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1. CMB 2011 (vol 54 pp. 663)
Admissible Sequences for Twisted Involutions in Weyl Groups
Let $W$ be a Weyl group, $\Sigma$ a set of simple reflections in $W$
related to a basis $\Delta$ for the root system $\Phi$ associated with
$W$ and $\theta$ an involution such that $\theta(\Delta) = \Delta$. We
show that the set of $\theta$-twisted involutions in $W$,
$\mathcal{I}_{\theta} = \{w\in W \mid \theta(w) = w^{-1}\}$ is in one
to one correspondence with the set of regular involutions
$\mathcal{I}_{\operatorname{Id}}$. The elements of $\mathcal{I}_{\theta}$ are
characterized by sequences in $\Sigma$ which induce an ordering called
the Richardson-Springer Poset. In particular, for $\Phi$ irreducible,
the ascending Richardson-Springer Poset of $\mathcal{I}_{\theta}$,
for nontrivial $\theta$ is identical to the descending
Richardson-Springer Poset of $\mathcal{I}_{\operatorname{Id}}$.
Categories:20G15, 20G20, 22E15, 22E46, 43A85 |
2. CMB 2003 (vol 46 pp. 332)
Some Questions about Semisimple Lie Groups Originating in Matrix Theory We generalize the well-known result that a square traceless complex
matrix is unitarily similar to a matrix with zero diagonal to
arbitrary connected semisimple complex Lie groups $G$ and their Lie
algebras $\mathfrak{g}$ under the action of a maximal compact subgroup
$K$ of $G$. We also introduce a natural partial order on
$\mathfrak{g}$: $x\le y$ if $f(K\cdot x) \subseteq f(K\cdot y)$ for
all $f\in \mathfrak{g}^*$, the complex dual of $\mathfrak{g}$. This
partial order is $K$-invariant and induces a partial order on the
orbit space $\mathfrak{g}/K$. We prove that, under some restrictions
on $\mathfrak{g}$, the set $f(K\cdot x)$ is star-shaped with respect
to the origin.
Categories:15A45, 20G20, 22E60 |