Point count of the variety of modules over the quantum plane over a finite field

Abstract: In 1960, Feit and Fine gave a formula for the number of pairs of commuting n by n matrices over a finite field. We consider a quantum deformation of the problem, namely, counting pairs (A,B) of n by n matrices over a finite field that satisfy AB=qBA for a fixed nonzero scalar q. We give a formula for this count in terms of the order of q as a root of unity, generalizing Feit and Fine's result. In this talk, after explaining the title and the results, we will discuss a curious phenomenon that one sees when comparing the commutative case (q=1) and the general case from a geometric viewpoint.

geometric topologynumber theoryoperator algebrasrepresentation theory

Audience: researchers in the topic

( paper | slides | video )

---

Noncommutative geometry in NYC

Series comments: Noncommutative Geometry studies an interplay between spatial forms and algebras with non-commutative multiplication. Our seminar welcomes talks in Number Theory, Geometric Topology and Representation Theory linked to the context of Operator Algebras. All talks are kept at the entry-level accessible to the graduate students and non-experts in the field. To join us click meet.google.com/zjd-ehrs-wtx (5 min in advance) and igor DOT v DOT nikolaev AT gmail DOT com to subscribe/unsubscribe for the mailing list, to propose a talk or to suggest a speaker. Pending speaker's consent, we record and publish all talks at the hyperlink "video" on speaker's profile at the "Past talks" section. The slides can be posted by providing the organizers with a link in the format "myschool.edu/~myfolder/myslides.pdf". The duration of talks is 1 hour plus or minus 10 minutes.

***** We're transitioning to a new platform google meet. Please bear with us and we apologize for the inconvenience! ****

---