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Register a new userBott-Samelson atlases, total positivity, and Poisson structures on some homogeneous spaces
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Let $G$ be a connected and simply connected complex semisimple Lie group. For a collection of homogeneous $G$-spaces $G/Q$, we construct a finite atlas ${mathcal{A}}_{rm BS}(G/Q)$ on $G/Q$, called the Bott-Samelson atlas, and we prove that all of its coordinate functions are positive with respect to the Lusztig positive structure on $G/Q$. We also show that the standard Poisson structure $pi_{G/Q}$ on $G/Q$ is presented, in each of the coordinate charts of ${mathcal{A}}_{rm BS}(G/Q)$, as a symmetric Poisson CGL extension (or a certain localization thereof) in the sense of Goodearl-Yakimov, making $(G/Q, pi_{G/Q}, {mathcal{A}}_{rm BS}(G/Q))$ into a Poisson-Ore variety. Examples of $G/Q$ include $G$ itself, $G/T$, $G/B$, and $G/N$, where $T subset G$ is a maximal torus, $B subset G$ a Borel subgroup, and $N$ the uniradical of $B$.
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In the framework of the problem of characterizing complete flag manifolds by their contractions, the complete flags of type $F_4$ and $G_2$ satisfy the property that any possible tower of Bott-Samelson varieties dominating them birationally deforms in a nontrivial moduli. In this paper we illustrate the fact that, at least in some cases, these deformations can be explained in terms of automorphisms of Schubert varieties, providing variations of certain isotropic structures on them. As a corollary, we provide a unified and completely algebraic proof of the characterization of complete flag manifolds in terms of their contractions.
We discuss properties of distributions that are multivariate totally positive of order two (MTP2) related to conditional independence. In particular, we show that any independence model generated by an MTP2 distribution is a compositional semigraphoid which is upward-stable and singleton-transitive. In addition, we prove that any MTP2 distribution satisfying an appropriate support condition is faithful to its concentration graph. Finally, we analyze factorization properties of MTP2 distributions and discuss ways of constructing MTP2 distributions; in particular we give conditions on the log-linear parameters of a discrete distribution which ensure MTP2 and characterize conditional Gaussian distributions which satisfy MTP2.
We introduce a notion of weakly log-canonical Poisson structures on positive varieties with potentials. Such a Poisson structure is log-canonical up to terms dominated by the potential. To a compatible real form of a weakly log-canonical Poisson variety we assign an integrable system on the product of a certain real convex polyhedral cone (the tropicalization of the variety) and a compact torus. We apply this theory to the dual Poisson-Lie group $G^*$ of a simply-connected semisimple complex Lie group $G$. We define a positive structure and potential on $G^*$ and show that the natural Poisson-Lie structure on $G^*$ is weakly log-canonical with respect to this positive structure and potential. For $K subset G$ the compact real form, we show that the real form $K^* subset G^*$ is compatible and prove that the corresponding integrable system is defined on the product of the decorated string cone and the compact torus of dimension $frac{1}{2}({rm dim} , G - {rm rank} , G)$.
For a compact Poisson-Lie group $K$, the homogeneous space $K/T$ carries a family of symplectic forms $omega_xi^s$, where $xi in mathfrak{t}^*_+$ is in the positive Weyl chamber and $s in mathbb{R}$. The symplectic form $omega_xi^0$ is identified with the natural $K$-invariant symplectic form on the $K$ coadjoint orbit corresponding to $xi$. The cohomology class of $omega_xi^s$ is independent of $s$ for a fixed value of $xi$. In this paper, we show that as $sto -infty$, the symplectic volume of $omega_xi^s$ concentrates in arbitrarily small neighbourhoods of the smallest Schubert cell in $K/T cong G/B$. This strengthens earlier results [9,10] and is a step towards a conjectured construction of global action-angle coordinates on $Lie(K)^*$ [4, Conjecture 1.1].
In this paper, we construct stable Bott--Samelson classes in the projective limit of the algebraic cobordism rings of full flag varieties, upon an initial choice of a reduced word in a given dimension. Each stable Bott--Samelson class is represented by a bounded formal power series modulo symmetric functions in positive degree. We make some explicit computations for those power series in the case of infinitesimal cohomology. We also obtain a formula of the restriction of Bott--Samelson classes to smaller flag varieties.
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