The symmetric functions catalog

#Schubert calculus

Schubert calculus is the use of Schubert varieties and their intersection products to solve enumerative problems. In the most classical case, this means computing in the cohomology ring of a Grassmannian. The combinatorics is governed by partitions, Young diagrams, Schur polynomials, Pieri rules, Giambelli determinants, and Littlewood–Richardson coefficients.

Good introductions include W. Fulton’s book on Young tableaux [Ful97] and M. Gillespie’s tutorial [Gil19], which is adapted from her Schubert calculus mini-course [Gil17]. For the flag-variety and polynomial side, see also Schubert polynomials. For a broader map of the geometric objects, see varieties in algebraic combinatorics.

#Grassmannians

Let \(V\) be an \(n\)-dimensional complex vector space. The Grassmannian \(\operatorname{Gr}(k,V),\) often written \(\operatorname{Gr}(k,n),\) is the space of \(k\)-dimensional linear subspaces \(W\subseteq V\); see also the varieties page for the Plücker-coordinate viewpoint.

The projective interpretation is often the most concrete. A \(k\)-dimensional linear subspace of \(V=\setC^n\) gives a projective \((k-1)\)-plane in \(\mathbb{P}^{n-1}.\) Thus \(\operatorname{Gr}(2,4)\) parametrizes lines in \(\mathbb{P}^3,\) and \(\operatorname{Gr}(3,6)\) parametrizes projective planes in \(\mathbb{P}^5.\)

The dimension is \[\dim \operatorname{Gr}(k,n)=k(n-k).\] One way to see this is that a generic \(k\)-plane can be represented by a \(k\times n\) matrix in row echelon form. After choosing pivot columns, the remaining free entries form a \(k(n-k)\)-dimensional affine chart.

#Schubert classes on the Grassmannian

Fix a complete flag \[0=F_0\subset F_1\subset F_2\subset\dotsb\subset F_n=V, \qquad \dim F_i=i.\] Schubert classes in \(\operatorname{Gr}(k,n)\) are indexed by partitions \(\lambda\) fitting in the \(k\times(n-k)\) rectangle: \[n-k\geq \lambda_1\geq \lambda_2\geq\dotsb\geq\lambda_k\geq 0.\] The corresponding Schubert variety is \[\Omega_\lambda(F_\bullet) \coloneqq \left\{ W\in\operatorname{Gr}(k,n): \dim(W\cap F_{n-k+i-\lambda_i})\geq i \text{ for } 1\leq i\leq k \right\}.\] Its cohomology class is denoted \(\sigma_\lambda,\) and its codimension is \(|\lambda|.\) These are the Grassmannian instances of the Schubert varieties described on the varieties page.

The Schubert classes form an additive basis: \[H^*(\operatorname{Gr}(k,n),\setZ) = \bigoplus_{\lambda\subseteq (n-k)^k}\setZ\,\sigma_\lambda.\] The top class is \(\sigma_{(n-k)^k},\) and integration over the Grassmannian extracts the coefficient of this class.

#Dictionary for \(\operatorname{Gr}(2,4)\)

The Grassmannian \(\operatorname{Gr}(2,4)\) has dimension \(4\) and parametrizes lines in \(\mathbb{P}^3.\) The partitions fit in a \(2\times 2\) box: \[\emptyset,\quad (1),\quad (2),\quad (1,1),\quad (2,1),\quad (2,2).\] With respect to a fixed flag in \(\setC^4,\) the corresponding geometric conditions are: \[\begin{array}{c|l} \lambda & \text{condition on the line } L\subseteq\mathbb{P}^3\\ \hline \emptyset & \text{no condition}\\ (1) & L \text{ meets a fixed line}\\ (2) & L \text{ contains a fixed point}\\ (1,1) & L \text{ lies in a fixed plane}\\ (2,1) & L \text{ contains a fixed point and lies in a fixed plane}\\ (2,2) & L \text{ is a fixed line.} \end{array}\] The codimension is the number of boxes in \(\lambda.\)

#Computation rules

For ordinary Grassmannians, multiplication of Schubert classes is computed by the same Littlewood–Richardson coefficients as multiplication of Schur polynomials, with the restriction that partitions must stay inside the \(k\times(n-k)\) rectangle: \[\sigma_\lambda\sigma_\mu = \sum_{\nu\subseteq(n-k)^k} c_{\lambda,\mu}^{\nu}\sigma_\nu.\] Equivalently, multiply \(\schurS_\lambda\schurS_\mu\) and discard or straighten terms which do not define classes in the Grassmannian. More precisely, one uses the usual quotient presentation of \(H^*(\operatorname{Gr}(k,n)),\) where the Schur classes outside the rectangle are reduced by the Grassmannian relations.

#Pieri rule

The special class \(\sigma_r\) corresponds to a single row of length \(r.\) The Pieri rule says \[\sigma_r\sigma_\lambda = \sum_\mu \sigma_\mu,\] where the sum is over partitions \(\mu\) such that \(\mu/\lambda\) is a horizontal strip of size \(r\) and \(\mu\) still fits in the \(k\times(n-k)\) rectangle. There is a dual version with \(\sigma_{1^r}\) and vertical strips.

#Giambelli formula

The Giambelli formula expresses every Schubert class in terms of the special classes \(\sigma_r:\) \[\sigma_\lambda = \det\left(\sigma_{\lambda_i+j-i}\right)_{1\leq i,j\leq k},\] where \(\sigma_0=1\) and \(\sigma_r=0\) if \(r\lt{}0\) or \(r\gt{}n-k.\) For the flag-variety polynomial analogue, compare the Giambelli formula for double Schubert polynomials.

