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arXiv · 1312.4315

Calculating the dimension of the universal embedding of the symplectic dual polar space using languages

Carlos Segovia·Monika Winklmeier
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Abstract

The main result of this paper is the construction of a bijection of the set of words in so-called standard order of length $n$ formed by four different letters and the set $\mathbb{N}^n$ of all subspaces of a fixed $n$-dimensional maximal isotropic subspace of the $2n$-dimensional symplectic space $V$ over $\mathbb{F}_2$ which are not maximal in a certain sense. Since the number of different words in standard order is known, this gives an alternative proof for the formula of the dimension of the universal embedding of a symplectic dual polar space $\mathcal G_n$. Along the way, we give formulas for the number of all $n$- and $(n-1)$-dimensional totally isotropic subspaces of $V$.

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Carlos Segovia, Monika Winklmeier. 2013-12-16. Calculating the dimension of the universal embedding of the symplectic dual polar space using languages. https://arxiv.org/abs/1312.4315

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