Interesting Esoterica

A stratification of the space of all $k$-planes in $\mathbb{C}_n$

Article by Knutson, A
  • Published in 2012
  • Added on
To each $k \times n$ matrix $\mathrm{M}$ of rank $k$, we associate a juggling pattern of periodicity $n$ with $k$ balls. The juggling pattern actually only depends on the $k$-plane spanned by the rows, so gives a decomposition of the “Grassmannian” of all $k$-planes in $n$-space. There are many connections between the geometry and the juggling. For example, the natural topology on the space of matrices induces a partial order on the space of juggling patterns, which indicates whether one pattern is “more excited” than another. This same decomposition turns out to naturally arise from totally positive geometry, characteristic $p$ geometry, and noncommutative geometry. It also arises by projection from the manifold of full flags in $n$-space, where there is no cyclic symmetry

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BibTeX entry

@article{Knutson2012,
	title = {A stratification of the space of all {\$}k{\$}-planes in {\$}\mathbb{\{}C{\}}{\_}n{\$}},
	author = {Knutson, A},
	url = {http://www.math.cornell.edu/{\~{}}allenk/joint.pdf},
	urldate = {2012-11-19},
	abstract = {To each {\$}k \times n{\$} matrix {\$}\mathrm{\{}M{\}}{\$} of rank {\$}k{\$}, we associate a juggling pattern of periodicity {\$}n{\$} with {\$}k{\$} balls. The juggling pattern actually only depends
on the {\$}k{\$}-plane spanned by the rows, so gives a decomposition of the “Grassmannian” of all {\$}k{\$}-planes in {\$}n{\$}-space.

There are many connections between the geometry and the juggling. For example, the natural topology on the space of matrices induces a partial order on the space of juggling patterns, which indicates whether one pattern is “more excited” than another. This same decomposition turns out to naturally arise from totally
positive geometry, characteristic {\$}p{\$} geometry, and noncommutative geometry. It also arises by projection from the manifold of full flags in {\$}n{\$}-space, where there is no cyclic symmetry},
	comment = {},
	year = 2012
}