Essential dimension of semisimple groups of type $B$

TL;DR

This paper calculates the essential dimension of certain semisimple groups of type B, specifically products of Spin groups modulo a central subgroup, over fields of characteristic zero, covering various cases including small ranks.

Contribution

It provides explicit formulas for the essential dimension of these groups for all ranks at least 7 and specific low-rank cases, extending previous knowledge in algebraic group theory.

Findings

Essential dimension formulas for groups with rank ≥ 7.

Explicit results for low-rank cases with specific central subgroups.

Complete classification of essential dimension in the considered cases.

Abstract

We determine the essential dimension of an arbitrary semisimple group of type $B$ of the form \[G=\big(\operatorname{\mathbf{Spin}}(2n_{1}+1)\times\cdots \times \operatorname{\mathbf{Spin}}(2n_{m}+1)\big)/\boldsymbol{\mu}\] over a field of characteristic $0$, for all $n_{1},\ldots, n_{m}\geq 7$, and a central subgroup $\boldsymbol{\mu}$ of $\operatorname{\mathbf{Spin}}(2n_{1}+1)\times\cdots \times \operatorname{\mathbf{Spin}}(2n_{m}+1)$ not containing the center of $\operatorname{\mathbf{Spin}}(2n_i+1)$ as a direct factor. We also find the essential dimension of $G$ for each of the following cases, where either $n_{i}=1$ for all $i$ or $m=2$, $n_{1}=1$, $2\leq n_{2}\leq 3$, $\boldsymbol{\mu}$ is the diagonal central subgroup for both cases.