Abstract
The number of shortest factorizations into reflections for a Singer cycle in $GL_n(\mathbb{F}_q)$ is shown to be $(q^n-1)^{n-1}$. Formulas counting factorizations of any length, and counting those with reflections of fixed conjugacy classes are also given. Nous prouvons que le nombre de factorisations de longueur minimale d’un cycle de Singer dans $GL_n(\mathbb{F}_q)$ comme un produit de réflexions est $(q^n-1)^{n-1}$. Nous présentons aussi des formules donnant le nombre de factorisations de toutes les longueurs ainsi que des formules pour le nombre de factorisations comme produit de réflexions ayant des classes de conjugaison fixes.
Highlights
Introduction and main resultOur goals are q-analogues, replacing the symmetric group Sn with the general linear group GLn(Fq), replacing transpositions with reflections, and replacing an n-cycle with a Singer cycle c: the image of a generator for the that comes from acycchloicicgeroofupFqF-×qvnec∼=toZr s/p(aqcne−is1o)mZournpdheisrmanFyqenm∼=beFdnqd.inTghFe ×qanna֒→logGyLbeFtqw(Feeqnn)S∼=ingGeLr cny(cFlqe)s in GLn(Fq) and n-cycles in Sn is reasonably well-established [17, §7], [18, §§8-9]
Introduction and main resultThis paper is motivated by two classic results on the number t(n, l) of ordered factorizations (t1, . . . , tl) of an n-cycle c = t1t2 · · · tl in the symmetric group Sn, where each ti is a transposition.Theorem (Hurwitz [8], Denes [2])
Our goals are q-analogues, replacing the symmetric group Sn with the general linear group GLn(Fq), replacing transpositions with reflections, and replacing an n-cycle with a Singer cycle c: the image of a generator for the that comes from acycchloicicgeroofupFqF-×qvnec∼=toZr s/p(aqcne−is1o)mZournpdheisrmanFyqenm∼=beFdnqd.inTghFe ×qanna֒→logGyLbeFtqw(Feeqnn)S∼=ingGeLr cny(cFlqe)s in GLn(Fq) and n-cycles in Sn is reasonably well-established [17, §7], [18, §§8-9]
Summary
Our goals are q-analogues, replacing the symmetric group Sn with the general linear group GLn(Fq), replacing transpositions with reflections, and replacing an n-cycle with a Singer cycle c: the image of a generator for the that comes from acycchloicicgeroofupFqF-×qvnec∼=toZr s/p(aqcne−is1o)mZournpdheisrmanFyqenm∼=beFdnqd.inTghFe ×qanna֒→logGyLbeFtqw(Feeqnn)S∼=ingGeLr cny(cFlqe)s in GLn(Fq) and n-cycles in Sn is reasonably well-established [17, §7], [18, §§8-9] Fixing such a Singer cycle c, denote by tq(n, l) the number of ordered factorizations Can one derive Theorem 1.1 bijectively, or by an overcount, in the spirit of Denes [2], that counts factorizations of all conjugates of a Singer cycle and divides by the conjugacy class size?
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