4,207 research outputs found
Schubert decompositions for ind-varieties of generalized flags
Let be one of the ind-groups , ,
and be a splitting parabolic
ind-subgroup. The ind-variety has been identified with
an ind-variety of generalized flags in the paper "Ind-varieties of generalized
flags as homogeneous spaces for classical ind-groups" (Int. Math. Res. Not.
2004, no. 55, 2935--2953) by I. Dimitrov and I. Penkov. In the present paper we
define a Schubert cell on as a -orbit on
, where is any Borel ind-subgroup of
which intersects in a maximal ind-torus. A
significant difference with the finite-dimensional case is that in general
is not conjugate to an ind-subgroup of , whence
admits many non-conjugate Schubert decompositions. We
study the basic properties of the Schubert cells, proving in particular that
they are usual finite-dimensional cells or are isomorphic to affine ind-spaces.
We then define Schubert ind-varieties as closures of Schubert cells and study
the smoothness of Schubert ind-varieties. Our approach to Schubert
ind-varieties differs from an earlier approach by H. Salmasian in "Direct
limits of Schubert varieties and global sections of line bundles" (J. Algebra
320 (2008), 3187--3198).Comment: Keywords: Classical ind-group, Bruhat decomposition, Schubert
decomposition, generalized flag, homogeneous ind-variety. [26 pages
Shallow decision-making analysis in General Video Game Playing
The General Video Game AI competitions have been the testing ground for
several techniques for game playing, such as evolutionary computation
techniques, tree search algorithms, hyper heuristic based or knowledge based
algorithms. So far the metrics used to evaluate the performance of agents have
been win ratio, game score and length of games. In this paper we provide a
wider set of metrics and a comparison method for evaluating and comparing
agents. The metrics and the comparison method give shallow introspection into
the agent's decision making process and they can be applied to any agent
regardless of its algorithmic nature. In this work, the metrics and the
comparison method are used to measure the impact of the terms that compose a
tree policy of an MCTS based agent, comparing with several baseline agents. The
results clearly show how promising such general approach is and how it can be
useful to understand the behaviour of an AI agent, in particular, how the
comparison with baseline agents can help understanding the shape of the agent
decision landscape. The presented metrics and comparison method represent a
step toward to more descriptive ways of logging and analysing agent's
behaviours
On homogeneous spaces for diagonal ind-groups
We study the homogeneous ind-spaces
where is a strict diagonal ind-group defined by a
supernatural number and is a parabolic ind-subgroup
of . We construct an explicit exhaustion of
by finite-dimensional partial flag
varieties. As an application, we characterize all locally projective
-homogeneous spaces, and some direct products of such
spaces, which are -homogeneous for a fixed
. The very possibility for a -homogeneous
space to be -homogeneous for a strict diagonal
ind-group arises from the fact that the automorphism
group of a -homogeneous space is much larger than
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