneqq \completeH_m(x_a,x_{a+1},\dotsc,x_b),$ one can show the Jacobi–Trudi type formula \[\schurS_{\lambda/\mu}(\avec,\bvec) = \left| \completeH_{\lambda_i-\mu_j - i + j}(a_j,b_i) \right|_{1 \leq i,j \leq \ell}.\] This was proved by I. Gessel and X. Viennot [GV89] and later in [Wac85]. An alternative proof is given in [Chu92], which relies on a recursion.

There are also operator constructions, but the cleanest statements usually embed the flagged skew family into a larger operator-defined family. For example, skew Schubert polynomials are defined using divided-difference operators, and their single-alphabet specialization gives the flagged skew Schur functions studied by Wachs and by Billey–Jockusch–Stanley [CYY04]. The flagged Schur module viewpoint gives another realization: for northwest diagrams, the corresponding characters are computed by Kohnert’s rule [AABE23]. Thus flagged skew Schur polynomials can be produced by operators after placing them in these frameworks, while the tableau and determinant definitions remain the most direct standalone definitions.

Unlike the classical Jacobi–Trudi identities, exchanging $\completeH \leftrightarrow \elementaryE$ does not in general yield the same determinant. Rather, the corresponding determinant of elementary symmetric polynomials is equal to a column-flagged Schur polynomial, where the tableaux in the generating set have bounds on the columns rather than the rows; see [Wac85]. An identity between determinants of this type is given in [McD23], thus giving a duality for row- and column-flagged Schur polynomials under certain conditions.

#Kostka coefficients

The flagged Kostka coefficients are defined as the coefficients in the monomial expansion of the flagged Schur functions. Given a skew shape $\lambda/\mu$ and a vector $(\nu_1,\dotsc,\nu_\ell)$ with $\nu_i \geq 0,$ we define the flagged Kostka coefficient $K_{\lambda/\mu,\nu}(\avec,\bvec)$ via the expansion \[\schurS_{\lambda/\mu}(\avec,\bvec) = \sum_{\nu} K_{\lambda/\mu,\nu}(\avec,\bvec) \cdot \xvec^\nu.\]

P. Alexandersson and G. Kantarci-Oğuz prove the following results [AO23].

Theorem (Alexandersson–Kantarci-Oğuz (2023)).

For any integer $k \geq 1,$ \[K_{\lambda/\mu,\nu}(\avec,\bvec) \gt 0 \iff K_{k\lambda/k\mu,k\nu}(\avec,\bvec) \gt 0\] where multiplication of a partition by $k$ is done elementwise. This is a generalization of saturation for Gelfand–Tsetlin polytopes.

Moreover, the map \[k \mapsto K_{k\lambda/k\mu,k\nu}(\avec,\bvec)\] is a polynomial in $k.$

Note that it is relatively easy to see that the map \[k \mapsto K_{k\lambda/k\mu,k\nu}(\avec,\bvec)\] is a quasi-polynomial as this is the Ehrhart function for a face of a (rational) Gelfand–Tsetlin polytope.

The following conjecture is then a flagged generalization of a conjecture by King, Tollu and Toumazet [KTT04].

Conjecture (Alexandersson–Kantarci-Oğuz (2023)).

The Ehrhart polynomial \[k \mapsto K_{k\lambda/k\mu,k\nu}(\avec,\bvec)\] has nonnegative coefficients.

K. Jochemko and K. Menon prove Ehrhart positivity for the corresponding unsliced skew GT polytopes and certain flagged faces. Precisely, in their indexing, the lower and upper flags $\mathbf a,\mathbf b\in\setZ_{\geq0}^k$ are weakly increasing, $0\leq a_i\lt{}b_i\leq m+1,$ and row $i$ is restricted to the interval $[a_i+1,b_i].$ The equalities excluding all other entries identify coordinates of the GT poset, so the face is itself a marked order polytope [JM26, Thm. 3.5 and Sec. 3.2].

This does not prove the conjecture above: fixing the content extracts \[[x^{k\nu}]\,\schurS_{k\lambda/k\mu}(\avec,\bvec)\] rather than summing over all weights. Geometrically, it intersects the flagged face with the affine weight subspace, producing a generally rational weight-restricted GT polytope.

#Key expansion

The flagged skew Schur polynomials expand positively into key polynomials, see [RS95]. R. A. Proctor and M. J. Willis also note that key polynomials indexed by $312$-avoiding permutations admit related determinant expressions [PW21].

#Twist by roots of unity

In [Kum23] the author generalizes a previous result by Kumari [Kum22], and [LO22, Thm. 16].

Bibliography

[AO23]Per Alexandersson and Ezgi Kantarci Oğuz. Cylindric Schur functions. arXiv:2311.07382, 2023.

.bib

@article{AlexanderssonOguz2023x,
Author = {Per Alexandersson and Ezgi Kantarci Oğuz},
Title = {Cylindric {S}chur functions},
Year = {2023},
Eprint = {2311.07382},
url = {https://arxiv.org/abs/2311.07382},
journal = {arXiv e-prints}
}

[AABE23]Sam Armon, Sami Assaf, Grant Bowling and Henry Ehrhard. Kohnert’s rule for flagged Schur modules. Journal of Algebra, 617:352–381, March 2023.

