The symmetric functions catalog

#K-theory

K-theory studies a space by looking at vector bundles, or more generally sheaves, on that space. The basic construction is a Grothendieck group: start with geometric objects, add formal differences, and impose the relation that a middle object is the sum of its subobject and quotient.

For algebraic combinatorics, the main reason to care is that K-theory is the natural home for Grothendieck polynomials. Ordinary Schubert calculus multiplies fundamental classes of Schubert varieties in cohomology. K-theoretic Schubert calculus instead multiplies the classes of their structure sheaves, and the resulting polynomial representatives are Grothendieck polynomials rather than Schubert polynomials.

#The Grothendieck group idea

Suppose we have a class of objects with exact sequences, for instance vector bundles on a space \(X.\) The Grothendieck group is generated by symbols \([E],\) one for each vector bundle \(E,\) with the relation \[[E]=[E']+[E'']\] whenever \[0\longrightarrow E'\longrightarrow E\longrightarrow E'' \longrightarrow 0\] is exact. Direct sum gives addition: \[[E\oplus F]=[E]+[F].\] Tensor product gives multiplication: \[[E]\cdot[F]=[E\otimes F].\] The resulting ring is often denoted \(K^0(X).\)

This should be compared with cohomology. Cohomology remembers cycle classes and intersection numbers. K-theory remembers bundle and sheaf classes. There are maps from K-theory to cohomology, such as the Chern character, but the multiplication in K-theory keeps information which is invisible if we only look at fundamental cohomology classes.

#Coherent sheaves

In algebraic geometry one often uses coherent sheaves instead of only vector bundles. The group built from coherent sheaves is usually denoted \(K_0(X).\) For a short exact sequence \[0\longrightarrow \mathcal F'\longrightarrow \mathcal F \longrightarrow \mathcal F''\longrightarrow 0\] we impose \[[\mathcal F]=[\mathcal F']+[\mathcal F''].\]

#K-theoretic Schubert calculus

Let \(X\) be a Grassmannian or flag variety, and let \(X_w\) be a Schubert variety. In cohomology we use the Schubert class \([X_w].\) In K-theory we use the class \[[\mathcal O_{X_w}]\] of its structure sheaf. These classes form a basis of the K-theory of the flag variety or Grassmannian. Since these varieties are smooth, the product of coherent-sheaf classes can be defined using the derived tensor product, and multiplication has the form \[[\mathcal O_{X_u}]\,[\mathcal O_{X_v}] = \sum_w c_{u,v}^w[\mathcal O_{X_w}].\] The constants \(c_{u,v}^w\) are K-theoretic analogues of Schubert structure constants.

In the Grassmannian case, these constants are computed by K-theoretic Littlewood–Richardson rules. A key combinatorial model is given by set-valued tableaux, introduced by A. S. Buch for the K-theory of Grassmannians [Buc02]. On the polynomial side, A. Lascoux’s Grothendieck polynomials represent K-theory Schubert classes, namely the structure-sheaf classes of Schubert varieties in the Grothendieck ring of the flag variety [Las90].

#How to read more

For a general entry point to topological K-theory, see M. F. Atiyah’s book [Ati67]. For Schubert calculus and the Grassmannian side, W. Fulton’s book on Young tableaux [Ful97] and M. Gillespie’s introduction [Gil19] are good first reads. For the K-theoretic Grassmannian story and set-valued tableaux, see Buch’s paper [Buc02]. For the polynomial representatives used throughout this site, continue with Grothendieck polynomials.

Bibliography

1. [Ati67]Michael F. Atiyah. K-theory. W. A. Benjamin, 1967.
   .bib

   Copy Download

   @book{Atiyah1967KTheory,
     author = {Atiyah, Michael F.},
     title = {K-Theory},
     year = {1967},
     publisher = {W. A. Benjamin},
     address = {New York}
   }
   

2. [Buc02]Anders Skovsted Buch. A Littlewood–Richardson rule for the K-theory of Grassmannians. Acta Mathematica, 189(1):37–78, 2002.
   .bib

   Copy Download

   @article{Buch2002,
   year = {2002},
   issn = {0001-5962},
   journal = {Acta Mathematica},
   volume = {189},
   number = {1},
   title = {A {L}ittlewood--{R}ichardson rule for the {K}-theory of {G}rassmannians},
   doi = {10.1007/BF02392644},
   publisher = {Kluwer Academic Publishers},
   author = {Anders Skovsted Buch},
   pages = {37--78},
   language = {English}
   }
   

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

   Copy Download

   @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}
   }
   

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

   Copy Download

   @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}
   }
   

5. [Las90]Alain Lascoux. Anneau de Grothendieck de la variété de drapeaux. Modern birkhäuser classics:1–34, 1990.
   .bib

   Copy Download

   @incollection{Lascoux1990Grothendieck,
     doi = {10.1007/978-0-8176-4576-2_1},
     url2 = {https://doi.org/10.1007/978-0-8176-4576-2_1},
     publisher = {Birkh{\"{a}}user Boston},
     pages = {1--34},
     year = {1990},
     author = {Alain Lascoux},
     title = {Anneau de {G}rothendieck de la vari{\'{e}}t{\'{e}} de drapeaux},
     booktitle = {Modern Birkh{\"{a}}user Classics}
   }