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#Varieties in algebraic combinatorics

Many constructions in algebraic combinatorics are shadows of geometric objects. A polynomial may be the character of a representation, the class of a subvariety, the Hilbert series of a coordinate ring, or the point-counting shadow of a variety over a finite field. This page gives a short map of the varieties that occur most often on this site.

The main examples are Grassmannians, flag varieties, Schubert varieties, Richardson varieties, Springer fibers, and Hessenberg varieties. They connect directly to Schubert calculus, Grothendieck polynomials, key polynomials, flagged Schur polynomials, positroids, and matroids.

#Affine and projective varieties

An affine variety is, roughly, the set of common zeros of polynomial equations in affine space: \[X=\{x\in\setC^n:f_1(x)=\dotsb=f_r(x)=0\}.\] A projective variety is the corresponding notion in projective space, where points are lines through the origin: \[\mathbb P^{n-1} = (\setC^n\setminus\{0\})/\setC^\ast.\] Projective varieties are the natural home for many compact geometric objects, including Grassmannians and flag varieties.

The coordinate ring viewpoint is often combinatorial. If \(X\) is affine with coordinate ring \(\setC[X],\) then a grading on \(\setC[X]\) gives a Hilbert series. Such Hilbert series often become symmetric functions, lattice-point generating functions, or Ehrhart-type series after a suitable choice of coordinates.

#Grassmannians

The Grassmannian \(\operatorname{Gr}(k,n)\) is the variety of \(k\)-dimensional subspaces of \(\setC^n.\) Equivalently, it parametrizes projective \((k-1)\)-planes in \(\mathbb P^{n-1}.\) For example, \(\operatorname{Gr}(2,4)\) parametrizes lines in \(\mathbb P^3.\)

Example

The Grassmannian \(\operatorname{Gr}(1,n)\) is projective space \(\mathbb P^{n-1},\) since a one-dimensional subspace of \(\setC^n\) is exactly a line through the origin.

The standard projective realization is the Plücker embedding \[\operatorname{Gr}(k,n)\hookrightarrow \mathbb P^{\binom nk-1}.\] If a \(k\)-plane is represented by a full-rank \(k\times n\) matrix, its maximal minors are the Plücker coordinates. These coordinates satisfy quadratic Plücker relations.

This is where matroids enter. Given a point of \(\operatorname{Gr}(k,n),\) the sets \(I\subseteq[n]\) with nonzero Plücker coordinate \(p_I\) form the bases of a rank \(k\) representable matroid. The totally nonnegative part of the Grassmannian gives positroids, introduced by A. Postnikov [Pos06]. Schubert matroids and lattice path matroids are important positroid families [Oh11].

#Flag varieties

A complete flag in \(\setC^n\) is a chain \[0=F_0\subset F_1\subset F_2\subset\dotsb\subset F_n=\setC^n, \qquad \dim F_i=i.\] The flag variety \(\operatorname{Fl}_n\) is the variety of all complete flags. A partial flag variety remembers only some of the subspaces in the chain. Grassmannians are partial flag varieties, since \(\operatorname{Gr}(k,n)\) remembers only the \(k\)-dimensional subspace.

The flag variety is the geometric source of many divided-difference constructions. Its Schubert classes are indexed by permutations, and the corresponding polynomial representatives are Schubert polynomials. Passing from cohomology to K-theory replaces these by Grothendieck polynomials.

#Schubert varieties and Schubert matroids

Fix a complete flag \(F_\bullet\) in \(\setC^n.\) Schubert varieties are subvarieties of a Grassmannian or flag variety defined by incidence conditions with respect to \(F_\bullet.\) In the Grassmannian, they are indexed by partitions in a rectangle, or equivalently by \(k\)-element subsets of \([n].\)

For \(I=\{i_1\lt{}\dotsb\lt{}i_k\}\subseteq[n],\) the corresponding Schubert matroid has bases \[\mathcal B_I = \{J=\{j_1\lt{}\dotsb\lt{}j_k\}: i_a\leq j_a\text{ for all }a\}.\] For example, if \(I=\{1,3\}\subseteq[4],\) then \[\mathcal B_I=\{13,14,23,24,34\}.\] Thus Schubert matroids are the matroid-theoretic shadow of Schubert cells in the Grassmannian. They are positroids, and Suho Oh showed that positroids can be characterized using Schubert matroids [Oh11].

