Root games on Grassmannians
Kevin Purbhoo
University of British Columbia Department of Mathematics 1984 Mathematics Rd. Vancouver BC V6T 1Z2 Canada 1984 Mathematics Rd. Vancouver BC V6T 1Z2 Canada
DOI: 10.1007/s10801-006-0033-4
Abstract
We recall the root game, introduced in [8], which gives a fairly powerful sufficient condition for non-vanishing of Schubert calculus on a generalised flag manifold G/ B. We show that it gives a necessary and sufficient rule for non-vanishing of Schubert calculus on Grassmannians. In particular, a Littlewood-Richardson number is non-zero if and only if it is possible to win the corresponding root game. More generally, the rule can be used to determine whether or not a product of several Schubert classes on Gr l (\Bbb C n ) is non-zero in a manifestly symmetric way. Finally, we give a geometric interpretation of root games for Grassmannian Schubert problems.
Pages: 239–258
Keywords: keywords Schubert calculus; Littlewood-Richardson numbers; grassmannians
Full Text: PDF
References
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2. I. Coskun, “A Littlewood-Richardson rule for two-step flag manifolds,” preprint: http://wwwmath.mit.edu/~coskun/.
3. W. Fulton, Young Tableaux with Applications to Representation Theory and Geometry, Cambridge U.P., New York,
1997. Springer J Algebr Comb (2007) 25:239-258
4. W. Fulton, “Eigenvalues, invariant factors, highest weights, and Schubert calculus,” Bull. Amer. Math. Soc. 37 (2000), 209-250.
5. S. Fomin and C. Greene, “A Littlewood-Richardson miscellany,” European J. Combin. 14(3) (1993), 191-212.
6. A. Knutson, T. Tao, and C. Woodward, “The honeycomb model of GLn(C) tensor products II. Puzzles determine facets of the Littlewood-Richardson cone,” J. Amer. Math. Soc. 17(1) (2004), 19-48 (electronic).
7. D.E. Littlewood and A.R. Richardson, “Group characters and algebra,” Philos. Trans. Roy. Soc. London. 233 (1934), 99-141.
8. K. Purbhoo, “Vanishing and non-vanishing criteria in Schubert calculus,” Int. Math. Res. Not. (2006), 24590, 1-38.
9. J.B. Remmel and R. Whitney, “Multiplying Schur functions,” J. of Algorithms. 5 (1984), 471-487.
10. R. Vakil, “A geometric Littlewood-Richardson rule,” to appear in Ann. Math, arXiv preprint: math.AG/0302294.
11. A.V. Zelevinsky, “A generalization of the Littlewood-Richardson rule and the Robinson-Schensted- Knuth correspondence,” J. Algebra. 69(1) (1981), 82-94.