The strong symmetric genus of the hyperoctahedral groups

Equi-distribution over Descent Classes of the Hyperoctahedral Group

A classical result of MacMahon shows that the length function and the major index are equi-distributed over the symmetric group. Foata and Schützenberger gave a remarkable refinement and proved that these parameters are equi-distributed over inverse descent classes, implying bivariate equi-distribution identities. Type $B$ analogues of these results, refinements and consequences are given in this paper.

A new statistic on the hyperoctahedral groups

We introduce a new statistic on the hyperoctahedral groups (Coxeter groups of type B), and give a conjectural formula for its signed distributions over arbitrary descent classes. The statistic is analogous to the classical Coxeter length function, and features a parity condition. For descent classes which are singletons the conjectured formula gives the Poincaré polynomials of the varieties of symmetric matrices of fixed rank. For several descent classes we prove the conjectural formula. For this we construct suitable "supporting sets" for the relevant generating functions. We prove cancellati…

The strong symmetric genus of the finite Coxeter groups

The strong symmetric genus of a finite group G is the smallest genus of a closed orientable topological surface on which G acts faithfully as a group of orientation preserving automorphisms. In this paper we complete the calculation of the strong symmetric genus for each finite Coxeter group excluding the group E8.

Classification of Groups with Strong Symmetric Genus up to Twenty-Five

The strong symmetric genus of a finite group is the minimum genus of a compact Riemann surface on which the group acts as a group of automorphisms preserving orientation. A characterization of the infinite number of groups with strong symmetric genus zero and one is well-known and the problem is finite for each strong symmetric genus greater than or equal to two. May and Zimmerman have published papers detailing the classification of all groups with strong symmetric genus two through four. Using the computer algebra system GAP, we extend these classifications to all groups of strong symmetric …

关于对称映射类群的同余子群性质

A congruence subgroup property for symmetric mapping class groups

本文研究了对称映射类群的同余子群性质，并证明了在特定条件下，torsion元素在共轭区分上的性质。

亏格4中一个对称折痕的算术Kontsevich--Zorich单值群

An arithmetic Kontsevich--Zorich monodromy of a symmetric origami in genus 4

本文证明了亏格4中存在一个对称折痕，其Kontsevich--Zorich单值群在Sarnak意义下是算术的，揭示了折痕的Veech群与单值群之间缺乏关联。