The fork-representation character conjecture for $\mathfrak{osp}(2m|2n)$: a positive supersymmetric Schur expansion for $[0,\ldots,0,r,p-r]$

The irreducible highest-weight representations of the orthosymplectic Lie superalgebra $\mathfrak{osp}(2m|2n)$ with Dynkin labels $[0,\ldots,0,r,p-r]$ (nonzero labels only on the two fork nodes of the distinguished Dynkin diagram) are infinite-dimensional; together with the chiral spinors and self-dual tensors $[0,\ldots,0,p]$ they realize a correspondence between superdimensions of $\mathfrak{osp}(m|n)$ modules and dimensions of $\mathfrak{so}(m-n)$ representations, motivated by covariant quantization in supergravity. Stoilova, Thierry-Mieg and Van der Jeugt proved a positive Schur-function expansion for the $\mathfrak{so}(2k)$ analogues of these fork representations and for the chiral $\mathfrak{osp}(2m|2n)$ case, and conjectured that the same pattern holds for the $\mathfrak{osp}(2m|2n)$ fork modules: the character equals a fixed prefactor times an infinite positive sum of supersymmetric Schur functions $s_\lambda(x/y)$ indexed by the partitions $\lambda$ of width at most $p$ that belong to $\mathcal{B}_r$ (partitions in which every part occurs an even number of times, with a horizontal strip of length $r$ attached), with $\mathcal{B}_r$ replaced by $\mathcal{B}_{p-r}$ when $|m-n|$ is odd. The problem is to prove the two character identities for all admissible $(m,n,p,r)$, or to refute them by an explicit counterexample.

Background

The orthosymplectic Lie superalgebra $\mathfrak{osp}(m|n)$ is the $\mathbb{Z}_2$-graded analogue of the orthogonal Lie algebras: its even part is $\mathfrak{so}(m)\oplus\mathfrak{sp}(n)$ (with $n$ even) and its odd part carries the tensor product of the defining $\mathfrak{so}(m)$ and $\mathfrak{sp}(n)$ modules. It is the natural symmetry algebra of supergravity models built on an antisymmetric tensor calculus, where spinor-type $\mathfrak{osp}(m|n)$ representations play the role that spinors and self-dual tensors play for $\mathfrak{so}(m)$. A module of a Lie superalgebra splits into an even and an odd part, $V=V_{\bar 0}\oplus V_{\bar 1}$, and its superdimension is $\operatorname{sdim}V=\dim V_{\bar 0}-\dim V_{\bar 1}$; even when $V$ is infinite-dimensional, the superdimension can be finite and is computed by specialization of the character (the $t$-superdimension, which weights each level of the induced $\mathbb{Z}$-gradation relative to the $\mathfrak{gl}(m|n)$ subalgebra by a factor $(-t)^{\text{level}}$).

Fix the distinguished Dynkin diagram of $\mathfrak{osp}(2m|2n)$, with simple roots $\delta_1-\delta_2,\ldots,\delta_{n-1}-\delta_n$, $\delta_n-\epsilon_1$, $\epsilon_1-\epsilon_2,\ldots,\epsilon_{m-1}-\epsilon_m$, $\epsilon_{m-1}+\epsilon_m$; a highest-weight module is specified by its list of Dynkin labels (the integer coefficients of its highest weight in the basis of fundamental weights). The spinor representation has labels $[0,\ldots,0,1]$ and the self-dual tensor $[0,\ldots,0,2]$; these, and more generally the modules $[0,\ldots,0,p]$, are atypical (singular with respect to an odd root) and hence infinite-dimensional. Stoilova, Thierry-Mieg and Van der Jeugt established an $\mathfrak{osp}(m|n)\sim\mathfrak{so}(m-n)$ correspondence for them: the superdimension of $[0,\ldots,0,p]$ of $\mathfrak{osp}(m|n)$ coincides with the dimension of a finite-dimensional $\mathfrak{so}(m-n)$ representation, a fact of interest for covariant quantization schemes in supergravity. Their technique rests on expanding characters as positive sums of supersymmetric Schur functions.

