ICTS:30471

Video URL

Reflection factorizations of Singer cycles in finite linear and unitary groups

Reflection factorizations of Singer cycles in finite linear and unitary groups

(2024). Reflection factorizations of Singer cycles in finite linear and unitary groups. SciVideos. https://youtube.com/live/m0bT2TEhqHo

Reflection factorizations of Singer cycles in finite linear and unitary groups. SciVideos, Dec. 09, 2024, https://youtube.com/live/m0bT2TEhqHo 

          @misc{ scivideos_ICTS:30471,
            doi = {},
            url = {https://youtube.com/live/m0bT2TEhqHo},
            author = {},
            keywords = {},
            language = {en},
            title = {Reflection factorizations of Singer cycles in finite linear and unitary groups},
            publisher = {},
            year = {2024},
            month = {dec},
            note = {ICTS:30471 see, \url{https://scivideos.org/index.php/icts-tifr/30471}}
          }
Joel Brewster Lewis
Talk numberICTS:30471
Source RepositoryICTS-TIFR
Collection

Abstract

In the symmetric group S_n, there are n^{n-2} ways to write each n-cycle as a product of the minimum number of transpositions. This theorem has numerous extensions: in the symmetric group, such questions are tied to the enumeration of embedded maps on surfaces and moduli spaces of curves, while in real and complex reflection groups the analogous theorem is one ingredient in the Catalan--Coxeter theory and the study of the lattice of W-noncrossing partitions.

About a decade ago, with Vic Reiner and Dennis Stanton, we studied the analogue of this result for the general linear group over a finite field F_q. In this setting, the role of the n-cycle is taken by a Singer cycle, and that of the transpositions by the reflections; we showed that the number of factorizations is (q^n - 1)^{n - 1}. In this talk, I will discuss ongoing work, joint with C. Ryan Vinroot, that extends this work to a larger family of linear and unitary groups over a finite field.