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Frobenius splitting and M\"obius inversion · Around

2009

Frobenius splitting and M\"obius inversion

Knutson, Allen

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We show that the fundamental class in K-homology of a Frobenius split scheme can be computed as a certain alternating sum over irreducible varieties, with the coefficients computed using M\"obius inversion on a certain poset.

If G/P is a generalized flag manifold and X is an irreducible subvariety homologous to a multiplicity-free union of Schubert varieties, then using a result of Brion we show how to compute the K_0-class [X] in K_0(G/P) from the Chow class in A_*(G/P).

Built on

A. Buch, A Littlewood-Richardson rule for the K K -theory of Grassmannians, Acta Math. 189 (2002), no. 1, 37–78. http://arxiv.org/abs/math.AG/0004137
2002
Earlier work this paper cites.
M. Brion, Multiplicity-free subvarieties of flag varieties, Contemporary Math. 331, 13-23, Amer. Math. Soc., Providence, 2003. http://arxiv.org/abs/math.AG/0211028
2003
Earlier work this paper cites.

Similar

by same author, S. Kumar, Frobenius Splitting Methods in Geometry and Representation Theory, Progress in Mathematics, 231. Birkh�user Boston, Inc., Boston, MA, 2005
2005
Cited alongside, same era.
L. Williams, Enumeration of totally positive Grassmann cells, Advances in Mathematics, Volume 190, Issue 2, January 2005, pages 319–342. http://arxiv.org/abs/math.CO/0307271
Original
2005
Cited alongside, same era.
A. Knutson, T. Lam, D. Speyer, Positroid varieties I: juggling and geometry. In preparation
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A. Postnikov, Total positivity, Grassmannians, and networks. http://arxiv.org/abs/math.CO/0609764
Cited in the paper.
by same author, D. Speyer, L. Williams, Matching polytopes, toric geometry, and the non-negative part of the Grassmannian. http://arxiv.org/abs/arXiv:0706.2501
Original
Cited in the paper.
M. Snider, A combinatorial approach to multiplicity-free Richardson subvarieties of the Grassmannian, preprint
Cited in the paper.

Then

T. Lam, L. Williams, Total positivity for cominuscule Grassmannians, New York Journal of Mathematics, Volume 14, 2008, 53–99. http://arxiv.org/abs/arXiv:0710.2932
Original
2008
Later among the works it cites.

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