Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · math/9909121

Cellini's descent algebra and semisimple conjugacy classes of finite groups of Lie type

Jason Fulman
View original source ↗Read source document ↗Export metadata ↓

Abstract

By algebraic group theory, there is a map from the semisimple conjugacy classes of a finite group of Lie type to the conjugacy classes of the Weyl group. Picking a semisimple class uniformly at random yields a probability measure on conjugacy classes of the Weyl group. We conjecture that this measure agrees with a second measure on conjugacy classes of the Weyl group induced by a construction of Cellini which uses the affine Weyl group. This conjecture is confirmed in special cases such as type C odd characteristic and the identity conjugacy class in type A. Models of card shuffling, old and new, arise naturally. Type A shuffles lead to interesting number theory involving Ramanujan sums. It is shown that a proof of our conjecture in type C even characteristic would give an alternate solution to a problem in dynamical systems. An idea is offered for how, at least in type A, to associate to a semisimple conjugacy class an element of the Weyl group, refining the map to conjugacy classes. This is confirmed for the simplest nontrivial example.

Explore related subjects

math.NTmath.DS

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Jason Fulman. 2000-03-30. Cellini's descent algebra and semisimple conjugacy classes of finite groups of Lie type. https://arxiv.org/abs/math/9909121

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Refined Counting of Geodesic Segments in the Hyperbolic Plane

For $Γ$ a cofinite Fuchsian group, and $l$ a fixed closed geodesic, we study the asymptotics of the number of those images of $l$ that have a prescribed orientation and distance from $l$ less than or equal to $X$. Using a new relative trace formula that we develop, we give a new concrete proof of the error bound $O(X^{2/3})$ that appears in the works of Good and Hejhal. Furthermore, we prove a new bound $O(X^{1/2}\log{X})$ for the mean square of the error. For particular arithmetic groups, we provide interpretations in terms of correlation sums of the number of ideals of norm at most $X$ in associated number fields, generalizing previous examples due to Hejhal.

math.NT↗
arXiv2026

A de Rham weight part of Serre's conjecture and generalized mod $p$ BGG decompositions

We propose the use of de Rham cohomology of special fibers of Shimura varieties to formulate a geometric version of the weight part of Serre's conjecture. We conjecture that this formulation is equivalent to the one using Serre weights and the étale cohomology of Shimura varieties. We treat in detail the cases of $G \in \{GL_3,GSp_4\}$, which are the simplest groups of semisimple rank at least $2$, where new phenomena occur. First we prove this equivalence for generic weights and generic non-Eisenstein eigensystems for a compact $U(2,1)$ Shimura variety such that $G_{\mathbb{Q}_p}=GL_3$. We do this by proving a generic concentration in middle degree of mod $p$ de Rham cohomology with coefficients. In turn, we prove this generic concentration by constructing generalized mod $p$ BGG decompositions for de Rham cohomology. After applying the results from our companion paper, this reduces to computing some BGG-like resolutions in a certain mod $p$ version of category $\mathcal{O}$. In the $GSp_4$ case we also compute some explicit BGG decompositions, which help us to upgrade the generic weak entailment from arxiv:2410.09602. to the expected generic entailment, as well as proving the equivalence above for one of the upper alcoves.

math.NT↗
arXiv2026

CM-liftability of simple superspecial abelian surfaces over prime fields

For any prime $p>0$, we prove that simple superspecial abelian surfaces over $\mathbb{F}_{p}$ admit CM liftings, by using the residual reflex condition (RRC) and Lie types. The CM-liftability of ordinary simple abelian surfaces is proved by Serre-Tate, and the CM-liftability of almost ordinary simple abelian surfaces is proved by Oswal-Shankar and Bergström-Karemaker-Marseglia, respectively. As there can only be ordinary, almost ordinary, or supersingular simple abelian surfaces over $\mathbb{F}_{p}$, our work is another step to complete the CM-liftability of simple abelian surfaces over $\mathbb{F}_{p}$.

math.NT↗