Eventual arm and leg widths in cocharacters of P. I. Algebras

Given a p.i. algebra $A$, we study which partitions $\lambda$correspond to characters with non-zero multiplicities in the cocharacter sequence of $A$. We define the $\omega_0(A)$, the eventual arm width to be the maximal $d$ so that such $\lambda$ can have $d$ parts arbitrarily large, and $\omega_1(A)$ to be the maximum $h$ so that the conjugate $\lambda'$ could have $h$ arbitrarily large parts. Our main result is that for any $A$, $\omega_0(A)\ge\omega_1(A)$.