.bib

@article{ArmonAssafBowlingEhrhard2023,
  doi = {10.1016/j.jalgebra.2022.10.032},
  url2 = {https://doi.org/10.1016/j.jalgebra.2022.10.032},
  year = {2023},
  month = mar,
  publisher = {Elsevier {BV}},
  volume = {617},
  pages = {352--381},
  author = {Sam Armon and Sami Assaf and Grant Bowling and Henry Ehrhard},
  title = {Kohnert's rule for flagged {S}chur modules},
  journal = {Journal of Algebra}
}

[CYY04]William Y. C. Chen, Guo-Guang Yan and Arthur L. B. Yang. The skew Schubert polynomials. European Journal of Combinatorics, 25(8):1181–1196, November 2004.

.bib

@article{ChenYanYang2004SkewSchubert,
  doi = {10.1016/j.ejc.2003.11.004},
  url2 = {https://doi.org/10.1016/j.ejc.2003.11.004},
  year = {2004},
  month = nov,
  publisher = {Elsevier {BV}},
  volume = {25},
  number = {8},
  pages = {1181--1196},
  author = {William Y. C. Chen and Guo-Guang Yan and Arthur L. B. Yang},
  title = {The skew {S}chubert polynomials},
  journal = {European Journal of Combinatorics}
}

[Chu92]Wenchang Chu. Direct proof of the determinant expression for flagged skew tableaux. Acta Mathematicae Applicatae Sinica, 8(4):318–322, October 1992.

.bib

@article{Chu1992,
  doi = {10.1007/bf02006740},
  url2 = {https://doi.org/10.1007/bf02006740},
  year = {1992},
  month = oct,
  publisher = {Springer Science and Business Media {LLC}},
  volume = {8},
  number = {4},
  pages = {318--322},
  author = {Wenchang Chu},
  title = {Direct proof of the determinant expression for flagged skew tableaux},
  journal = {Acta Mathematicae Applicatae Sinica}
}

[GV89]I. M. Gessel and X. G. Viennot. Determinants, paths, and plane partitions. 1989. Preprint

.bib

@misc{GesselViennot1989,
	author = {I. M. Gessel and X. G. Viennot},
	title = {Determinants, Paths, and Plane Partitions},
	year = {1989},
	note ={Preprint},
	url = {https://people.brandeis.edu/~gessel/homepage/papers/pp.pdf}
}

[JM26]Katharina Jochemko and Krishna Menon. Ehrhart positivity for marked order polytopes. arXiv:2604.08394v2, 2026.

.bib

@article{JochemkoMenon2026x,
  author = {Katharina Jochemko and Krishna Menon},
  title = {Ehrhart positivity for marked order polytopes},
  year = {2026},
  eprint = {2604.08394v2},
  url = {https://arxiv.org/abs/2604.08394v2},
  journal = {arXiv e-prints}
}

[KTT04]R. C. King, C. Tollu and F. Toumazet. Stretched Littlewood–Richardson coefficients and Kostka coefficients. Symmetry in physics: In memory of Robert T. Sharp, 34:99–112, 2004.

.bib

@incollection{KingTolluToumazet2004,
  author = {R. C. King and C. Tollu and F. Toumazet},
  title = {Stretched {L}ittlewood--{R}ichardson coefficients and {K}ostka coefficients},
  journal = {CRM Proceedings and Lecture Notes},
  year = {2004},
  pages = {99--112},
  volume = {34},  
  booktitle = {Symmetry in Physics: In Memory of {R}obert {T}. {S}harp},
  isbn = {978-0-8218-3409-1},
  editor = {P. Winternitz and J. Harnard and C. S. Lam and J. Patera},
  publisher = {AMS / OUP},
  url = {http://eprints.soton.ac.uk/41202/}
}

[Kum23]V. Sathish Kumar. Flagged skew Schur polynomials twisted by roots of unity. arXiv:2306.10289, 2023.

.bib

@article{Kumar2023x,
Author = {V. Sathish Kumar},
Title = {Flagged Skew {S}chur Polynomials Twisted By Roots Of Unity},
Year = {2023},
Eprint = {2306.10289},
  url = {https://arxiv.org/abs/2306.10289},
journal = {arXiv e-prints}
}

[Kum22]Nishu Kumari. Skew hook Schur functions and the cyclic sieving phenomenon. arXiv:2211.14093, 2022.

.bib

@article{Kumari2022x,
Author = {Nishu Kumari},
Title = {Skew hook {S}chur functions and the cyclic sieving phenomenon},
Year = {2022},
Eprint = {2211.14093},
  url = {https://arxiv.org/abs/2211.14093},
journal = {arXiv e-prints}
}

[LS82]Alain Lascoux and Marcel-Paul Schützenberger. Polynômes de Schubert. C. R. Math. Acad. Sci. Paris, Sér. I Math., 294(13):447–450, 1982.