This viewpoint helps explain why matroid polytopes, positroid polytopes, Schubert calculus, and Grassmannian coordinates often appear together.

#Richardson varieties

A Richardson variety is an intersection of a Schubert variety and an opposite Schubert variety. In the complete flag variety, if \(X_w\) is the Schubert variety indexed by \(w\) and \(X^v\) is the opposite Schubert variety indexed by \(v,\) then \[X_w^v\coloneqq X_w\cap X^v.\] This intersection is nonempty precisely when \(v\leq w\) in Bruhat order.

Richardson varieties are useful because they isolate the geometry of an interval in Bruhat order. They also appear naturally in total positivity, positroid geometry, and the study of components of Springer fibers. For a recent tableau connection, see the Richardson tableaux of S. N. Karp and M. E. Precup [KP25].

#Hessenberg varieties

Hessenberg varieties are subvarieties of the flag variety cut out by a linear operator and a Hessenberg function. In type \(A,\) one starts with a weakly increasing sequence \(m=(m_1,\dotsc,m_n)\) with \(i\leq m_i\leq n\) and considers flags satisfying \[T F_i\subseteq F_{m_i}.\] The precise behavior depends on the choice of the operator \(T.\)

They were introduced by F. D. Mari, C. Procesi, and M. A. Shayman [MPS92]. Regular semisimple Hessenberg varieties are the varieties behind the dot action and the Shareshian–Wachs connection with chromatic quasisymmetric functions.

#Springer fibers and the Peterson variety

A Springer fiber is a subvariety of the flag variety attached to a nilpotent linear operator \(N.\) In type \(A,\) it is \[\mathcal B_N = \{F_\bullet\in\operatorname{Fl}_n:N F_i\subseteq F_i \text{ for all }i\}.\] The cohomology of Springer fibers carries deep Weyl-group representations; this is the starting point of Springer theory [Spr78]. Springer fibers also interact with tableaux, Hall–Littlewood theory, and the \(\Delta\)-Springer varieties appearing near the Delta conjecture.

The Peterson variety is a special regular nilpotent Hessenberg variety. In type \(A,\) it is obtained from a regular nilpotent operator \(N\) and the Hessenberg function \[m(i)=i+1\quad(1\leq i\lt{}n),\qquad m(n)=n.\] Thus its flags satisfy \[N F_i\subseteq F_{i+1}\qquad\text{for }1\leq i\lt{}n.\] It sits between the ordinary Springer-fiber world and the broader Hessenberg world, and it is important in the Schubert-calculus literature because of its relationship with quantum cohomology of flag varieties.

#Demazure characters, keys, and flagged Schur polynomials

Let \(G=\operatorname{GL}_n(\setC)\) and let \(B\) be the subgroup of upper triangular matrices. The flag variety can be written as the quotient \(G/B.\) The subgroup \(B\) is a Borel subgroup, and the irreducible closed \(B\)-stable subvarieties inside \(G/B\) are precisely Schubert varieties. General closed \(B\)-stable subvarieties are unions of Schubert varieties.

This is the geometric background for key polynomials. A dominant weight \(\lambda\) gives a line bundle on \(G/B,\) and taking sections over a Schubert variety gives a Demazure module. Its character is a Demazure character, also called a key polynomial [Dem74, Dem74].

The flagged Schur polynomials are closely related. They are Schur-type generating functions with row bounds, and they occur as key polynomials in important cases. In particular, every flagged Schur polynomial is a key polynomial [Thm. 23, RS95], and vexillary Schubert polynomials are flagged Schur functions.

#Where to go next

For computations in the cohomology of Grassmannians, see Schubert calculus. For the K-theoretic version, see K-theory and Grothendieck polynomials. For the matroid side of Grassmannians, see positroids and matroid base polytopes. For Hessenberg varieties and representation theory, see the dot action page. For Springer-fiber and Richardson-tableau connections to insertion algorithms, see the RSK page.

