Cameron-Liebler sets of generators in finite classical polar spaces

Abstract

Cameron-Liebler sets were originally defined as collections of lines ("line classes") in PG(3, q) sharing certain properties with line classes of symmetric tactical decompositions. While there are many equivalent characterisations, these objects are defined as sets of lines whose characteristic vector lies in the image of the transpose of the point-line incidence matrix of PG(3, q), and so combinatorially they behave like a union of pairwise disjoint point-pencils. Recently, the concept of a Cameron-Liebler set has been generalised to several other settings. In this article we introduce Cameron-Liebler sets of generators in finite classical polar spaces. For each of the polar spaces we give a list of characterisations that mirrors those for Cameron-Liebler line sets, and also prove some classification results.

Equivalent definitions for (degree one) Cameron–Liebler classes of generators in finite classical polar spaces

“…In this article, we consider three different types of polar spaces, see Table 1. Type I and II corresponds with type I and II respectively, defined in [9], while type III in this paper corresponds with the union of type III and IV in [9], as we handle the symplectic polar spaces W (4n + 1, q), for both q odd and q even, in the same way. Definition 1.2 and Definition 1.1 are equivalent for the polar spaces of type I by [9,Theorem 3.7,Theorem 3.15].…”

“…Type I and II corresponds with type I and II respectively, defined in [9], while type III in this paper corresponds with the union of type III and IV in [9], as we handle the symplectic polar spaces W (4n + 1, q), for both q odd and q even, in the same way. Definition 1.2 and Definition 1.1 are equivalent for the polar spaces of type I by [9,Theorem 3.7,Theorem 3.15]. For the polar spaces of type II we can consider the (degree one) Cameron-Liebler sets of one class of generators; we see that Cameron-Liebler sets and degree one Cameron-Liebler sets coincide when we only consider one class (see [9,Theorem 3.16]).…”

“…After many results about those Cameron-Liebler line sets in PG (3, q), the Cameron-Liebler set concept has been generalized to many other contexts: Cameron-Liebler line sets in PG(n, q) [6], Cameron-Liebler sets of k-spaces in PG(2k+1, q) [23], Cameron-Liebler sets of k-spaces in PG(n, q) [2], Cameron-Liebler classes in finite sets [10,14,21] and Cameron-Liebler sets of generators in finite classical polar spaces [9] were defined. The central problem for Cameron-Liebler sets is to find for which parameter x a Cameron-Liebler set exists, and finding examples with this parameter [6,13,16,17,19,20].…”

“…In their article, they define Boolean degree one functions, or Cameron-Liebler sets in projective and polar spaces by the fact that the corresponding characteristic vector lies in V 0 ⊥ V 1 , which are eigenspaces of the related association scheme (see Section 2). In [9], M. De Boeck, M. Rodgers, L. Storme and A. Švob introduced Cameron-Liebler sets of generators in the finite classical polar spaces. In this article, Cameron-Liebler set of generators in the polar spaces are defined by the disjointness-definition and the authors give several equivalent definitions for these Cameron-Liebler sets.…”

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In this article, we study degree one Cameron-Liebler sets of generators in all finite classical polar spaces, which is a particular type of a Cameron-Liebler set of generators in this polar space, [9]. These degree one Cameron-Liebler sets are defined similar to the Boolean degree one functions, [15].

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