Representations of dimensions $(p^n\pm 1)/2$ of the symplectic group of degree $2n$ over a field of characteristic $p$

TL;DR

This paper investigates specific irreducible representations of the symplectic group over a field of characteristic p, determining their dimensions, weight multiplicities, and behavior under restriction and reduction modulo p.

Contribution

It explicitly computes the dimensions and multiplicities of these representations and describes their restriction properties and relation to complex representations.

Findings

Dimensions of the representations are $(p^n ext{+/-} 1)/2$.

All weight multiplicities are equal to 1.

Restrictions to subgroups are completely reducible with known constituents.

Abstract

The irreducible representations $\phi_n^1$ and $\phi_n^2$ of the symplectic group $G_n=Sp_{2n}(P)$ over an algebraically closednfield $P$ of characteristic $p>2$ with highest weights $\omega_{n-1}+\frac{p-3}{2}\omega_n$ and $\frac{p-1}{2}\omega_n$, respectively, are investigated. It is proved that the dimension of $\phi_n^i$ ($i=1,2$) is equal to $(p^n+(-1)^i )/2$, all weight multiplicities of these representations are equal to $1$, their restrictions to the group $G_k$ naturally embedded into $G_n$ are completely reducible with irreducible constituents $\phi_k^1$ and $\phi_k^2$, and their restrictions to $Sp_{2n}(p)$ can be obtained as the result of the reduction modulo $p$ of certain complex irreducible representations of the group $Sp_{2n}(p)$.