nLab Schubert calculus

Idea

Schubert calculus is a formal calculus in enumerative geometry, which geometrically reduces to the combinatorics and intersection theory of so-called Schubert cells in Grassmannians.

Schubert calculus is concerned with the ring structure on the cohomology of flag varieties and Schubert varieties. Traditionally this is considered for ordinary cohomology (see References – traditional) later also for generalized cohomology theories (see References – In generalized cohomology), notably in complex oriented cohomology theory such as K-theory, elliptic cohomology and algebraic cobordism.

The rigorous foundations of Schubert calculus is the content of the 15th of Hilbert's problems.

Details

Flag varieties and Schubert varieties

The basic data to be fixed is a sequence of inclusions

$$T \subset B \subset G$$

where

• $G$ is a connected complex reductive algebraic group

• $B$ is a Borel subgroup

• $T$ is a maximal torus.

This induces

• the Weyl group $W_0 = N(T)/T$;

• the character lattice $\mathfrak{h}_{\mathbb{Z}}^\ast = Hom(T, \mathbb{C}^\times)$;

• the cocharacter lattice $\mathfrak{h}_{\mathbb{Z}} = Hom(\mathbb{C}^\times, T)$.

• a standard parabolic subgroup of $G$ is a subgroup $P_J$ including $B$ such that $G/P$ is a projective variety;

  parabolic subgroup is one conjugate to the standard parabolic subgroup.

• the flag variety $G/B$;

• the partial flag varieties $G/P_J$

• the Bruhat decomposition is the coproduct decomposition

  $$G = \underset{w \in W_0}{\coprod} B w B$$
  $$G = \underset{u \in W^J}{\coprod} B u P_j$$

  with

  • $W_J \coloneqq \{v \in W_0 | v T \subset P_J\}$

  • $W^J \coloneqq \{coset\; representatives\; u \; of \; cosets \; in W_0/W_J\}$

• into the Schubert varieties

  $$X_w \coloneqq \overline{B w B} \subset G/B$$
  $$X_u^J \coloneqq \overline{B u P_J} \subset G/P_J \,.$$

Correspondences, pull-push and Schubert classes

From the above data one obtains homomorphisms of spaces with $G$-action forming correspondences (“generalized twistor correspondence”)

$$\array{ && G/B \\ & {}^{\mathllap{p_1}}\swarrow && \searrow^{\mathrlap{p_2}} \\ G/P_1 && && G/P_2 } \,.$$

e.g. (Ganter-Ram 12, p.4)

For fiber integration $(p_i)_!$ in generalized cohomology theories along these maps see (Ganter-Ram 12, 4.1)

Similarly, let

$$\sigma_w \;\colon\; X_w \hookrightarrow G/B$$

be the inclusion of the Schubert varieties, then push-forward of the unit classes allong these inclusions defined Schubert classes

$$[X_w] \coloneqq (\sigma_w)_!(w)$$

(Ganter-Ram 12, 5)

For equivariant K-theory this is discussed in (Ganter 12, 8.2). For equivariant elliptic cohomology in (Ganter 12, 8.3)

Schubert products

With Schubert classes $[X_w]$ defines as above in a multiplicative cohomology theory, the Schubert product formula is

$$[X_u][X_v] = \underset{w \in W_0}{\sum} c^w_{u v} [X_w]$$

for some coefficients $\{c^w_{u v}\}$, to be determined.

(Ganter-Ram 12, 6)

Related concepts

• Borel-Weil-Bott theorem

  • orbit method

  • Dirac induction

• quantum Schubert calculus

• twistor space,

• twistor correspondence, horocycle correspondence, Grothendieck-Springer correspondence

References

Traditional

• [eom]: Frank Sottile, Schubert calculus

• wikipedia Schubert calculus

• H. Schubert, Kalkül der abzählenden Geometrie, Springer (1879) (Reprinted (with an introduction by S. Kleiman) 1979), MR0555576

• S.L. Kleiman, D. Laksov, Schubert calculus, Amer. Math. Monthly 79 (1972) pp. 1061–1082, MR0323796, jstor

In generalized cohomology theory

Discussion of Schubert calculus in generalized cohomology theories is in

• Paul Bressler, Sam Evens. The Schubert calculus, braid relations,

  and generalized cohomology_. Trans. Amer. Math. Soc., 317(2):799–811, 1990

• Paul Bressler, Sam Evens, Schubert calculus in complex cobordism Trans. Amer. Math. Soc., 331(2):799–813, 1992

• Baptiste Calmès, Victor Petrov, Kirill Zainoulline, Invariants, torsion indices and oriented cohomology of complete flags May 200 (web)

• Jens Hornbostel, Valentina Kiritchenko, Schubert calculus for algebraic cobordism. J. Reine Angew. Math., 656:59–85, 2011

• Nora Ganter, Arun Ram, Generalized Schubert calculus, J. Ramanujan Math. Soc. 28A (Spec. Issue-2013) 1-42 arxiv/1212.5742

• Nora Ganter, The elliptic Weyl character formula (arXiv:1206.0528)

category: combinatorics, algebraic geometry