Minimum codimension of eigenspaces in irreducible representations of simple linear algebraic groups

TL;DR

This paper computes the minimum codimension of eigenspaces for non-central elements acting on irreducible modules of simple classical algebraic groups, providing explicit values and bounds for various group types and dimensions.

Contribution

It explicitly calculates or bounds the minimum eigenspace codimension for irreducible modules of classical groups of certain ranks and dimensions, extending understanding of eigenspace structures.

Findings

Calculated $ u_{G}(V)$ for type A$_{ ext{ell}}$ with $ ext{ell} extgreater= 16$ and $ ext{dim}(V) extless= rac{ ext{ell}^3}{2}$.

Determined $ u_{G}(V)$ for types B$_{ ext{ell}}$, C$_{ ext{ell}}$ with $ ext{ell} extgreater= 14$ and $ ext{dim}(V) extless= 4 ext{ell}^3$.

Provided lower bounds for $ u_{G}(V)$ for smaller rank groups within specified dimension bounds.

Abstract

Let $k$ be an algebraically closed field of characteristic $p \geq 0$, let $G$ be a simple simply connected classical linear algebraic group of rank $\ell$ and let $T$ be a maximal torus in $G$ with rational character group $X(T)$. For a nonzero $p$-restricted dominant weight $\lambda \in X(T)$, let $V$ be the associated irreducible $kG$-module. Define $\nu_{G}(V)$ to be the minimum codimension of eigenspaces corresponding to non-central elements of $G$ on $V$. In this paper, we calculate $\nu_{G}(V)$ for $G$ of type $A_{\ell}$, $\ell \geq 16$, and $dim(V) \leq \frac{\ell^{3}}{2}$; for $G$ of type $B_{\ell}$, respectively $C_{\ell}$, $\ell \geq 14$, and $dim(V) \leq 4\ell^{3}$; and for $G$ of type $D_{\ell}$, $\ell \geq 16$, and $dim(V) \leq 4\ell^{3}$. Moreover, for the groups of smaller rank and their corresponding irreducible modules with dimension satisfying the above bounds, we determine lower-bounds for $\nu_{G}(V)$.