The classification of thick representations of the symmetric group
We give the classification of thick representations and dense representations of the symmetric group over a field of characteristic zero.
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arXiv subjects
Shingo Okuyama
Publications and source records attributed to Shingo Okuyama.
Partially additive rings and group schemes over ${\mathbb F}_1$
We develop an elementary theory of partially additive rings as a foundation of ${\mathbb F}_1$-geometry. Our approach is so concrete that an analog of classical algebraic geometry is established very straightforwardly. As applications, (1) we construct a kind of group scheme ${\mathbb GL}_n$ whose value at a commutative ring $R$ is the group of $n\times n$ invertible matrices over $R$ and at ${\mathbb F}_1$ is the $n$-th symmetric group, and (2) we construct a projective space $\mathbb P^n$ as a kind of scheme and count the number of points of ${\mathbb P}^n({\mathbb F}_q)$ for $q=1$ or $q=p^n$ a power of a rational prime, then we explain a reason of number 1 in the subscript of ${\mathbb F}_1$ even though it has two elements.
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Configuration space of intervals with partially summable labels
A configuration space of intervals in $\mathbb R^1$ with partially summable labels is constructed. It is a kind of an extension of the configuration space with partially summable labels constructed by the second author and at the same time a generalization of the configuration space of intervals with labels in a based space constructed by the first author. An approximation theorem of the preceding configuration space is generalized to our case. When partially summable labels are given by a partial abelian monoid $M,$ we prove that it is weakly homotopy equivalent to the space of based loops on the classifying space of $M$ under some assumptions on $M$.
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Interactions of strings and equivariant homology theories
We introduce the notion of the space of parallel strings with partially summable labels, which can be viewed as a geometrically constructed group completion of the space of particles with labels. We utilize this to construct a machinery which produces equivariant generalized homology theories from such simple and abundant data as partial monoids.
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The space of intervals in a Euclidean space
For a path-connected space X, a well-known theorem of Segal, May and Milgram asserts that the configuration space of finite points in R^n with labels in X is weakly homotopy equivalent to the n-th loop-suspension of X. In this paper, we introduce a space I_n(X) of intervals suitably topologized in R^n with labels in a space X and show that it is weakly homotopy equivalent to n-th loop-suspension of X without the assumption on path-connectivity.
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