A Pieri-type formula for isotropic flag manifolds (Q2782658)

An integral basis of the cohomology of a flag manifold \(G/B\) is the Schubert classes \(\mathfrak S_w\) indexed by the elements \(w\) of the Weyl group of \(G\). Hence there are, for all pairs of elements \(u,v\) in the Weyl group, integers \(c_{uv}^w\) such that NEWLINE\[NEWLINE\mathfrak S_u\mathfrak S_v =\sum_wc_{uv}^w \mathfrak S_w,NEWLINE\]NEWLINE where the summation is over the elements in the Weyl group. A Pieri type formula is a formula that describes the structure constants \(c_{uv}^w\) when \(\mathfrak S _v\) is a special Schubert class pulled back from the projection \(G/B\to G/P\), where \(P\) is a maximal parabolic subgroup. When \(G=\text{Gl}_n(\mathbb{C})\) the classical Pieri formula gives such a description. For other \(G\) there are formulas by \textit{H. Hiller} and \textit{B. Boe} [Adv. Math. 62, 49-67 (1986; Zbl 0611.14036)] and by \textit{P. Pragacz} and \textit{J. Ratajski} [J. Reine Angew. Math. 476, 143-189 (1996; Zbl 0847.14029); C. R. Acad. Sci., Paris, Sér. I 317, 1035-1040 (1993; Zbl 0812.14034); Manuscr. Math. 79, 127-151 (1993; Zbl 0789.14041)]. When \(G=\text{Gl}_n(\mathbb{C})\) an interpretation of the structure constants \(c_{uv}^w\) in Pieri's formula, in terms of chains in the Bruhat order, was conjectured by \textit{N. Bergeron} and \textit{S. Billey} [Exp. Math. 2, 257-269 (1993; Zbl 0803.05054)] and given algebraic, geometric and combinatorial proofs by \textit{A. Postnikov} [Prog. Math. 172, 371-383 (1999; Zbl 0944.14019)], \textit{F. Sottile} [Ann. Inst. Fourier 46, 89-110 (1996; Zbl 0837.14041)], and \textit{M. Kogan} and \textit{A. Kumar} [Proc. Am. Math. Soc. 130, 2525-2534 (2002; Zbl 1001.05121)], respectively. The main result of the present article are analogous Pieri type formulas when \(G\) is \(\text{Sp}_{2n}(\mathbb{C})\) and \(\text{SO}_{2n+1}(\mathbb{C})\). One of the techniques used is to explicitly determine triple intersections of Schubert varieties. \textit{F. Sottile} has used this technique with success earlier [see, e.g., Colloq. Math. 82, 49-63 (1999; Zbl 0977.14023)], and shows that the coefficients in the Pieri type formulas are the intersection number of a linear space with a collection of quadrics, and thus are either \(0\) or a power of \(2\).