group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
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 15th of Hilbert's problems.
The basic data to be fixed is a sequence of inclusions
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
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
From the above data one obtains homomorphisms of spaces with $G$-action forming correspondences (“generalized twistor correspondence”)
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
be the inclusion of the Schubert varieties, then push-forward of the unit classes allong these inclusions defined Schubert classes
For equivariant K-theory this is discussed in (Ganter 12, 8.2). For equivariant elliptic cohomology in (Ganter 12, 8.3)
With Schubert classes $[X_w]$ defines as above in a multiplicative cohomology theory, the Schubert product formula is
for some coefficients $\{c^w_{u v}\}$, to be determined.
[eom]: Frank Sotile, 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
Discussion of Schubert calculus in generalized cohomology theories is in
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, (arxiv/1212.5742)
Nora Ganter, The elliptic Weyl character formula (arXiv:1206.0528)