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Erschienen in:
2017 | OriginalPaper | Buchkapitel
5. The Deodhar Decomposition for the Grassmannian and the Positivity
verfasst von : Yuji Kodama
Erschienen in: KP Solitons and the Grassmannians
Verlag: Springer Singapore
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Abstract
In this chapter, we start to review the flag variety \(G/B^+\) for \(G=\mathrm {SL}_M(\mathbb R)\) and the Borel subgroup \(B^+\) of upper triangular matrices, and introduce the Deodhar decomposition of \(G/B^+\) [33, 34]. Then we give a refinement of the Schubert decomposition of \(\mathrm{Gr}(N, M)\) as a projection of the Deodhar decomposition, and parametrize each component of the refinement by introducing Go-diagram, which is a Young diagram decorated with black and white stones. In particular, if the Go-diagram has only white stones, it represents a projected Deodhar component of the totally nonnegative (TNN) Grassmannian \(\mathrm{Gr}(N, M)_{\ge 0}\). The Go-diagram in this case is the
-diagram introduced by Postnikov in [103]. We also construct an explicit form of matrix \(A\in \mathrm{Gr}(N, M)_{\ge 0}\) as a point of the projected Deodhar component by using the parameterizations of the flag variety due to Marsh and Rietsch [83]. We conclude this section to give an algorithm to compute an explicit form of the matrix A and to discuss the positivity of A. Most of the materials presented here can be also found in [72].
-diagram introduced by Postnikov in [103]. We also construct an explicit form of matrix \(A\in \mathrm{Gr}(N, M)_{\ge 0}\) as a point of the projected Deodhar component by using the parameterizations of the flag variety due to Marsh and Rietsch [83]. We conclude this section to give an algorithm to compute an explicit form of the matrix A and to discuss the positivity of A. Most of the materials presented here can be also found in [72].