A Pieri-Type Formula for H*T(SLn( )/B)

The singular cohomology of the Grassmann variety of k-planes in n has a basis {sν} indexed by partitions. The classical Pieri formula is an explicit rule for determining the coefficients in the expansion of the cup products1m sλ = ∑ c1m,λμsμ,where 1m is a column of length m and s1m is the mth Chern class of the tautological bundle. Lascoux and Schutzenberger [C. R. Acad. Sci. Paris294 (1982), 447–450] formulated a generalization of Pieriʹs formula to the cohomology of the flag variety SLn( )/B and briefly indicated an algebraic proof. (Manivel [Cours Spécialisés 3 (1998)] provides details of this proof.) A geometric proof was given by Sottile [Ann. Inst. Fourier (Grenoble) 46 (1996), 89–110]. In this paper we state and prove a generalization of this Pieri-type formula for the T-equivariant cohomology of the flag variety. We use the algebraic description of the T-equivariant cohomology of the flag variety due to Kostant and Kumar [Adv. Math.62 (1986), 187–237] and Arabia [Bull. Soc. Math. France117 (1989), 129–165], and our new formula exposes an equality of certain structure Constants in this algebra. Our proof is an induction based on the original idea in Lascoux and Schützenberger.