Schur polynomials and weighted Grassmannians

In this paper, we introduce a family of symmetric polynomials by specializing the factorial Schur polynomials. We show that it represents the weighted Schubert classes of the cohomology of the weighted Grassmannian introduced by Corti–Reid, and we regard it as an analogue of the Schur polynomials. F...

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Bibliographic Details
Published in:Journal of algebraic combinatorics 2015-11, Vol.42 (3), p.875-892
Main Authors: Abe, Hiraku, Matsumura, Tomoo
Format: Article
Language:English
Subjects:
Combinatorics
Computer Science
Convex and Discrete Geometry
Group Theory and Generalizations
Lattices
Mathematics
Mathematics and Statistics
Order
Ordered Algebraic Structures
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Summary:In this paper, we introduce a family of symmetric polynomials by specializing the factorial Schur polynomials. We show that it represents the weighted Schubert classes of the cohomology of the weighted Grassmannian introduced by Corti–Reid, and we regard it as an analogue of the Schur polynomials. Furthermore, we prove that these polynomials are the characters of certain representations, and hence, we give an interpretation of the Schubert structure constants of the weighted Grassmannians as the (rational) multiplicities of the tensor products of the representations. We also derive two determinantal formulas for the weighted Schubert classes: One is in terms of the special weighted Schubert classes, and the other is in terms of the Chern classes of the tautological orbi-bundles.
ISSN:0925-9899
1572-9192
DOI:10.1007/s10801-015-0608-z