Parity unimodality of an sl2 character ($c_j\ge c_{j+2}(j\ge 0)$)

For a finite-dimensional sl2 Lie algebra representation, write $\chi (q)={\sum }_j{c}_j{q}^j$. Each irreducible summand has character $q^m+q^{m-2}+\cdots +q^{-m}$, so $c_j=c_{-j}$ and $c_j\ge c_{j+2}$ for nonnegative $j$. Thus each parity subsequence is symmetric and unimodal. If all weights have one parity, removing the unused exponents gives an ordinarily unimodal coefficient sequence. The full sequence including every integer exponent need not be unimodal: $q+q^{-1}$ has coefficients $1,0,1$.