The supersymmetric Schur function $s_\lambda(x/y)$ in the variables $x=(x_1,\ldots,x_m)$ and $y=(y_1,\ldots,y_n)$, introduced by Berele and Regev, is the character of the covariant $\mathfrak{gl}(m|n)$ module labelled by the partition $\lambda$ (its Young diagram), and it vanishes unless $\lambda$ lies in the $(m,n)$-hook, i.e. $\lambda_{m+1}\le n$. Expanding an $\mathfrak{osp}(2m|2n)$ character in the functions $s_\lambda(x/y)$ is therefore equivalent to describing the branching to the subalgebra $\mathfrak{gl}(m|n)$. For the chiral case the following formula is proven (without any restriction on the parity of $|m-n|$): $\operatorname{char}[0,\ldots,0,p]_{\mathfrak{osp}(2m|2n)}=\left(\frac{y_1\cdots y_n}{x_1\cdots x_m}\right)^{p/2}\sum_{\lambda\in\mathcal{B},\ \lambda_1\le p}s_\lambda(x/y),$ where $\mathcal{B}$ is the set of partitions in which every part occurs an even number of times (equivalently, partitions of the form $(\mu_1,\mu_1,\mu_2,\mu_2,\ldots)$ obtained by writing each part of a partition $\mu$ twice; this includes the zero partition). The same authors also proved the purely Lie-algebraic fork theorem: for the $\mathfrak{so}(2k)$ module with Dynkin labels $[0,\ldots,0,r,p-r]$, whose only nonzero labels sit on the two fork nodes $\epsilon_{k-1}-\epsilon_k$ and $\epsilon_{k-1}+\epsilon_k$ of the $D_k$ diagram, one has, for $k$ even, $\operatorname{char}[0,\ldots,0,r,p-r]_{\mathfrak{so}(2k)}=(x_1\cdots x_k)^{-p/2}\sum_{\lambda_1\le p,\ \ell(\lambda)\le k,\ \lambda\in\mathcal{B}_r}s_\lambda(x),$ and for $k$ odd the same identity with $\mathcal{B}_r$ replaced by $\mathcal{B}_{p-r}$. Here $\mathcal{B}_r$ consists of the partitions of $\mathcal{B}$ to which a horizontal strip (a skew Young diagram with at most one box in each column) of length $r$ is attached, and the sum runs over diagrams fitting inside the $k\times p$ rectangle. Its proof, via the $\mathfrak{so}(2k)\supset\mathfrak{gl}(k)$ branching rules of King and Wybourne, is technical but standard. The conjecture asserts that the identical pattern survives the passage from $\mathfrak{so}(2k)$ to the infinite-dimensional $\mathfrak{osp}(2m|2n)$ fork modules: one drops the rectangle constraint $\ell(\lambda)\le k$ and replaces ordinary by supersymmetric Schur functions. This is valuable because the alternative character formulas in the literature, notably the Howe-duality combinatorial character formula of Cheng and Zhang for orthosymplectic Lie superalgebras, give alternating sums of S-functions (a traditional name for Schur functions) that are hard to handle, whereas the conjectured expansion is a positive (non-alternating) sum.

Problem Statement

Prove or refute the following conjecture of Stoilova, Thierry-Mieg and Van der Jeugt (Conjecture, equation (26) of arXiv:1904.00074; Conjecture 1, equation (32) of arXiv:1904.00067).

Let $m\ge 2$, $n\ge 1$, $p\ge 1$, and $0\le r\le p$. Consider the irreducible highest-weight representation $[0,\ldots,0,r,p-r]$ of $\mathfrak{osp}(2m|2n)$ with respect to the distinguished Dynkin diagram, i.e. the module whose only nonzero Dynkin labels are $r$ on the node $\epsilon_{m-1}-\epsilon_m$ and $p-r$ on the node $\epsilon_{m-1}+\epsilon_m$. Then:

• for $|m-n|$ even, $\operatorname{char}[0,\ldots,0,r,p-r]_{\mathfrak{osp}(2m|2n)}=\left(\frac{y_1\cdots y_n}{x_1\cdots x_m}\right)^{p/2}\sum_{\lambda\in\mathcal{B}_r,\ \lambda_1\le p}s_\lambda(x/y);$
• for $|m-n|$ odd and $1\le r\le p$, the same identity holds with $\mathcal{B}_r$ replaced by $\mathcal{B}_{p-r}$. (At $r=0$ the module $[0,\ldots,0,0,p]$ is the chiral module $[0,\ldots,0,p]$, whose character is given for every $(m,n)$ by the proven chiral formula of the Background, i.e. by the $\mathcal{B}$-indexed sum; the source states the odd-parity replacement without restricting $r$, but at $r=0$ it is superseded by that proven formula.)

