The character formula for even-rank fork representations of orthosymplectic Lie superalgebras

Let $m,n$ be nonnegative integers with $|m-n|$ even. Let $p$ be a nonnegative integer and $r$ an integer with $0\leq r\leq p$. Consider the representation of $\mathfrak{osp}(2m|2n)$ with Dynkin labels $[0,\ldots,0,r,p-r]$. Let ${\cal B}_r$ be the set of partitions specified in the paper, let $s_\lambda(x/y)$ denote the supersymmetric Schur function, and let $x_1,\ldots,x_m$ and $y_1,\ldots,y_n$ be the relevant character variables. The even-rank fork character conjecture. For $|m-n|$ even, one has

$$\operatorname{char} [0,\ldots,0,r,p-r]_{\mathfrak{osp}(2m|2n)} = (y_1\cdots y_n/x_1\cdots x_m)^{p/2} \sum_{\lambda_1\leq p,\; \lambda\in{\cal B}_r} s_\lambda (x/y).$$

Thus the character expands as an infinite sum of supersymmetric Schur functions indexed by partitions inside the $(m,n)$-hook, of width at most $p$, and belonging to ${\cal B}_r$. This formula is the explicit version of the proposed correspondence for fork representations. The analogous odd-$|m-n|$ case is described separately in the paper with ${\cal B}_r$ replaced by ${\cal B}_{p-r}$; the displayed even case is presented as a conjecture.