Bibliography

  1. [Dem74]Michel Demazure. Désingularisation des variétés de schubert généralisées. Annales scientifiques de l’École normale supérieure, 7(1):53–88, 1974.
    .bib
    @article{Demazure1974,
  doi = {10.24033/asens.1261},
  url2 = {https://doi.org/10.24033/asens.1261},
  year  = {1974},
  publisher = {Societe Mathematique de France},
  volume = {7},
  number = {1},
  pages = {53--88},
  author = {Michel Demazure},
  title = {D{\'{e}}singularisation des vari{\'{e}}t{\'{e}}s de Schubert g{\'{e}}n{\'{e}}ralis{\'{e}}es},
  journal = {Annales scientifiques de l'{\'{E}}cole normale sup{\'{e}}rieure}
}
  2. [Dem74]Michel Demazure. Une nouvelle formule des caractères. Bulletin des Sciences Mathématiques, 98(3):163–172, 1974.
    .bib
    @article{Demazure1974nouvelle,
title = {Une nouvelle formule des caract{\`{e}}res},
journal = {Bulletin des Sciences Math{\'{e}}matiques},
volume = {98},
number = {3},
pages = {163--172},
year = {1974},
author = {Michel Demazure}
}
  3. [KP25]Steven N. Karp and Martha E. Precup. Richardson tableaux and components of Springer fibers equal to Richardson varieties. arXiv:2506.20792, 2025.
    .bib
    @article{KarpPrecup2025x,
  author = {Steven N. Karp and Martha E. Precup},
  title = {Richardson tableaux and components of {S}pringer fibers equal to
    {R}ichardson varieties},
  year = {2025},
  eprint = {2506.20792},
  url = {https://arxiv.org/abs/2506.20792},
  journal = {arXiv e-prints}
}
  4. [MPS92]F. De Mari, C. Procesi and M. A. Shayman. Hessenberg varieties. Transactions of the American Mathematical Society, 332(2):529, August 1992.
    .bib
    @article{MariProcesiShayman1992,
  doi = {10.2307/2154181},
  url2 = {https://doi.org/10.2307/2154181},
  year = {1992},
  month = aug,
  publisher = {{JSTOR}},
  volume = {332},
  number = {2},
  pages = {529},
  author = {F. De Mari and C. Procesi and M. A. Shayman},
  title = {Hessenberg Varieties},
  journal = {Transactions of the American Mathematical Society}
}
  5. [Oh11]Suho Oh. Positroids and Schubert matroids. Journal of Combinatorial Theory. Series A, 118(8):2426–2435, 2011.
    .bib
    @article{Oh2011,
  author = {Suho Oh},
  title = {Positroids and {S}chubert matroids},
  journal = {Journal of Combinatorial Theory. Series A},
  volume = {118},
  number = {8},
  pages = {2426--2435},
  year = {2011},
  doi = {10.1016/j.jcta.2011.06.006},
  url = {https://doi.org/10.1016/j.jcta.2011.06.006}
}
  6. [Pos06]Alexander Postnikov. Total positivity, Grassmannians, and networks. arXiv:math/0609764, 2006.
    .bib
    @article{Postnikov2006x,
Author = {Alexander Postnikov},
Title = {Total positivity, {G}rassmannians, and networks},
Year = {2006},
Eprint = {math/0609764},
  url = {https://arxiv.org/abs/math/0609764},
journal = {arXiv e-prints}
}
  7. [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}
}
  8. [Spr78]Tonny A. Springer. A construction of representations of Weyl groups. Inventiones Mathematicae, 44:279–293, 1978.
    .bib
    @article{Springer1978,
  doi = {10.1007/BF01403165},
  url = {https://doi.org/10.1007/BF01403165},
  year = {1978},
  publisher = {Springer Nature},
  volume = {44},
  pages = {279--293},
  author = {Tonny A. Springer},
  title = {A construction of representations of {W}eyl groups},
  journal = {Inventiones Mathematicae}
}

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