.bib

@article{LascouxSchutzenberger1982,
  year  = {1982},
  volume = {294},
  number = {13},
  pages = {447--450},
  author = {Alain Lascoux and Marcel-Paul Sch{\"{u}}tzenberger},
  title = {Polyn{\^{o}}mes de {S}chubert},
  journal = {C. R. Math. Acad. Sci. Paris, S{\'{e}}r. I Math.}
}

[LO22]So-Yeon Lee and Young-Tak Oh. Skew Schur polynomials and cyclic sieving phenomenon. The Electronic Journal of Combinatorics, 29(4), October 2022.

.bib

@article{LeeOh2022,
  doi = {10.37236/10941},
  url2 = {https://doi.org/10.37236/10941},
  year = {2022},
  month = oct,
  publisher = {The Electronic Journal of Combinatorics},
  volume = {29},
  number = {4},
  author = {So-Yeon Lee and Young-Tak Oh},
  title = {Skew {S}chur polynomials and cyclic sieving phenomenon},
  journal = {The Electronic Journal of Combinatorics}
}

[Man01]Laurent Manivel. Symmetric functions, Schubert polynomials and degeneracy loci. SMF/AMS texts & monographs. American Mathematical Society, 2001.

.bib

@book{Manivel2001,
  Author        = {Laurent Manivel},
  Title         = {Symmetric Functions, {S}chubert Polynomials and Degeneracy Loci},
  Year          = {2001},
  Publisher     = {American Mathematical Society},
  Series        = {SMF/AMS Texts \& Monographs},
  Address       = {Providence, RI},
  Month         = {aug}
}

[McD23]Eoghan McDowell. Flagged Schur polynomial duality via a lattice path bijection. The Electronic Journal of Combinatorics, 30(1), January 2023.

.bib

@article{McDowell2023,
  title = {Flagged {S}chur Polynomial Duality via a Lattice Path Bijection},
  volume = {30},
  ISSN = {1077-8926},
  url = {http://dx.doi.org/10.37236/11200},
  DOI = {10.37236/11200},
  number = {1},
  journal = {The Electronic Journal of Combinatorics},
  publisher = {The Electronic Journal of Combinatorics},
  author = {McDowell,  Eoghan},
  year = {2023},
  month = jan 
}

[MS15]Grigory Merzon and Evgeny Smirnov. Determinantal identities for flagged Schur and Schubert polynomials. European Journal of Mathematics, 2(1):227–245, October 2015.

.bib

@article{MerzonSmirnov2015,
  doi = {10.1007/s40879-015-0078-9},
  url2 = {https://doi.org/10.1007/s40879-015-0078-9},
  year  = {2015},
  month = oct,
  publisher = {Springer Nature},
  volume = {2},
  number = {1},
  pages = {227--245},
  author = {Grigory Merzon and Evgeny Smirnov},
  title = {Determinantal identities for flagged {S}chur and {S}chubert polynomials},
  journal = {European Journal of Mathematics}
}

[PW21]Robert A. Proctor and Matthew J. Willis. Row bounds needed to justifiably express flagged Schur functions with Gessel–Viennot determinants. Discrete Mathematics & Theoretical Computer Science, 23(1), 2021.

.bib

@article{ProctorWillis2021,
  author = {Robert A. Proctor and Matthew J. Willis},
  title = {Row bounds needed to justifiably express flagged {S}chur functions
    with {G}essel--{V}iennot determinants},
  year = {2021},
  journal = {Discrete Mathematics \& Theoretical Computer Science},
  volume = {23},
  number = {1},
  doi = {10.46298/dmtcs.6632},
  eprint = {1701.01182}
}

[RS95]Victor Reiner and Mark Shimozono. Key polynomials and a flagged Littlewood–Richardson rule. J. Combin. Theory Ser. A, 70(1):107–143, 1995.

.bib

@article{ReinerShimozono1995,
    AUTHOR = {Victor Reiner and Mark Shimozono},
     TITLE = {Key polynomials and a flagged {L}ittlewood--{R}ichardson rule},
   JOURNAL = {J. Combin. Theory Ser. A},
  FJOURNAL = {Journal of Combinatorial Theory. Series A},
    VOLUME = {70},
      YEAR = {1995},
    NUMBER = {1},
     PAGES = {107--143},
      ISSN = {0097-3165},
      DOI = {10.1016/0097-3165(95)90083-7}
}

[Wac85]Michelle L. Wachs. Flagged Schur functions, Schubert polynomials, and symmetrizing operators. Journal of Combinatorial Theory, Series A, 40(2):276–289, November 1985.

.bib

@article{Wachs1985,
  doi = {10.1016/0097-3165(85)90091-3},
  url2 = {https://doi.org/10.1016/0097-3165(85)90091-3},
  year  = {1985},
  month = nov,
  publisher = {Elsevier {BV}},
  volume = {40},
  number = {2},
  pages = {276--289},
  author = {Michelle L. Wachs},
  title = {Flagged {S}chur functions,  {S}chubert polynomials, and symmetrizing operators},
  journal = {Journal of Combinatorial Theory,  Series A}
}

The symmetric functions catalog