#Littlewood–Richardson truncation

Littlewood–Richardson multiplication can be used directly, provided we keep only the classes which fit in the Grassmannian rectangle.

#The ring of \(\operatorname{Gr}(2,4)\)

Here is the complete multiplication table for Schubert classes in \(H^*(\operatorname{Gr}(2,4)).\) Products not visible by symmetry are obtained by commutativity. \[\begin{array}{c|cccccc} \cdot & 1 & \sigma_1 & \sigma_2 & \sigma_{1,1} & \sigma_{2,1} & \sigma_{2,2}\\ \hline 1 & 1 & \sigma_1 & \sigma_2 & \sigma_{1,1} & \sigma_{2,1} & \sigma_{2,2}\\ \sigma_1 & \sigma_1 & \sigma_2+\sigma_{1,1} & \sigma_{2,1} & \sigma_{2,1} & \sigma_{2,2} & 0\\ \sigma_2 & \sigma_2 & \sigma_{2,1} & \sigma_{2,2} & \sigma_{2,2} & 0 & 0\\ \sigma_{1,1} & \sigma_{1,1} & \sigma_{2,1} & \sigma_{2,2} & \sigma_{2,2} & 0 & 0\\ \sigma_{2,1} & \sigma_{2,1} & \sigma_{2,2} & 0 & 0 & 0 & 0\\ \sigma_{2,2} & \sigma_{2,2} & 0 & 0 & 0 & 0 & 0 \end{array}\]

For example, \[\sigma_1^2=\sigma_2+\sigma_{1,1}, \qquad \sigma_1^3=2\sigma_{2,1}, \qquad \sigma_1^4=2\sigma_{2,2}.\] Since \(\sigma_{2,2}\) is the class of a point, integration gives \[\int_{\operatorname{Gr}(2,4)}\sigma_1^4=2.\]

#Poincaré duality

Let \(\lambda\subseteq(n-k)^k.\) The Poincaré dual partition is \[\lambda^\vee = (n-k-\lambda_k,\dotsc,n-k-\lambda_2,n-k-\lambda_1).\] Then \[\int_{\operatorname{Gr}(k,n)} \sigma_\lambda\sigma_\mu = \begin{cases} 1, & \mu=\lambda^\vee,\\ 0, & \text{otherwise.} \end{cases}\]

#Flag varieties and Schubert polynomials

Schubert calculus on the complete flag variety is indexed by permutations rather than by partitions. The polynomial representatives are the Schubert polynomials \(\schubert_w(\xvec)\) introduced by A. Lascoux and M.-P. Schützenberger [LS82].

For the full flag variety in \(\setC^3,\) with permutations written in one-line notation, the six divided-difference Schubert polynomials are \[\begin{array}{c|c} w & \schubert_w(x_1,x_2)\\ \hline 123 & 1\\ 213 & x_1\\ 132 & x_1+x_2\\ 231 & x_1x_2\\ 312 & x_1^2\\ 321 & x_1^2x_2. \end{array}\] Therefore \[\schubert_{213}\schubert_{132} = x_1(x_1+x_2) = \schubert_{312}+\schubert_{231}.\] This is the flag-variety analogue of a Schubert-class multiplication computation; Grassmannian permutations recover the Schur-polynomial subcase, as explained on the Schubert polynomial page.

More generally, the problem of finding a positive combinatorial rule for the structure constants \[\schubert_u(\xvec)\schubert_v(\xvec) = \sum_w c_{u,v}^w\schubert_w(\xvec)\] is a central open problem; see Schubert Littlewood–Richardson coefficients.

Bibliography

1. [Ful97]William Fulton. Young tableaux: With applications to representation theory and geometry. London mathematical society student texts (book 35). Cambridge University Press, 1997.
   .bib

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   @book{Fulton1997,
     title = {Young Tableaux: With Applications to Representation Theory and Geometry},
     doi = {10.1017/cbo9780511626241},
     author = {William Fulton},
     isbn = {978-0511626241},
     series = {London Mathematical Society Student Texts (Book 35)},
     year = {1997},
     publisher = {Cambridge University Press}
   }
   

2. [Gil17]Maria Gillespie. Schubert calculus mini-course. 2017. Mathematical Gemstones blog post
   .bib

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   @misc{Gillespie2017SchubertMiniCourse,
     author = {Maria Gillespie},
     title = {Schubert Calculus Mini-Course},
     year = {2017},
     month = sep,
     url = {https://www.mathematicalgemstones.com/gemstones/sapphire/schubert-calculus-mini-course/},
     note = {Mathematical Gemstones blog post}
   }
   

3. [Gil19]Maria Gillespie. Variations on a Theme of Schubert Calculus. Recent trends in algebraic combinatorics:115–158, 2019.
   .bib

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   @inbook{Gillespie2019,
     author = {Gillespie, Maria},
     title = {Variations on a {T}heme of {S}chubert {C}alculus},
     year = {2019},
     pages = {115--158},
     publisher = {Springer International Publishing},
     doi = {10.1007/978-3-030-05141-9_4},
     url = {http://dx.doi.org/10.1007/978-3-030-05141-9_4},
     issn = {2364-5741},
     isbn = {9783030051419},
     booktitle = {Recent Trends in Algebraic Combinatorics},
     eprint = {1804.08164}
   }
   

4. [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

   Copy Download

   @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.}
   }