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Schubert decompositions for ind-varieties of generalized flags
- Source :
- Asian Journal of Mathematics, Asian Journal of Mathematics, International Press, 2017, ⟨10.4310/AJM.2017.v21.n4.a1⟩
- Publication Year :
- 2015
-
Abstract
- Let $\mathbf{G}$ be one of the ind-groups $GL(\infty)$, $O(\infty)$, $Sp(\infty)$ and $\mathbf{P}\subset \mathbf{G}$ be a splitting parabolic ind-subgroup. The ind-variety $\mathbf{G}/\mathbf{P}$ has been identified with an ind-variety of generalized flags in the paper "Ind-varieties of generalized flags as homogeneous spaces for classical ind-groups" (Int. Math. Res. Not. 2004, no. 55, 2935--2953) by I. Dimitrov and I. Penkov. In the present paper we define a Schubert cell on $\mathbf{G}/\mathbf{P}$ as a $\mathbf{B}$-orbit on $\mathbf{G}/\mathbf{P}$, where $\mathbf{B}$ is any Borel ind-subgroup of $\mathbf{G}$ which intersects $\mathbf{P}$ in a maximal ind-torus. A significant difference with the finite-dimensional case is that in general $\mathbf{B}$ is not conjugate to an ind-subgroup of $\mathbf{P}$, whence $\mathbf{G}/\mathbf{P}$ admits many non-conjugate Schubert decompositions. We study the basic properties of the Schubert cells, proving in particular that they are usual finite-dimensional cells or are isomorphic to affine ind-spaces. We then define Schubert ind-varieties as closures of Schubert cells and study the smoothness of Schubert ind-varieties. Our approach to Schubert ind-varieties differs from an earlier approach by H. Salmasian in "Direct limits of Schubert varieties and global sections of line bundles" (J. Algebra 320 (2008), 3187--3198).<br />Keywords: Classical ind-group, Bruhat decomposition, Schubert decomposition, generalized flag, homogeneous ind-variety. [26 pages]
- Subjects :
- Schubert decomposition
General Mathematics
01 natural sciences
[MATH.MATH-GR]Mathematics [math]/Group Theory [math.GR]
Combinatorics
Mathematics::Algebraic Geometry
MSC (2010): 14M15, 14M17, 20G99
0103 physical sciences
FOS: Mathematics
Algebraic Topology (math.AT)
Mathematics - Algebraic Topology
0101 mathematics
Algebra over a field
Representation Theory (math.RT)
Mathematics::Representation Theory
Mathematics
[MATH.MATH-RT]Mathematics [math]/Representation Theory [math.RT]
Classical ind-group
Applied Mathematics
010102 general mathematics
Significant difference
14M15, 14M17, 20G99
16. Peace & justice
homogeneous ind-variety
Homogeneous
Bruhat decomposition
[MATH.MATH-AG]Mathematics [math]/Algebraic Geometry [math.AG]
010307 mathematical physics
Mathematics - Representation Theory
generalized flag
Subjects
Details
- Language :
- English
- ISSN :
- 10936106
- Database :
- OpenAIRE
- Journal :
- Asian Journal of Mathematics, Asian Journal of Mathematics, International Press, 2017, ⟨10.4310/AJM.2017.v21.n4.a1⟩
- Accession number :
- edsair.doi.dedup.....b043f8d6d521fd6c74524f086e1c98db