{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,2]],"date-time":"2026-06-02T07:46:25Z","timestamp":1780386385181,"version":"3.54.1"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"4","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>We study Cameron-Liebler $k$-sets in the affine geometry, so sets of $k$-spaces in $\\mathrm{AG}(n,q)$. This generalizes research on Cameron-Liebler $k$-sets in the projective geometry $\\mathrm{PG}(n,q)$. Note that in algebraic combinatorics, Cameron-Liebler $k$-sets of $\\mathrm{AG}(n,q)$ correspond to certain equitable bipartitions of the association scheme of $k$-spaces in $\\mathrm{AG}(n,q)$, while in the analysis of Boolean functions, they correspond to Boolean degree $1$ functions of $\\mathrm{AG}(n,q)$.\r\nWe define Cameron-Liebler $k$-sets in $\\mathrm{AG}(n,q)$ by intersection properties with $k$-spreads and show the equivalence of several definitions. In particular, we investigate the relationship between Cameron-Liebler $k$-sets in $\\mathrm{AG}(n,q)$ and $\\mathrm{PG}(n,q)$. As a by-product, we calculate the character table of the association scheme of affine lines. Furthermore, we characterize the smallest examples of Cameron-Liebler $k$-sets.\r\nThis paper focuses on $\\mathrm{AG}(n,q)$ for $n &gt; 3$, while the case for Cameron-Liebler line classes in $\\mathrm{AG}(3,q)$ was already treated separately.<\/jats:p>","DOI":"10.37236\/9469","type":"journal-article","created":{"date-parts":[[2021,10,21]],"date-time":"2021-10-21T10:53:31Z","timestamp":1634813611000},"source":"Crossref","is-referenced-by-count":4,"title":["Cameron-Liebler $k$-sets in $\\mathrm{AG}(n,q)$"],"prefix":"10.37236","volume":"28","author":[{"given":"Jozefien","family":"D'haeseleer","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Ferdinand","family":"Ihringer","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Jonathan","family":"Mannaert","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Leo","family":"Storme","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"23455","published-online":{"date-parts":[[2021,10,22]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v28i4p11\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v28i4p11\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,10,21]],"date-time":"2021-10-21T10:53:31Z","timestamp":1634813611000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v28i4p11"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,10,22]]},"references-count":0,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2021,10,8]]}},"URL":"https:\/\/doi.org\/10.37236\/9469","relation":{},"ISSN":["1077-8926"],"issn-type":[{"value":"1077-8926","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,10,22]]},"article-number":"P4.11"}}