A Littlewood-Richardson rule for Grassmannian permutations

Authors:
Kevin Purbhoo and Frank Sottile

Journal:
Proc. Amer. Math. Soc. **137** (2009), 1875-1882

MSC (2000):
Primary 14N15; Secondary 05E10

Published electronically:
January 8, 2009

MathSciNet review:
2480266

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Abstract | References | Similar Articles | Additional Information

Abstract: We give a combinatorial rule for computing intersection numbers on a flag manifold which come from products of Schubert classes pulled back from Grassmannian projections. This rule generalizes the known rule for Grassmannians.

**[BS98]**Nantel Bergeron and Frank Sottile,*Schubert polynomials, the Bruhat order, and the geometry of flag manifolds*, Duke Math. J.**95**(1998), no. 2, 373–423. MR**1652021**, 10.1215/S0012-7094-98-09511-4**[Co07]**I. Coskun,*A Littlewood-Richardson rule for two-step flag varieties*, manuscript, 2007.**[Ful97]**William Fulton,*Young tableaux*, London Mathematical Society Student Texts, vol. 35, Cambridge University Press, Cambridge, 1997. With applications to representation theory and geometry. MR**1464693****[Knu00]**Allen Knutson,*Descent-cycling in Schubert calculus*, Experiment. Math.**10**(2001), no. 3, 345–353. MR**1917423****[Kog01]**Mikhail Kogan,*RC-graphs and a generalized Littlewood-Richardson rule*, Internat. Math. Res. Notices**15**(2001), 765–782. MR**1849481**, 10.1155/S1073792801000393**[Mon59]**D. Monk,*The geometry of flag manifolds*, Proc. London Math. Soc. (3)**9**(1959), 253–286. MR**0106911****[RSSS06]**Jim Ruffo, Yuval Sivan, Evgenia Soprunova, and Frank Sottile,*Experimentation and conjectures in the real Schubert calculus for flag manifolds*, Experiment. Math.**15**(2006), no. 2, 199–221. MR**2253007****[Sot96]**Frank Sottile,*Pieri’s formula for flag manifolds and Schubert polynomials*, Ann. Inst. Fourier (Grenoble)**46**(1996), no. 1, 89–110 (English, with English and French summaries). MR**1385512****[Sta00]**Richard P. Stanley,*Positivity problems and conjectures in algebraic combinatorics*, Mathematics: frontiers and perspectives, Amer. Math. Soc., Providence, RI, 2000, pp. 295–319. MR**1754784**

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Additional Information

**Kevin Purbhoo**

Affiliation:
Department of Combinatorics and Optimization, University of Waterloo, Waterloo, ON, N2L 3G1 Canada

Email:
kpurbhoo@math.uwaterloo.ca

**Frank Sottile**

Affiliation:
Department of Mathematics, Texas A&M University, College Station, Texas 77843

Email:
sottile@math.tamu.edu

DOI:
https://doi.org/10.1090/S0002-9939-09-09637-3

Keywords:
Flag manifold,
Grassmannian,
Littlewood-Richardson rule

Received by editor(s):
September 14, 2007

Received by editor(s) in revised form:
May 2, 2008

Published electronically:
January 8, 2009

Additional Notes:
The work of the second author was supported by NSF CAREER grant DMS-0538734

Communicated by:
Jim Haglund

Article copyright:
© Copyright 2009
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.