Geodesic
Affine partitions and affine Grassmannians
The electronic journal of combinatorics, The Björner Festschrift volume, Tome 16 (2009) no. 2
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We give a bijection between certain colored partitions and the elements in the quotient of an affine Weyl group modulo its Weyl group. By Bott's formula these colored partitions give rise to some partition identities. In certain types, these identities have previously appeared in the work of Bousquet-Melou-Eriksson, Eriksson-Eriksson and Reiner. In other types the identities appear to be new. For type $A_{n}$, the affine colored partitions form another family of combinatorial objects in bijection with $(n+1)$-core partitions and $n$-bounded partitions. Our main application is to characterize the rationally smooth Schubert varieties in the affine Grassmannians in terms of affine partitions and a generalization of Young's lattice which refines weak order and is a subposet of Bruhat order. Several of the proofs are computer assisted.
DOI : 10.37236/84
Classification : 14M15
Mots-clés : rationally smooth Schubert varieties, Young's lattice 
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     author = {Sara C. Billey and Stephen A. Mitchell},
     title = {Affine partitions and affine {Grassmannians}},
     journal = {The electronic journal of combinatorics},
     year = {2009},
     volume = {16},
     number = {2},
     doi = {10.37236/84},
     zbl = {1286.05179},
     url = {http://geodesic.mathdoc.fr/articles/10.37236/84/}
}
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Sara C. Billey; Stephen A. Mitchell. Affine partitions and affine Grassmannians. The electronic journal of combinatorics, The Björner Festschrift volume, Tome 16 (2009) no. 2. doi: 10.37236/84

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