Cameron–Liebler k-sets in subspaces and non-existence conditions

In this article we generalize the concepts that were used in the PhD thesis of Drudge to classify Cameron-Liebler line classes in PG(n, q), n >= 3, to Cameron-Liebler sets of k-spaces in PG(n, q) and AG(n, q). In his PhD thesis, Drudge proved that every Cameron-Liebler line class in PG(n, q) intersects every 3-dimensional subspace in a Cameron-Liebler line class in that subspace. We are using the generalization of this result for sets of k-spaces in PG(n, q) and AG (n , q). Together with a basic counting argument this gives a very strong non-existence condition, n >= 3k + 3. This condition can also be improved for k-sets in AG(n, q), with n >= 2k + 2.

Publication year:2022

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