Here $x_i=e^{\epsilon_i}$ and $y_j=e^{\delta_j}$ are the character variables, $s_\lambda(x/y)$ are the supersymmetric Schur functions (the sum effectively runs over partitions in the $(m,n)$-hook, since $s_\lambda(x/y)=0$ outside it), $\mathcal{B}$ is the set of partitions in which every part occurs an even number of times (equivalently, partitions of the form $(\mu_1,\mu_1,\mu_2,\mu_2,\ldots)$ obtained by writing each part of a partition $\mu$ twice; this includes the zero partition), and $\mathcal{B}_r$ is the set of partitions of $\mathcal{B}$ to which a horizontal strip of length $r$ is attached.

An explicit counterexample is a specific admissible quadruple $(m,n,p,r)$ for which the two formal characters differ, exhibited at an identifiable weight or $\mathfrak{gl}(m|n)$-degree.

References

1. N.I. Stoilova, J. Thierry-Mieg, J. Van der Jeugt, "On characters and superdimensions of some infinite-dimensional irreducible representations of $\mathfrak{osp}(m|n)$", Physics of Atomic Nuclei 81 (2018) 939–944; arXiv:1904.00074; DOI 10.1134/S1063778818060285, https://arxiv.org/abs/1904.00074
2. N.I. Stoilova, J. Thierry-Mieg, J. Van der Jeugt, "On superdimensions of some infinite-dimensional irreducible representations of $osp(m|n)$", Springer Proceedings in Mathematics & Statistics 263 (2018) 165–176; arXiv:1904.00067; DOI 10.1007/978-981-13-2715-5_9, https://arxiv.org/abs/1904.00067
3. N.I. Stoilova, J. Thierry-Mieg, J. Van der Jeugt, "Extension of the $\mathfrak{osp}(m|n)\sim\mathfrak{so}(m-n)$ correspondence to the infinite-dimensional chiral spinors and self dual tensors", Journal of Physics A: Mathematical and Theoretical 50 (2017) 155201; arXiv:1609.06350; DOI 10.1088/1751-8121/aa5e8f, https://arxiv.org/abs/1609.06350
4. Shun-Jen Cheng, R.B. Zhang, "Howe duality and combinatorial character formula for orthosymplectic Lie superalgebras", Advances in Mathematics 182 (2004) 124–172; DOI 10.1016/S0001-8708(03)00076-8, https://doi.org/10.1016/S0001-8708(03)00076-8
5. A. Berele, A. Regev, "Hook young diagrams with applications to combinatorics and to representations of Lie superalgebras", Advances in Mathematics 64 (1987) 118–175; DOI 10.1016/0001-8708(87)90007-7, https://doi.org/10.1016/0001-8708(87)90007-7
6. R.C. King, B.G. Wybourne, "Analogies between finite-dimensional irreducible representations of SO(2n) and infinite-dimensional irreducible representations of Sp(2n,R). I. Characters and products", Journal of Mathematical Physics 41 (2000) 5002–5019; DOI 10.1063/1.533389, https://doi.org/10.1063/1.533389
7. Steven V. Sam, Keller VandeBogert, "An infinite-dimensional character coincidence between Lie algebras of type B and BC", arXiv:2503.18918 (2025), https://arxiv.org/abs/2503.18918
8. Jean Thierry-Mieg, Peter D. Jarvis, "Conformal invariance of antisymmetric tensor field theories in any even dimension", arXiv:2311.01701 (2023, revised 2025), https://arxiv.org/abs/2311.01701