8 problems

Let $\tilde{\alpha}$ be the highest root of the Lie algebra of type $A_r$, and write $\alpha_{i,j}=\alpha_i+\alpha_{i+1}+\cdots+\alpha_j$ for $1\leq i\leq j\leq r$. Let…

Parabolic branching polynomial positivity conjecture. The polynomial $b_{\nu,\overline{\lambda}+\gamma}(p)$ has nonnegative integer coefficients at least when…

Let $m_q(\lambda,\mu)$ denote the $q$-multiplicity associated with a pair of weights $(\lambda,\mu)$ from the list provided by Berenshteu{}in and Zelevinskiu{}. Harris et al.'s con…

Let $\Lambda$ be the integral weight lattice of $\mathfrak{sl}_m(\mathbb{C})$, let $\mathrm{V}(\lambda)$ be the irreducible module with highest weight $\lambda$, and let…

Let $\mathfrak{g}$ be a simple Lie algebra of rank $r$, and let $\lambda$ and $\mu$ be weights such that the multiplicity $m(\lambda,\mu)$ of $\mu$ in the irreducible representatio…

Let $\mathfrak g=B_n^{(1)}$, let $\ell\geq 2$, and let ${\rm max}^+(\lambda+\Lambda\mid\ell)$ denote the set of dominant maximal weights at the indicated level. Dominant maximal we…

Let $P^+(n,m)$ be the set of dominant weights, and let $K_{\lambda,\mu}^{\mathrm{stab}}(q)$ denote the stable $q$-analog of the weight multiplicity for the orthosymplectic Lie supe…

Let $P^+(n,m)$ be the set of dominant weights, let $\rho$ be the relevant Weyl vector, and let $K_{\lambda,\mu}(q)$ denote the $q$-analog of the weight multiplicity for…