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''Schubert class''

The cycle class of a [[Schubert variety|Schubert variety]] in the [[Cohomology ring|cohomology ring]] of a complex flag manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130100/s1301001.png" /> (cf. also [[Flag structure|Flag structure]]), also called a Schubert class. Here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130100/s1301002.png" /> is a semi-simple [[Linear algebraic group|linear algebraic group]] and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130100/s1301003.png" /> is a [[Parabolic subgroup|parabolic subgroup]]. Schubert cycles form a basis for the cohomology groups [[#References|[a4]]], [[#References|[a1]]], 14.12, of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130100/s1301004.png" /> (cf. also [[Cohomology group|Cohomology group]]). They arose [[#References|[a4]]] as representatives of Schubert conditions on linear subspaces of a vector space in the [[Schubert calculus|Schubert calculus]] for enumerative geometry [[#References|[a3]]]. The justification of Schubert's calculus in this context by C. Ehresmann [[#References|[a2]]] realized Schubert cycles as cohomology classes Poincaré dual to the fundamental homology cycles of Schubert varieties in the Grassmannian. While Schubert, Ehresmann and others worked primarily on the Grassmannian, the pertinent features of the Grassmannian extend to general flag varieties <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130100/s1301005.png" />, giving Schubert cycles as above.

More generally, when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130100/s1301006.png" /> is a semi-simple linear algebraic group over a field, there are Schubert cycles associated to Schubert varieties in each of the following theories for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130100/s1301007.png" />: singular (or de Rham) cohomology, the Chow ring, K-theory, or equivariant or quantum versions of these theories. For each, the Schubert cycles form a basis over the base ring. For the cohomology or the Chow ring, the Schubert cycles are universal characteristic classes for (flagged) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130100/s1301008.png" />-bundles. In particular, certain special Schubert cycles for the Grassmannian are universal Chern classes for vector bundles (cf. also [[Chern class|Chern class]]).

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> A. Borel, "Linear algebraic groups" , ''Grad. Texts Math.'' , '''126''' , Springer (1991) (Edition: Second) {{MR|1102012}} {{ZBL|0726.20030}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> C. Ehresmann, "Sur la topologie de certains espaces homogènes" ''Ann. Math.'' , '''35''' (1934) pp. 396–443 {{MR|1503170}} {{ZBL|0009.32903}} {{ZBL|60.1223.05}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> H. Schubert, "Kälkul der abzählenden Geometrie" , Springer (1879) (Reprinted (with an introduction by S. Kleiman): 1979) {{MR|0555576}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> H. Schubert, "Losüng des Charakteristiken-Problems für lineare Räume beliebiger Dimension" ''Mitt. Math. Ges. Hamburg'' (1886) pp. 135–155 {{MR|}} {{ZBL|}} </TD></TR></table>