Saturday, February 2, 2008

Poset Realization

Let $Sub(n)$ be the poset of proper nontrivial linear subspaces of $\mathbb{R}^n$. I've asked before if anybody knew the realization of this poset (topological category), and haven't heard anything. Listening to my advisor, it looks like the realization should be homeomorphic to the unit sphere in the following vector space $V$: Let $Sym(n)$ be the symmetric $n$ by $n$ matrices, and include $\mathbb{R}$ via constant diagonal matrices. The quotient $Sym(n)/\mathbb{R}$ is what I will call $V$. To get the unit sphere, we take $(V-0)/\mathbb{R}^+$, where the positive reals act by scalar multiplication (perhaps this could be said better).

Anyway, the correspondence goes something like this. Recall that symmetric matrices have all real eigenvalues, and that the eigenspaces form a direct sum decomposition of $\mathbb{R}^n$. Modding out by $\mathbb{R}$ in $Sym(n)$ shifts eigenvalues (if $e_1,\ldots,e_n$ are the eigenvalues for $M$, then the $e_i+r$ are the eigenvalues for $M+rI$), so we might as well say V is the those symmetric matrices that have 0 as their least eigenvalue. Next, modding by $\mathbb{R}^+$ scales the eigenvalues, so we think of points in the unit sphere of $V$ as symmetric matrices with eigenvalues $0<\lambda_1<\cdots<\lambda_l<1$. The associated eigenspaces $E_1,\ldots,E_l$ let us define a flag $E_1 < E_1+E_2 < \cdots < \sum E_i$, and the eigenvalues themselves pick out a unique point in the simplex corresponding to this flag, in the geometric realization of the poset.

This was pretty brief, and in fact, when I've been messing with it, I generally write the homeomorphism going from the poset to the sphere, but whatever. Apparently its a homeomorphism, so it doesn't matter which way we go :).

Anyway, the main reason I'm posting this is fishing for people who already knew this result (or can see an error - I'm happy to provide a more rigorously written account, on request). My advisor expects the result is 'known to the experts', which clearly doesn't include me. And I can't find a paper with it online. Somebody should really make a theorems database... [Update: Found it]

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